Research Article On the Multispecies Delayed Gurtin...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 908768 11 pageshttpdxdoiorg1011552013908768

Research ArticleOn the Multispecies Delayed Gurtin-MacCamy Model

Anna Poskrobko1 and Antoni Leon Dawidowicz2

1 Faculty of Computer Science Bialystok University of Technology Ulica Wiejska 45A 15-351 Białystok Poland2 Faculty of Mathematics and Computer Science Jagiellonian University Ulica Łojasiewicza 6 30-348 Krakow Poland

Correspondence should be addressed to Anna Poskrobko aposkrobkopbedupl

Received 8 November 2012 Revised 9 March 2013 Accepted 29 March 2013

Academic Editor Carlos Vazquez

Copyright copy 2013 A Poskrobko and A L Dawidowicz This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The paper deals with the description of multispecies model with delayed dependence on the size of population It is based on theGurtin and MacCamy model The existence and uniqueness of the solution for the new problem of n populations dynamics areproved as well as the asymptotical stability of the equilibrium age distribution

1 Introduction

In this paper we consider mutual influence of 119899 populationsWe assume that each population develops differently affect-ing each other Populations do not destroy each other How-ever they share the same natural resources and the space Inour model 119906 denotes the density of the ecosystem consistingof 119899 different populations Therefore 119906 = (119906

1 119906

119899) is the

vector function Each component 119906119894 for 119894 = 1 119899 denotesthe 119894th population densityThe development of 119894th populationcan be expressed by the following system of the equations

119863119906119894(119886 119905) = minus120582

119894(119886 119911

119905) 119906

119894(119886 119905) (1)

119906119894(0 119905) = int

infin

0

int

[minus1199030]

120573119894(119886 119911 (119905 + 119904)) 119906

119894(119886 119905 + 119904

119894) 119889119904 119889119886 (2)

119911119894(119905) = int

infin

0

119906119894(119886 119905) 119889119886 (3)

with the initial condition

119906119894(119886 119904) = 120593

119894(119886 119904) for 119904 isin [minus119903119894 0] (4)

where

[minus119903 0] = [minus1199031 0] times [minus119903

2 0] times sdot sdot sdot times [minus119903

119899 0]

119911 (119905 + 119904) = (1199111(119905 + 119904

1) 119911

2(119905 + 119904

2) 119911

119899(119905 + 119904

119899))

119911119894

119905 [minus119903

119894 0] 997888rarr R

+ 119903

119894⩾ 0 R

+= [0infin)

119911119894

119905(119904) = 119911

119894(119905 + 119904)

(5)

The rate

119863119906119894(119886 119905) = lim

ℎrarr0

119906119894(119886 + ℎ 119905 + ℎ) minus 119906

119894(119886 119905)

ℎ

(6)

denotes the intensity of changing of the 119894th population intime In particular if 119906119894 is differentiable then119863119906119894 = 120597119906119894120597119886 +120597119906

119894120597119905 The quantity 119911119894(119905) is the total 119894th population at time

119905 We can express total population of the whole ecosystem attime 119905 by

119911 (119905) = (1199111(119905) 119911

119899(119905)) (7)

Thus

119911119905= (119911

1

119905 119911

119899

119905) (8)

2 Abstract and Applied Analysis

We assume that

119911119894

0(0) = int

infin

0

120593119894(119886 0) 119889119886 (9)

The birth and death processes of the 119894th population aredescribed by coefficients 120582119894 and 120573119894 We assume that both pro-cesses depend on the population size not only at the moment119905 but also at any preceding period of time (so-called historysegment) Introducing the delay parameter to the model hasdeep biological approach (see for instance [1]) All naturalprocesses occur with some delay with respect to the momentof their initiation We can take into consideration for exam-ple the period of pregnancy (constant delay) or morbidity(variable delay) In our model considered processes occurwith 119899 various delays typical for 119899 populations It is a novelapproach to the population dynamics research Moreoverthe system (1)ndash(4) consists of the general description of 119899populations dynamics If populations develop independentlythen the functions 120582119894 and 120573119894 depend on 119894th coordinate of thefunction 119911

119905only and the system consists of 119899 independent

von Foerster equations However if there are populationsrelationships we can consider three cases

(1) Competition for food in this case 120573119894 is decreas-

ing function of variables describing the numbers ofspecies competing for food with the 119894th one Forexample 120573119894

(119886 119911) = max119886(119865minussum119895isin119868

119896119895119911119895) 0 where 119865

denotes total amount of food and 119868 is the set of speciescompeting with the 119894th one

(2) Schema predator prey if 119894th species feeds on 119895thspecies individuals then 120582

119895 and 120573119894 are increasing

functions of variables 119911119895 and 119911119894 respectively The

classical models show that balance state between twospecies competing for food is not possible In thiscase the model does not have nonzero equilibriumage distribution However if we additionally considerpredator feeding on one competing species then weget balance between three considered species (see forinstance [2ndash5])

(3) Symbiosis if 119894th and 119895th species live in symbiosis witheach other then 120573119894 can be an increasing function of119911119895 (when the presence of 119895th species individuals is thereproduction of 119894th species favour) or 120582119895 is a decreas-ing function of 119911119894 (when the presence of 119894th speciesprotects 119895th species individuals from death)

In this paper we develop the idea of Gurtin and MacCamymodel [6] for one age-dependent population

119863119906 (119886 119905) = minus120582 (119886 119911 (119905)) 119906 (119886 119905)

119906 (0 119905) = int

infin

0

120573 (119886 119911 (119905)) 119906 (119886 119905) 119889119886

119911 (119905) = int

infin

0

119906 (119886 119905) 119889119886

119906 (119886 0) = 120593 (119886)

(10)

This classical model has many generalizations [7ndash16] Con-siderations of the dynamics of age-structured populations

have received substantial treatment on various fields [17ndash22] In particular our latest extension of this theme waspresented in the article [10] where we considered an age-dependent population dynamics with a delayed dependenceon the structure The right-hand side of the equation in ourmodel is not in the form 120582119906 butΛ119906 HereΛ is an operator thatdoes not apply to the value of the function 119906 as in the classicalmodel but to its restriction to some particular space

119863119906 (119886 119905) = minusΛ (119886 119911119905) 119906 (119886 119905 + sdot)

119906 (0 119905) = int

infin

0

int

0

minus119903

120573 (119886 119911 (119905 + 119904)) 119906 (119886 119905 + 119904) 119889119904 119889119886

119911 (119905) = int

infin

0

119906 (119886 119905) 119889119886 for 119905 isin [minus119903 119879]

119906 (119886 119904) = 120593 (119886 119904) for 119904 isin [minus119903 0]

(11)

In our paper [10] we present the proof of the existence anduniqueness of the new problem solution Conditions of theexponential asymptotical stability for this model still remainto be formulated On account of the significant difficultieswith formulating conditions of the stability we considerslightly simpler version of the model than the one describedin [10] We resign from the delay in the structure during adeliberation on mortality However the delayed dependenceof the structure still remains the element of the birth processdescription In this paper we aim to state stability conditionsfor the ecosystem of 119899 populations The plan of the paperis as follows In Section 2 we formulate the problem in theterms of operator equations Section 3 contains the proof ofthe existence and uniqueness of the solution We also studyequilibriumage distributions that is solutions to the problemwhich are independent of time In Section 4 exponentialasymptotic stability of the equilibrium age distributions isanalyzed We use generalized Laplace transform to study thestability

We consider our model (1)ndash(4) under the followingassumptions

(1198671) 120593 = (120593

1 120593

119899) where 120593119894

isin 119862(R+times [minus119903

119894 0]) for 119894 =

1 119899(119867

2) Φ

119894= int

infin

0sup

119904isin[minus1199031198940]120593119894(119886 119904)119889119886 lt infin for 119894 = 1 119899

(1198673) The function [minus119903119894 0] ni 119905 997891rarr int

infin

0120593119894(119886 119905) 119889119886 is contin-

uous(119867

4) 120582 = (120582

1 120582

119899) 120573 = (120573

1 120573

119899) where 120582119894 isin

119862(R+times ⨉

119899

119894=1119862([minus119903

119894 0])) 120573119894

isin 119862(R119899+1

+) the Frechet

derivatives 119863120582119894 of 120582119894(119886 120595) with respect to 120595 exist forall 119886 ⩾ 0 and 120595 ⩾ 0

(1198675) The components of the function 120582(sdot 120595) = (120582

1(sdot 120595)

120582119899(sdot 120595)) belong to 119862(119862([minus119903

119894 0]) 119871

infin(R

+)) re-

spectively for 119894 = 1 119899(119867

6) The components of Frechet derivative 119863

1205950120582 =

(11986312059501205821 119863

1205950120582119899) in the point 120595

0as functions of

1205950belong to 119862(119862([minus119903119894 0])L(119862([minus119903

119894 0]) 119871

infin(R

+)))

119894 = 1 119899 Here L(119883 119884) denotes the Banach spaceof all bounded linear operators from119883 to 119884

Abstract and Applied Analysis 3

(1198677) 120593

119894⩾ 0 120582

119894⩾ 0 120573

119894⩾ 0 for 119894 = 1 119899

(1198678) The functions 120582 and 120573 are bounded that is120582(119886 120595) ⩽ 120582

0and 120573(119886 119911) ⩽ 120573

0where 120582

0and 120573

0

are finite quantities(119867

9) There exist 120582

1and 120573

1such that

1003817100381710038171003817120582 (119886 120595

1) minus 120582 (119886 120595

2)1003817100381710038171003817⩽ 120582

1

10038171003817100381710038171205951minus 120595

2

1003817100381710038171003817

(12)

for every 119886 1205951 120595

2and

119899

sum

119894=1

119899

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

120597120573119894(119886 119911)

120597119911119895

100381610038161003816100381610038161003816100381610038161003816

⩽ 1205731 (13)

for every 119886 119911

2 Equivalent Formulation of the Problem

In this section we will formulate problem (1)ndash(4) in termsof operator equations Thanks to this we will prove localand global existences of the presented problem solution Thenext theorem is analogous to these well-known results forexample von Foerster [23] or Gurtin and MacCamy models

Theorem 1 Nonnegative continuous functions 119911119905= (119911

1

119905

119911119899

119905) and 119861 = (1198611

119861119899) where

119861119894(119905) = int

[minus1199030]

int

119905+119904119894

0

120573119894(119905 + 119904

119894minus 119886 119911 (119905 + 119904))

times 119861119894(119886) 119890

minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

119889119886 119889119904

+ int

[minus1199030]

int

infin

0

120573119894(119886 + 119905 + 119904

119894 119911 (119905 + 119904))

times 120593119894(119886) 119890

minusint119905+119904119894

0120582119894(119886+120591119911120591)119889120591

119889119886 119889119904

(14)

119911119894

119905(119904

119894) = int

119905+119904119894

0

119861119894(119886) 119890

minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

119889119886

+ int

infin

0

120593119894(119886 0) 119890

minusint119905+119904119894

0120582119894(119886+120591119911120591)119889120591

119889119886 for 119905 + 119904119894⩾ 0

119911119894

119905(119904

119894) = 119911

119894

0(119905 + 119904

119894) for 119905 + 119904

119894lt 0

(15)

for 119894 = 1 119899 are the solutions of the problems (2) and (3) upto time 119879 gt 0 if and only if the function 119906 = (1199061 119906119899) is thesolution of the age-dependent 119899 populations problem (1)ndash(4) on[0 119879] and 119906 is defined by the formula

119906119894(119886 119905) =

120593119894(119886 minus 119905 0) 119890

minusint119905

0120582119894(119886minus119905+120591119911120591)119889120591 for 119886 ⩾ 119905

119861119894(119905 minus 119886) 119890

minusint119886

0120582119894(120572119911119905minus119886+120572)119889120572 for 119905 gt 119886

(16)

where 119861119894(119905) = 119906

119894(0 119905) for each 119894 = 1 119899

Proof Theidea of the proof is analogous to the result includedin the paper [9] treating the theme of the age-dependent

population problem for one species Let 119906 = (1199061 119906

119899) be

a solution of the problem up to time 119879 Let 119906119894(ℎ) = 119906119894(119886

0+

ℎ 1199050+ ℎ) 120582

119894

(ℎ) = 120582119894(119886

0+ ℎ 119911

1199050+ℎ) then we can rewrite (1) as

the equation 119889119906119894119889ℎ + 120582119894

(ℎ)119906119894= 0 with the unique solution

119906119894(119886

0+ ℎ 119905

0+ ℎ) = 119906

119894(119886

0 119905

0) 119890

minusintℎ

0120582119894

(120578)119889120578 (17)

Substituting (1198860 119905

0) = (119886 minus 119905 0) ℎ = 119905 and (119886

0 119905

0) = (0 119905 minus 119886)

ℎ = 119886 into (17) yields the formula (16) Applying (16) to (2)and (3) we obtain the operator equations (14) and (15)

To prove the second part of the theorem we shouldassume that 119911119894

119905⩾ 0 and 119861119894

⩾ 0 for 119894 = 1 119899 are continuousfunctions on the interval [0 119879] fulfilling conditions (14) and(15) Let 119906 for each component 119906119894 be defined on R

+times [0 119879]

by the formula (16)The function 119906 is nonnegative because of(119867

7) An easy computation shows that (4) holds and 119906(0 119905) =

119861(119905) for 119905 gt 0 119906 isin 1198711(R+) because 120582 120573 and 119911

119905are continuous

and 120593 isin 1198711(R) It follows from (14)ndash(16) that (2) and (3) are

satisfied To complete the proof let us notice that (6) and(16) imply existing 119863119906 on R

+times [0 119879] and the equality (1)

holds

Let us make some estimation for the necessity of the nextsection We turn back to the operator equation (14) By theassumptions (119867

2) and (119867

8) we get

119861119894(119905)⩽120573

01199031sdot sdot sdot 119903

119899int

119905

0

sup119904isin[minus119903

1198940]

119861119894(119886 + 119904) 119889119886+120573


119899Φ

119894 (18)

Denoting

B119894(119905) = sup

120591isin[minus119903119894119905]

119861119894(120591) (19)

we obtain

B119894(119905) ⩽ 120573


119899int

119905

0

B119894(119886) 119889119886 + 120573


119899Φ

119894 (20)

and by Gronwallrsquos inequality we get

B119894(119905) ⩽ 120573


119899Φ

11989411989012057301199031sdotsdotsdot119903119899119905 (21)

3 Existence and Uniqueness

According to the theorem in Section 2 to solve the populationproblem up to time 119879 it is sufficient to find functions 119911 =

(1199111 119911

119899)isin ⨉

119899

119894=1119862+([minus119903

119894 119879]) and 119861 = (119861

1 119861

119899) isin

⨉119899

119894=1119862+([minus119903

119894 119879]) satisfying the system of operator equations

(14) and (15) We can notice that (14) is a Volterra equationof 119861119894 with a unique solution B119894

119879(119911) Let us define on

⨉119899

119894=1119862+([minus119903

119894 119879]) the new operator Z

119879= (Z1

119879 Z119899

119879)

where

Z119894

119879(119911) (119905) = int

119905

0

B119894

119879(119911) (119886) 119890

minusint119905

119886120582119894(120591minus119886119911120591)119889120591

119889119886

+ int

infin

0

120593119894(119886 0) 119890

minusint119905

0120582119894(119886+120591119911120591)119889120591

119889119886

(22)

We define the previous operator using the system of (15)for 119894 = 1 119899 with 119861 = (119861

1 119861

119899) replaced by B

119879=

(B1

119879 B119899

119879)


Theorem 2 The operator Z119879 ⨉

119899

119894=1119862+[minus119903

119894 119879] rarr

⨉119899

119894=1119862+

[minus119903119894 119879] defined by (22) has a unique fixed point

for any 119879 gt 0

Proof We prove that the operator Z119879is contracting and

then the assertion will be a consequence of the Banach fixedpoint theorem Let us consider the Banach space 119862[minus119903119894 119879]with the Bielecki norm 119891

119894119879= sup

119905isin[minus119903119894119879]119890minus119870119905

|119891119894(119905)| for any

119870 gt 0 Such norm is equivalent to classical supremum norm sdot in 119862[minus119903119894 119879] Choose 119911 isin ⨉119899

119894=1119862[minus119903

119894 119879] Applying the

definition (22) we obtain

10038171003817100381710038171003817Z

119894

119879(119911) minusZ

119894

119879()

10038171003817100381710038171003817119879

⩽ sup119905isin[minus119903

119894119879]

119890minus119870119905

times int

119905

0

1003816100381610038161003816100381610038161003816

119890minusint119905

119886120582119894(120591minus119886119911120591)119889120591

minus 119890minusint119905

119886120582119894(120591minus119886120591)119889120591

1003816100381610038161003816100381610038161003816

B119894

119879(119911) (119886) 119889119886

+ sup119905isin[minus119903

119894119879]

119890minus119870119905

times int

119905

0


119886120582119894(120591minus119886120591)119889120591

10038161003816100381610038161003816B

119894

119879(119911) (119886) minusB

119894

119879() (119886)

10038161003816100381610038161003816119889119886


119894119879]

119890minus119870119905

times int

infin

0

1003816100381610038161003816100381610038161003816


0120582119894(119886+120591119911120591)119889120591


0120582119894(119886+120591120591)119889120591

1003816100381610038161003816100381610038161003816

120593119894(119886 0) 119889119886

= 1198681+ 119868

2+ 119868

3

(23)

Estimating the quantities 1198681 119868

2 and 119868

3we use the assump-

tions (1198672) (119867

7) and (119867

8) inequalities (21) and |119890119896 minus 1| ⩽

|119896|119890|119896| Thus

1198681⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

times int

119905

0

1003816100381610038161003816100381610038161003816

1 minus 119890minusint119905

119886[120582119894(120591minus119886119911120591)minus120582

119894(120591minus119886120591)]119889120591

1003816100381610038161003816100381610038161003816

B119894

119879(119911) (119886) 119889119886

⩽ 1205821119911 minus 119879


119894119879]

119890minus119870119905

times int

119905

0

int

119905

119886

11989021205820(119905minus119886)

119890119870120591B

119894

119879(119911) (119886) 119889120591 119889119886

⩽

1205821

119870

12057301199031sdot sdot sdot 119903

119899Φ

119894

12057301199031sdot sdot sdot 119903

119899minus 2120582

0

11989012057301199031sdotsdotsdot119903119899119879119911 minus 119879

1198683⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

times int

infin

0

int

119905

119886

11989021205820(119905minus119886)

1198901198701205911205821119911 minus 119879

120593119894(119886 0) 119889120591 119889119886

⩽

1205821

119870

Φ11989411989021205820119879

119911 minus 119879

(24)

From (14) and the definition ofB119879we have

B119894

119879(119911) (119905) minusB

119894

119879() (119905)

= int

[minus1199030]

int

119905+119904119894

0

120573119894(119905+119904

119894minus119886 119911 (119905+119904)) sdot 119890

minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

times (B119894

119879(119911) (119886) minusB

119894

119879() (119886)) 119889119886 119889119904

(25)

+ int

[minus1199030]

int

119905+119904119894

0

B119894

119879() (119886)

times (120573119894(119905 + 119904

119894minus 119886 119911 (119905 + 119904))

times119890minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

minus120573119894(119905+ 119904

119894minus119886 (119905 + 119904))

times 119890minusint119905+119904119894

119886120582119894(120591minus119886120591)119889120591

)119889119886 119889119904

+ int

[minus1199030]

int

infin

0

120593119894(119886 0)

times (120573119894(119886 + 119905 + 119904

119894 119911 (119905 + 119904))

times119890minusint119905+119904119894

0120582119894(119886+120591119911120591)119889120591

minus120573119894(119886 + 119905 + 119904

119894 (119905 + 119904))

times 119890minusint119905+119904119894

0120582119894(119886+120591120591)119889120591

)119889119886 119889119904

(26)

Let us denote (26) by 119891119894(119905) then

B119894

119879(119911) (119905) minusB

119894

119879() (119905)

⩽ 12057301199031sdot sdot sdot 119903

119899int

119905

0

(B119894

119879(119911) (119886) minusB

119894

119879() (119886)) 119889119886

+

10038161003816100381610038161003816119891119894(119905)

10038161003816100381610038161003816

(27)

and hence by Gronwallrsquos inequality we have

B119894

119879(119911) (119905) minusB

119894

119879() (119905)

⩽

10038161003816100381610038161003816119891119894(119905)

10038161003816100381610038161003816+ 120573


119899int

119905

0

10038161003816100381610038161003816119891119894(119886)

1003816100381610038161003816100381611989012057301199031sdotsdotsdot119903119899(119905minus119886)

119889119886

(28)

Let us estimate |119891119894(119905)|

10038161003816100381610038161003816119891119894(119905)

10038161003816100381610038161003816

⩽ int

[minus1199030]

int

119905+119904119894

0

1198621B

119894

119879() (119886)

times (120573119894(119905 + 119904

119894minus 119886 119911 (119905 + 119904))

minus120573119894(119905 + 119904

119894minus 119886 (119905 + 119904))) 119889119886 119889119904


+ int

[minus1199030]

int

infin

0

1198622120593119894(119886 0)

times (120573119894(119886 + 119905 + 119904

119894 119911 (119905 + 119904))

minus120573119894(119886 + 119905 + 119904

119894 (119905 + 119904))) 119889119886 119889119904

⩽ 11986212057311199031sdot sdot sdot 119903

119899119890119870119905119911 minus 119879

times (Φ119894int

119905

0

12057301199031sdot sdot sdot 119903

11989911989012057301199031sdotsdotsdot119903119899119886119889119886 + int

infin

0

120593119894(119886 0) 119889119886)

⩽ 11986212057311199031sdot sdot sdot 119903

119899Φ


(29)

where1198621and119862

2are positive constants and119862 = max119862

1 119862

2

Finally we have

1198682⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

timesint

119905

0

(

10038161003816100381610038161003816119891119894(119886)

10038161003816100381610038161003816+ 120573


119899int

119886

0

10038161003816100381610038161003816119891119894(120585)


119889120585) 119889119886

⩽ 11986212057311199031sdot sdot sdot 119903

119899Φ

119894119890119870119905119911 minus 119879

times sup119905isin[minus119903

119894119879]

119890minus119870119905

(int

119905

0

119890(119870+1205730119903

1sdotsdotsdot119903119899)119886119889119886

+12057301199031sdot sdot sdot 119903

119899int

119905

0

int

119886

0

11989011987012058511989012057301199031sdotsdotsdot119903119899119886119889120585 119889119886)

⩽

11986212057311199031sdot sdot sdot 119903

119899Φ

119894

119870 + 12057301199031sdot sdot sdot 119903

119899119911 minus 119879

11989012057301199031sdotsdotsdot119903119899119879(1 +

12057301199031sdot sdot sdot 119903

119899

119870

)

(30)

Therefore we can choose sufficiently large constant 119870 forfixed 119879 that 119868

1 119868

2 and 119868

3are less than 119862119911 minus

119879with the

constant119862 isin [0 (13119899)) (independent of 119911 and )This showsthe contraction ofZ

119879and completes the proof

According to the previous mentioned there exists theexact one solution for any interval [0 119879] so the solutionsdefined on two different intervals coincide on their intersec-tion By the extension property we have the existence anduniqueness of the 119899 populations problem solution for alltimes

4 Stability of Equilibrium Age Distribution

A stationary solution 119906(119886) = (1199061(119886) 119906

119899(119886)) of the model

(1)ndash(4) satisfies the following system of the equations

(119906119894

0)

1015840

(119886) + 120582119894(119886 119911

0) 119906

119894

0= 0

119911119894

0= int

infin

0

119906119894

0(119886) 119889119886

119906119894

0(0) = 119903


119899int

infin

0

120573119894(119886 119911

0) 119906

119894

0(119886) 119889119886

(31)

for 119894 = 1 119899

The population of the whole ecosystem 1199110= (119911

1

0 119911

119899

0)

and its birthrate 1198610= 119906

0(0) = (119906

1

0(0) 119906

119899

0(0)) are con-

stantsThe quantity 1199060isin 119862

1(R

+) is the solution of the system

(31) and it will be referred to the equilibrium age distributionThe probability that an individual of 119894th population survivesto age 119886 if the population of the ecosystem is on the constantsize level 119911

0can be expressed by

120587119894

0(119886) = 119890

minusint119886

0120582119894(1205721199110)119889120572

(32)

The quantity

119877119894(119911

0) = 119903


119899int

infin

0

120573119894(119886 119911

0) 120587

119894

0(119886) 119889119886 (33)

is the number of offsprings expected to be born to anindividual of 119894th populationwhen the population of thewholeecosystem equals 119911

0We can formulate the following theorem

describing the connection between these three quantities

Theorem 3 Let 1199110= (119911

1

0 119911

119899

0) and 119911119894

0gt 0 for 119894 = 1 119899

and assume that 120587119894

0(sdot) 120573119894

(sdot 1199110)120587

119894

0(sdot) isin 119871

1(R

+) for each 119894 =

1 119899 Then

119877 (1199110) = (119877

1(119911

0) 119877

119899(119911

0)) (34)

with

119877119894(119911

0) = 1 119894 = 1 119899 (35)

is a necessary and sufficient condition that an equilibrium agedistribution exists The unique equilibrium age distribution1199060= (119906

1

0 119906

119899

0) corresponding to 119911

0= (119911

1

0 119911

119899

0) is given

by

119906119894

0(119886) = 119861

119894

0120587119894

0(119886) 119894 = 1 119899 (36)

where

119861119894

0=

119911119894

0

int

infin

0120587119894

0(119886) 119889119886

(37)

Proof The function (36) is the unique solution of (31)1with

the initial condition 119906119894

0(0) = 119861

119894

0 By (31)

2we obtain the

formula (37) for 119861119894

0 An easy computation shows that (35) is

equivalent to (31)3for each 119894 = 1 119899

We now turn to the problem of the equilibrium agedistribution stabilityWe consider ldquoperturbationsrdquo 120585119894(119886 119905) and119901119894(119905) Let us write

119906 (119886 119905) = 1199060(119886) + 120585 (119886 119905)

= (1199061

0(119886) + 120585

1(119886 119905) 119906

2

0(119886)

+1205852(119886 119905) 119906

119899

0(119886) + 120585

119899(119886 119905))

(38)

119906119894(119886 119905) = 119906

119894

0(119886) + 120585

119894(119886 119905) (39)


where 120585119894(119886 119905) = 120593119894(119886 119905) minus 119906

119894

0(119886) for 119905 isin [minus119903119894 0] And

119911 (119905) = 1199110+ 119901 (119905)

= (1199111

0+ 119901

1(119905) 119911

2

0+ 119901

2(119905) 119911

119899

0+ 119901

119899(119905))

119911119905= 119911

0+ 119901

119905= (119911

1

0+ 119901

1

119905 119911

2

0+ 119901

2

119905 119911

119899

0+ 119901

119899

119905)

(40)

119911119894(119905) = 119911

119894

0+ 119901

119894(119905)

119911119894

119905= 119911

119894

0+ 119901

119894

119905

(41)

where119901119894

119905(119904) = 119901

119894(119905+119904) for 119904 isin [minus119903119894 0] Our goal is to formulate

relations for 120585119894 and 119901119894 which guarantee that 119906119894 and 1199111198940obey the

basic equations (1)ndash(4) From (1) we obtain

119863120585119894(119886 119905) + 120582

119894

0(119886) 120585

119894(119886 119905) + 120596

119894(119886) 119901

119894

119905= 119909

119894(119886 119905) (42)

for

119909119894(119886 119905) = minus 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905120585119894(119886 119905)

minus Λ119894(119886 119901

119905) [119861

119894

0120587119894

0(119886) + 120585

119894(119886 119905)]

120582119894

0(119886) = 120582

119894(119886 119911

0)

(43)

where 120596119894 R

+times 119862([minus119903

119894 0]) rarr R is the functional

120596119894(119886) 119901

119894

119905= 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905(119861

119894

0120587119894

0(119886))

Λ119894(119886 119901

119905) = 120582

119894(119886 119911

0+ 119901

119905) minus 120582

119894

0(119886) minus 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905

(44)

We conclude from (2) that

120585119894(0 119905) = int

infin

0

int

[minus1199030]

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119904 119889119886 + 120581

119894119901119894

119905+ 120595

119894(119905)

(45)

where

120573119894(119886 119911

0) = 120573

119894

0(119886)

120595119894(119905) = int

infin

0

int

[minus1199030]

119863119911119894120573

119894(119886 119911

0) 119901

119894(119905 + 119904

119894) 120585

119894(119886 119905 + 119904

119894) 119889119904 119889119886

+ int

infin

0

int

[minus1199030]

Ω119894(119886 119901 (119905 + 119904))

times [119861119894

0120587119894

0(119886) + 120585

119894(119886 119905 + 119904

119894)] 119889119904 119889119886

120581119894119901119894

119905= 119861

119894

0int

infin

0

int

[minus1199030]

119863119911119894120573

119894(119886 119911

0) 119901

119894

119905120587119894

0(119886) 119889119904 119889119886

(46)

Here 120581119894 119862([minus119903119894 0]) rarr R is the functional Moreover wehave

Ω119894(119886 119901 (119905 + 119904)) = 120573

119894(119886 119911

0+ 119901 (119905 + 119904))

minus 120573119894

0(119886) minus 119863

119911119894120573

119894(119886 119911

0) 119901

119894(119905 + 119904

119894)

(47)

A trivial verification shows that (3) gives the condition for 119901in the form

119901119894(119905) = int

infin

0

120585119894(119886 119905) 119889119886 (48)

We consider the system (42)ndash(48) with the initial condition

120585119894(119886 0) = 120578

119894(119886) (49)

where

120578119894(119886) = 120593

119894(119886 0) minus 119861

119894

0 (50)

We first express the solution of (42)ndash(48) in the form of thematrix equation Let 120585119894 be the solution of (42) up to time 119879Let (119886

0 119905

0) isin R

+times [0 119879] 120585119894(ℎ) = 120585

119894(119886

0+ ℎ 119905

0+ ℎ) 120582119894

0(ℎ) =

120582119894

0(119886

0+ ℎ) 120596119894

(ℎ) = 120596119894(119886

0+ ℎ) 119901119894

ℎ= 119901

119894

119905+ℎ and 119909119894(ℎ) = 119909119894(119886

0+

ℎ 1199050+ ℎ) Then we can rewrite (42) as the equation

119889

119889ℎ

120585119894(ℎ) + 120582

119894

0(ℎ) 120585

119894(ℎ) = 119909

119894(ℎ) minus 120596

119894(ℎ) 119901

119894

ℎ(51)

with the unique solution

120585119894(119886

0+ ℎ 119905

0+ ℎ)

= 120585119894(119886

0 119905

0) 119890

minusintℎ

0120582119894

0(1198860+120591)119889120591

+ 119890minusintℎ

0120582119894

0(1198860+120591)119889120591

times int

ℎ

0

[119909119894(119886

0+ 120591 119905

0+ 120591) minus 120596

119894(119886

0+ 120591) 119901

119894

1199050+120591]

times 119890int120591

0120582119894

0(1198860+120572)119889120572

119889120591

(52)


0) = (119886 minus 119905 0) ℎ = 119905 for 119886 ⩾ 119905 and (119886

0 119905

0) =

(0 119905 minus 119886) ℎ = 119886 for 119886 lt 119905 yields the formulas

120585119894(119886 119905) = 120578

119894(119886 minus 119905)

119894(119886 minus 119905 119886)

+ int

119905

0

[119909119894(119886 minus 119905 + 120591 120591) minus 120596

119894(119886 minus 119905 + 120591) 119901

119894

120591]

times 119894

0(119886 minus 119905 + 120591 119886) 119889120591 for 119886 ⩾ 119905

120585119894(119886 119905) = 119887

119894(119905 minus 119886) 120587

119894

0(119886)

+ int

119905

119905minus119886

[119909119894(120591 + 119886 minus 119905 120591) minus 120596

119894(120591 + 119886 minus 119905) 119901

119894

120591]

times 119894

0(120591 + 119886 minus 119905 119886) 119889120591 for 119886 lt 119905

(53)

where

119894

0(119886 minus 119905 119886) =

120587119894

0(119886)

120587119894

0(119886 minus 119905)

(54)

119887119894(119905 minus 119886) = 120585

119894(0 119905 minus 119886) (55)

Furthermore (45) implies that

119901119894(119905) = int

119905

0

120585119894(119886 119905) 119889119886 + int

infin

119905

120585119894(119886 119905) 119889119886 (56)


It follows that

119901119894(119905) + int

119905

0

int

infin

0

120596119894(119886) 119901

119894

120591119894

0(119886 119886 + 119905 minus 120591) 119889119886 119889120591

minus int

119905

0

119887119894(120591) 120587

119894

0(119905 minus 120591) 119889120591

= int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591

(57)


119887119894(119905) = int

[minus1199030]

int

119905+119904119894

0

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ int

[minus1199030]

int

infin

119905+119904119894

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ 120581119894(119905) 119901

119894

119905+ 120595

119894(119905)

(58)

and hence

minus 120581119894119901119894

119905+ 119887

119894(119905)

+ int

119905

0

int

minus[1199030]

int

infin

0

120573119894

0(119886 minus 120591 + 119905 + 119904

119894) 120596

119894(119886) 119901

119894

120591

times 119894

0(119886 119886 minus 120591 + 119905 + 119904

119894) 119889119886 119889119904 119889120591

minus int

119905

0

int

[minus1199030]

120573119894

0(119905 + 119904

119894minus 120591) 119887

119894(120591) 120587

119894

0(119905 + 119904

119894minus 120591) 119889119904 119889120591

= int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(59)

Conditions (57) and (59) imply that

y (119905) = [[[

y1 (119905)

y119899 (119905)

]

]

]

(60)

where

y119894 (119905) = [119901119894(119905)

119887119894(119905)

] (61)

y119894119905(119904) = y119894 (119905 + 119904) for 119904 isin [minus119903119894 0] 119894 = 1 2 119899 (62)

satisfy the matrix equation

Ay119905+ int

119905

0

K (119905 minus 120591) y120591119889120591 = f (119905) (63)

where A and K are block diagonal matrices We have

A =

[

[

[

[

[

A1 O OO A2

O

d

O O A119899

]

]

]

]

]

K =

[

[

[

[

[

K1 O OO K2

O

d

O O K119899

]

]

]

]

]

(64)

for zero matrix O A119894= [

1 0

minus1205811198941] A119894y

119905= [

119901119894

119905(0) 0

minus120581119894119901119894

119905119887119894

119905(0)] =

[119901119894(119905) 0

minus120581119894119901119894

119905119887119894(119905)] and

K119894(119905) =

[

[

[

[

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) 119889119886 minus120587

119894

0(119905)

int

0

minus119903119894

int

infin

0

120573119894

0(119886 + 119905 + 119904)

119894

0(119886 119886 + 119905 + 119904) 120596

119894(119886) 119889119886 119889119904 minusint

0

minus119903119894

120573119894

0(119905 + 119904) 120587

119894

0(119905 + 119904) 119889119904

]

]

]

]

(65)

Since 120596(119886) is the functional so in the formula ofK(119905) we con-sider 120596 as the function with functional values Furthermorein the notation (63) looks like the ordinary matrix equationbut in fact it is the operator one Moreover

f (119905) = [[[

f1 (119905)

f119899 (119905)

]

]

]

(66)

for

f 119894 (119905) = [119891119894

1(119905)

119891119894

2(119905)

] minus A119894y1198941119905minus int

119905

0

K119894(119905 minus 120591) y119894

1120591119889120591

119891119894

1(119905) = int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591


119891119894

2(119905) = int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(67)

and y1198941(119905) = [

119901119894(119905)

0] |

[minus1199031198940] y119894

1119905(119904) = y119894

1(119905 + 119904) for 119904 isin [minus119903119894 0]

We can notice that the last two elements of the expressiondefining f 119894(119905) equal zero for 119905 gt 119903

We return to the previous deliberation in the nexttheorem We take some additional assumptions

(11986710) The Frechet derivatives 119863

119911119894120582

119894(119886 119911

0) and the deriva-

tion 119863119911119894120573

119894(119886 119911

0) for 119894 = 1 2 119899 as functions of the

variable 119886 belong to 119871infin(R+)

(11986711) (Λ

119894(119886 120588)120588

0) and (Ω

119894(119886 120588)120588

0) tend to zero as

1205880rarr 0 uniformly for 119886 rarr 0 here sdot

0denotes

the supremum norm in the Banach space 119862([minus119903119894 0])(119867

12) 120582

lowast= inf

119886⩾0

120582119894(119886 119911

0) 119894 = 1 2 119899 gt 0

Denote 119890120579 (minusinfin 0] rarr R by the formula 119890

120579(119904) = 119890

120579119904 Theexponential asymptotical stability of the model is establishedby the following theorem

Theorem 4 Let 119894

1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 +

119904119894)119889119904 and let 120587119894

1(119905) = int

[minus1199030]120573119894

0(119905+119904

119894)120587

119894

0(119905+119904

119894)119889119904 Let us assume

that there exists some 120583 gt 0 that the equation

1 = int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905

minus int

infin

0

119890minus119905120574

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905

sdot (1 minus int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905)

minus int

infin

0

119890minus119905120574120587119894

0(119905) 119889119905

sdot (int

infin

0

119890minus119905120574

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905 minus 120581

119894(119890

120574))

(68)

has no solution 120574 with Re(120574) ⩾ minus120583 Then there exist realnumbers 120575 gt 0 and 120583 gt 0 such that for any initial data 120593 with 120593 minus 119906

01198711 lt 120575 the corresponding solution of the population

problem (1)ndash(4) if it exists for 119905 gt 0 satisfies1003817100381710038171003817119911119905minus 119911

0

1003817100381710038171003817= 119874 (119890

minus120583119905) (69)

1003817100381710038171003817119906 (119886 119905) minus 119906

0(119886)1003817100381710038171003817= 119874 (119890

minus120583119905) for each 119886

as 119905 997888rarr infin

(70)

Proof Let 119906 be the solution of the population problem (1)ndash(4) for 119905 gt 0 In the proof we will use the properties of thegeneralized Laplace transform Let 984858 [0infin) rarr 119862([minus119903

119894 0])

lowast

be the function with measure value Define

984858 (120579) = int

infin

0

984858 (119905) (119890120579) 119890

minus120579119905119889119905 (71)

If we define the convolution by the formula

(984858 lowast V120591) (119905) = int

119905

0

984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)

where V(119905) = 0 for 119905 lt 0 then we have the equality analogousto the property of classical Laplace transform that is

984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)

Using generalized Laplace transform to (63) we get

A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)

where

A (120579) =[

[

[

[

[

A1(120579) O sdot sdot sdot OO A2

(120579) sdot sdot sdot O

d

O O sdot sdot sdot A119899(120579)

]

]

]

]

]

A119894(120579) = [

1 0

minus120581119894(119890

120579) 1

]

K (120579) =[

[

[

[

[

K1(120579) O sdot sdot sdot OO K2


d

O O sdot sdot sdotK119899

(120579)

]

]

]

]

]

K119894(120579)

=[

[

[

int

infin

0

119890minus119905120579

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

0(120579)

int

infin

0

119890minus119905120579

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

1(120579)

]

]

]

(75)

for 119894

1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 + 119904

119894)119889119904 and

120587119894

1(119905) = int

[minus1199030]120573119894

0(119905 + 119904

119894)120587

119894

0(119905 + 119904

119894)119889119904 From the equality (68)

we conclude that the matrix A + K has an analytic inverse forRe(120579) ⩾ minus120583 Moreover we have

Aminus1(120579) =

[

[

[

[

[

[

[

(A1)

minus1

(120579) O sdot sdot sdot OO (

A2)

minus1


d

O O sdot sdot sdot (A119899)

minus1

(120579)

]

]

]

]

]

]

]

(A119894)

minus1

(120579) = [

1 0

120581119894(119890

120579) 1

]

(76)

so A is analytic for120579 isin C


The solution of the matrix equation (63) exists and isgiven by

y (119905) = Aminus1f119905+ int

119905

0

J (119905 minus 120591) f120591119889120591 (77)

where f120591(119904) =

[

[

f1(120591+119904)

f119899(120591+119904)

]

]

f 119894120591(119904) = f 119894(120591 + 119904) for 119904 isin [minus119903119894 0] and

J (119905) = 1

2120587

119890minus120583119905

int

infin

minusinfin

119890119894120577119905J (minus120583 + 119894120577) 119889120577 (78)

J = (A(120579) + K(120579))minus1 minus Aminus1(120579) is the Laplace transform of the

function J and

|J (119905)| ⩽ 1198621119890minus120583119905 (79)

where |J| denotes the sum of moduli of the elements of thematrix J Here and in the whole proof 119862

1 119862

2 119862

3 denote

positive constantsBy (119867

12) (32) and (54)

120587119894

0(119886) ⩽ 119890

minus120582lowast119886

119894

0(119886

1015840 119886) ⩽ 119890

minus120582lowast(119886minus1198861015840)

(1198861015840⩽ 119886) (80)

For 120582lowastgt 120583 and 120573

0119871infin lt infin we can estimate that

f (119905) ⩽ 1198622int

119905

0

119890minus120582lowast(119905minus120591)

119909 (sdot 120591)1198711119889120591

+100381710038171003817100381712057810038171003817100381710038171198711119890

minus120582lowast119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

(81)

1003817100381710038171003817y (119905)100381710038171003817

1003817⩽ 119862

3100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

+int

119905

0

119890minus120583(119905minus120591)

[1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(82)

where y(119905) = (sum119899

119894=1(119901

119894)2+ (119887

119894)2)

12

1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 ⩽ int

119905

0

119890minus120582lowast120591

119887 (119905 minus 120591) 119889120591 +100381710038171003817100381712057810038171003817100381710038171198711119890


+ int

119905

0


(119909 (sdot 120591)1198711 + 120596119871

1

1003817100381710038171003817119901120591

10038171003817100381710038170) 119889120591

⩽ 1198624100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905

+ int

119905

0

119890minus120583(119905minus120591)

times [1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(83)

Let 120576 gt 0 By (11986711) there exists 120575 =

120575(120576) such that1003817100381710038171003817Λ (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

1003817100381710038171003817Ω (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(84)

for 1199011199050lt120575 By the previous inequalities and the fact that

119863119911120582(119886 119911

0)

119871infin 119863119911

120573(119886 1199110)

119871infin and 11990601198711 are finite we can

estimate that

119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

1003817100381710038171003817119901119905

10038171003817100381710038170sdot1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 + 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(85)

We require that 120576 lt 1 and 120575 lt 120576 Hence 1199011199050lt 120576 According

to the previous remarks we have

119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

120576120590 (119905)

120590 (119905) =1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 +

1003817100381710038171003817119901119905

10038171003817100381710038170

(86)

Thus (82)ndash(86) imply that

120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+ 2120576int

119905

0

119890minus120583(119905minus120591)

120590 (120591) 119889120591 (87)

where119872 gt 0 Gronwallrsquos inequality yields

120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

(minus120583+2119872120576)119905 (88)

Let us choose 120575 and 120576 gt 0 such that 120583 = 120583 minus 2119872120576 gt 0 and 120575 ltmin (120575(120576) 120575(120576)119872) Assume that 120593 minus 119906

01198711 = 120578

1198711 lt 120575 In

that case from what has already been proved it follows that 119909(sdot 119905)

1198711 120595(119905)

1198711 120590(119905) and 119887(119905) are 119874(119890minus120583119905) Therefore

(41) (86)2imply (69) and (53)

2 (80) and (39) imply (70)

Example 5 Let 119899 = 2 Let us consider

120582119894(119886 119911) = 120582

119894(119886 119911

1

1 119911

1

2 119911

2

1 119911

2

2)

120573119894(119886 119911) = 120573

119894(119886 119911

1 119911

2)

(89)

where

120582119894(119886 119911) = 120582

119894(119911

1(0) 119911

1(minus119903

1) 119911

2(0) 119911

2(minus119903

2))

120573119894(119886 119911) = 120573

119894(119911

1 119911

2) 119890

minus120572119894119886

(90)

with 120582119894 gt 0 120573119894gt 0 119903119894 gt 0 120572119894 gt 0 for 119894 = 1 2 Let us define

120582119894

0= 120582

119894(119911

1

0 119911

1

0 119911

2

0 119911

2

0)

120573119894

0= 120573

119894(119911

1

0 119911

2

0) 119890

minus120572119894119886

(91)

for an arbitrary 1199110= (119911

1

0 119911

2

0) isin R2 From (32) and (33) we

conclude that

120587119894

0(119886) = 119890

minus119886120582119894

0

119877119894(119911

0) =

11990311199032120573119894

0

120572119894+ 120582

119894

0

(92)

By Theorem 3 there exists an equilibrium age distribution119906119894

0(119886) = 119911

119894

0120582119894

0119890minus120582119894

0119886 where 119861

0= 119911

119894

0120582119894

0if and only if 11990311199032120573119894

0(120572

119894+

120582119894

0) = 1 To investigate the stability of the equilibrium age


distributionwe consider (68) Let 1199110= (119911

1

0 119911

2

0) be the solution

of the system of the equations

119877119894(119911

0) = 1 119894 = 1 2 (93)

First let us notice that

120596119894(119890

120574) = (

120597120582119894

120597119911119894

1

+

120597120582119894

120597119911119894

2

119890minus120574119903119894

)120582119894

0119911119894

0119890minus119886120582119894

0 (94)

with the derivatives in the point 1199110 It is easy to notice that

|120597120582119894120597119911

119894

1+(120597120582

119894120597119911

119894

2)119890

minus120574119903119894

| is bounded on half-plane Re(120574) gt 120583for every real 120583 (also negative) Analogously

120581119894(119890

120574) =

120582119894

0

120573119894

0

sdot

119911119894

0

119903119894sdot

120597120573119894

120597119911119894(119911

0) int

0

minus119903119894

119890120574119904119889119904 (95)

It is also possible to prove that for every real 120583 the function|120581

119894(119890

120574)| is bounded on half-plane Re(120574) gt 120583 Consider the

right-hand side of (68) All components can be presented inthe form

119860

119861 + 120574

(96)

where 119860 is bounded and 119861 ⩾ max120572119894 1205821198940 Let 120583 lt 119861 and let

Re(120574) gt minus120583

10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + 120574

10038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + Re (120574)

100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860

2radic119861 minus 120583radic120583 + Re (120574)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(97)

Fix120590 gt 0 For119861 sufficiently large there exist some120583 and 120583 lt 120583that

100381610038161003816100381610038161003816100381610038161003816

119860

2radic119861 minus 120583radic120583 minus 120583

100381610038161003816100381610038161003816100381610038161003816

lt 120590 (98)

For Re(120574) gt minus120583 we have

10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

⩽

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⩽

100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590

(99)

In consequence for 120572119894 and 1205821198940sufficiently large we can find

that 120583 gt 0 such that the modulus of the right-hand side of(68) is less than 1 for every 120574 for which Re(120574) gt minus120583

Acknowledgment

The first author acknowledges the support from BialystokUniversity of Technology (Grant no SWI22011)

References

[1] A Hastings ldquoInteracting age structured populationsrdquo inMath-ematical Ecology vol 17 of Biomathematics pp 287ndash294Springer Berlin Germany 1986

[2] K R Fister and S Lenhart ldquoOptimal harvesting in an age-structured predator-prey modelrdquo Applied Mathematics andOptimization vol 54 no 1 pp 1ndash15 2006

[3] Z He and H Wang ldquoControl problems of an age-dependentpredator-prey systemrdquo Applied Mathematics vol 24 no 3 pp253ndash262 2009

[4] D S Levine ldquoBifurcating periodic solutions for a class ofage-structured predator-prey systemsrdquoBulletin ofMathematicalBiology vol 45 no 6 pp 901ndash915 1983

[5] D J Wollkind A Hastings and J A Logan ldquoFunctional-response numerical response and stability in arthropodpredator-prey ecosystems involving age structurerdquo Researcheson Population Ecology vol 22 pp 323ndash338 1980

[6] M E Gurtin and R C MacCamy ldquoNon-linear age-dependentpopulation dynamicsrdquo Archive for Rational Mechanics andAnalysis vol 54 pp 281ndash300 1974

[7] D Breda M Iannelli S Maset and R Vermiglio ldquoStabilityanalysis of the Gurtin-MacCamy modelrdquo SIAM Journal onNumerical Analysis vol 46 no 2 pp 980ndash995 2008

[8] J M Cushing ldquoThe dynamics of hierarchical age-structuredpopulationsrdquo Journal of Mathematical Biology vol 32 no 7 pp705ndash729 1994

[9] A L Dawidowicz and A Poskrobko ldquoAge-dependent single-species population dynamics with delayed argumentrdquo Mathe-matical Methods in the Applied Sciences vol 33 no 9 pp 1122ndash1135 2010

[10] A L Dawidowicz and A Poskrobko ldquoOn the age-dependentpopulation dynamics with delayed dependence of the struc-turerdquo Nonlinear Analysis Theory Methods amp Applications vol71 no 12 pp e2657ndashe2664 2009

[11] G Di Blasio ldquoNonlinear age-dependent population growthwith history-dependent birth raterdquo Mathematical Biosciencesvol 46 no 3-4 pp 279ndash291 1979

[12] V G Matsenko ldquoA nonlinear model of the dynamics of the agestructure of populationsrdquoNelınıını Kolivannya vol 6 no 3 pp357ndash367 2003

[13] S Piazzera ldquoAn age-dependent population equation withdelayed birth processrdquo Mathematical Methods in the AppliedSciences vol 27 no 4 pp 427ndash439 2004

[14] K E Swick ldquoA nonlinear age-dependentmodel of single speciespopulation dynamicsrdquo SIAM Journal on Applied Mathematicsvol 32 no 2 pp 484ndash498 1977

[15] K E Swick ldquoPeriodic solutions of a nonlinear age-dependentmodel of single species population dynamicsrdquo SIAM Journal onMathematical Analysis vol 11 no 5 pp 901ndash910 1980

[16] Z G Bao and W L Chan ldquoA semigroup approach to age-dependent population dynamics with time delayrdquo Communica-tions in Partial Differential Equations vol 14 no 6 pp 809ndash8321989

[17] O Arino E Sanchez and G F Webb ldquoNecessary and sufficientconditions for asynchronous exponential growth in age struc-tured cell populationswith quiescencerdquo Journal ofMathematicalAnalysis and Applications vol 215 no 2 pp 499ndash513 1997

[18] O Arino E Sanchez and G F Webb ldquoPolynomial growthdynamics of telomere loss in a heterogeneous cell populationrdquoDynamics of Continuous Discrete and Impulsive Systems vol 3no 3 pp 263ndash282 1997


[19] F Billy J Clairambault O Fercoq et al ldquoSynchronisation andcontrol of proliferation in cycling cell population models withage structurerdquoMathematics and Computers in Simulation 2012

[20] H Inaba ldquoStrong ergodicity for perturbed dual semigroups andapplication to age-dependent population dynamicsrdquo Journal ofMathematical Analysis and Applications vol 165 no 1 pp 102ndash132 1992

[21] I Roeder M Herberg and M Horn ldquoAn ldquoagerdquo-structuredmodel of hematopoietic stem cell organization with applicationto chronic myeloid leukemiardquo Bulletin of Mathematical Biologyvol 71 no 3 pp 602ndash626 2009

[22] R Rundnicki and M C Mackey ldquoAsymptotic similarity andMalthusian growth in autonomous and nonautonomous popu-lationsrdquo Journal of Mathematical Analysis and Applications vol187 no 2 pp 548ndash566 1994

[23] J von Foerster Some Remarks on Changing Populations theKinetics of Cell Proliferation Grune amp Stratton New York NYUSA 1959

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Algebra

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Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


We assume that

119911119894

0(0) = int

infin

0

120593119894(119886 0) 119889119886 (9)

The birth and death processes of the 119894th population aredescribed by coefficients 120582119894 and 120573119894 We assume that both pro-cesses depend on the population size not only at the moment119905 but also at any preceding period of time (so-called historysegment) Introducing the delay parameter to the model hasdeep biological approach (see for instance [1]) All naturalprocesses occur with some delay with respect to the momentof their initiation We can take into consideration for exam-ple the period of pregnancy (constant delay) or morbidity(variable delay) In our model considered processes occurwith 119899 various delays typical for 119899 populations It is a novelapproach to the population dynamics research Moreoverthe system (1)ndash(4) consists of the general description of 119899populations dynamics If populations develop independentlythen the functions 120582119894 and 120573119894 depend on 119894th coordinate of thefunction 119911

119905only and the system consists of 119899 independent

von Foerster equations However if there are populationsrelationships we can consider three cases

(1) Competition for food in this case 120573119894 is decreas-

ing function of variables describing the numbers ofspecies competing for food with the 119894th one Forexample 120573119894

(119886 119911) = max119886(119865minussum119895isin119868

119896119895119911119895) 0 where 119865

denotes total amount of food and 119868 is the set of speciescompeting with the 119894th one

(2) Schema predator prey if 119894th species feeds on 119895thspecies individuals then 120582

119895 and 120573119894 are increasing

functions of variables 119911119895 and 119911119894 respectively The

classical models show that balance state between twospecies competing for food is not possible In thiscase the model does not have nonzero equilibriumage distribution However if we additionally considerpredator feeding on one competing species then weget balance between three considered species (see forinstance [2ndash5])

(3) Symbiosis if 119894th and 119895th species live in symbiosis witheach other then 120573119894 can be an increasing function of119911119895 (when the presence of 119895th species individuals is thereproduction of 119894th species favour) or 120582119895 is a decreas-ing function of 119911119894 (when the presence of 119894th speciesprotects 119895th species individuals from death)

In this paper we develop the idea of Gurtin and MacCamymodel [6] for one age-dependent population

119863119906 (119886 119905) = minus120582 (119886 119911 (119905)) 119906 (119886 119905)

119906 (0 119905) = int

infin

0

120573 (119886 119911 (119905)) 119906 (119886 119905) 119889119886

119911 (119905) = int

infin

0

119906 (119886 119905) 119889119886

119906 (119886 0) = 120593 (119886)

(10)

This classical model has many generalizations [7ndash16] Con-siderations of the dynamics of age-structured populations

have received substantial treatment on various fields [17ndash22] In particular our latest extension of this theme waspresented in the article [10] where we considered an age-dependent population dynamics with a delayed dependenceon the structure The right-hand side of the equation in ourmodel is not in the form 120582119906 butΛ119906 HereΛ is an operator thatdoes not apply to the value of the function 119906 as in the classicalmodel but to its restriction to some particular space

119863119906 (119886 119905) = minusΛ (119886 119911119905) 119906 (119886 119905 + sdot)

119906 (0 119905) = int

infin

0

int

0

minus119903

120573 (119886 119911 (119905 + 119904)) 119906 (119886 119905 + 119904) 119889119904 119889119886

119911 (119905) = int

infin

0

119906 (119886 119905) 119889119886 for 119905 isin [minus119903 119879]

119906 (119886 119904) = 120593 (119886 119904) for 119904 isin [minus119903 0]

(11)

In our paper [10] we present the proof of the existence anduniqueness of the new problem solution Conditions of theexponential asymptotical stability for this model still remainto be formulated On account of the significant difficultieswith formulating conditions of the stability we considerslightly simpler version of the model than the one describedin [10] We resign from the delay in the structure during adeliberation on mortality However the delayed dependenceof the structure still remains the element of the birth processdescription In this paper we aim to state stability conditionsfor the ecosystem of 119899 populations The plan of the paperis as follows In Section 2 we formulate the problem in theterms of operator equations Section 3 contains the proof ofthe existence and uniqueness of the solution We also studyequilibriumage distributions that is solutions to the problemwhich are independent of time In Section 4 exponentialasymptotic stability of the equilibrium age distributions isanalyzed We use generalized Laplace transform to study thestability

We consider our model (1)ndash(4) under the followingassumptions

(1198671) 120593 = (120593

1 120593

119899) where 120593119894

isin 119862(R+times [minus119903

119894 0]) for 119894 =

1 119899(119867

2) Φ

119894= int

infin

0sup

119904isin[minus1199031198940]120593119894(119886 119904)119889119886 lt infin for 119894 = 1 119899

(1198673) The function [minus119903119894 0] ni 119905 997891rarr int

infin

0120593119894(119886 119905) 119889119886 is contin-

uous(119867

4) 120582 = (120582

1 120582

119899) 120573 = (120573

1 120573

119899) where 120582119894 isin

119862(R+times ⨉

119899

119894=1119862([minus119903

119894 0])) 120573119894

isin 119862(R119899+1

+) the Frechet

derivatives 119863120582119894 of 120582119894(119886 120595) with respect to 120595 exist forall 119886 ⩾ 0 and 120595 ⩾ 0

(1198675) The components of the function 120582(sdot 120595) = (120582

1(sdot 120595)

120582119899(sdot 120595)) belong to 119862(119862([minus119903

119894 0]) 119871

infin(R

+)) re-

spectively for 119894 = 1 119899(119867

6) The components of Frechet derivative 119863

1205950120582 =

(11986312059501205821 119863

1205950120582119899) in the point 120595

0as functions of

1205950belong to 119862(119862([minus119903119894 0])L(119862([minus119903

119894 0]) 119871

infin(R

+)))

119894 = 1 119899 Here L(119883 119884) denotes the Banach spaceof all bounded linear operators from119883 to 119884


(1198677) 120593

119894⩾ 0 120582

119894⩾ 0 120573

119894⩾ 0 for 119894 = 1 119899


0and 120573(119886 119911) ⩽ 120573

0where 120582

0and 120573

0



1and 120573

1such that

1003817100381710038171003817120582 (119886 120595

1) minus 120582 (119886 120595

2)1003817100381710038171003817⩽ 120582

1

10038171003817100381710038171205951minus 120595

2

1003817100381710038171003817

(12)

for every 119886 1205951 120595

2and

119899

sum

119894=1

119899

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

120597120573119894(119886 119911)

120597119911119895

100381610038161003816100381610038161003816100381610038161003816

⩽ 1205731 (13)

for every 119886 119911




1

119905

119911119899

119905) and 119861 = (1198611

119861119899) where

119861119894(119905) = int

[minus1199030]

int

119905+119904119894

0

120573119894(119905 + 119904

119894minus 119886 119911 (119905 + 119904))

times 119861119894(119886) 119890

minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

119889119886 119889119904

+ int

[minus1199030]

int

infin

0

120573119894(119886 + 119905 + 119904

119894 119911 (119905 + 119904))

times 120593119894(119886) 119890

minusint119905+119904119894

0120582119894(119886+120591119911120591)119889120591

119889119886 119889119904

(14)

119911119894

119905(119904

119894) = int

119905+119904119894

0

119861119894(119886) 119890

minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

119889119886

+ int

infin

0

120593119894(119886 0) 119890

minusint119905+119904119894

0120582119894(119886+120591119911120591)119889120591

119889119886 for 119905 + 119904119894⩾ 0

119911119894

119905(119904

119894) = 119911

119894

0(119905 + 119904

119894) for 119905 + 119904

119894lt 0

(15)


119906119894(119886 119905) =

120593119894(119886 minus 119905 0) 119890

minusint119905

0120582119894(119886minus119905+120591119911120591)119889120591 for 119886 ⩾ 119905

119861119894(119905 minus 119886) 119890

minusint119886

0120582119894(120572119911119905minus119886+120572)119889120572 for 119905 gt 119886

(16)

where 119861119894(119905) = 119906

119894(0 119905) for each 119894 = 1 119899



119899) be


0+

ℎ 1199050+ ℎ) 120582

119894

(ℎ) = 120582119894(119886

0+ ℎ 119911


the equation 119889119906119894119889ℎ + 120582119894


119906119894(119886

0+ ℎ 119905

0+ ℎ) = 119906

119894(119886

0 119905

0) 119890

minusintℎ

0120582119894

(120578)119889120578 (17)


0) = (119886 minus 119905 0) ℎ = 119905 and (119886

0 119905

0) = (0 119905 minus 119886)



119905⩾ 0 and 119861119894


+times [0 119879]








holds


2) and (119867

8) we get

119861119894(119905)⩽120573


119899int

119905

0


1198940]

119861119894(119886 + 119904) 119889119886+120573


119899Φ

119894 (18)

Denoting

B119894(119905) = sup

120591isin[minus119903119894119905]

119861119894(120591) (19)

we obtain

B119894(119905) ⩽ 120573


119899int

119905

0

B119894(119886) 119889119886 + 120573


119899Φ

119894 (20)


B119894(119905) ⩽ 120573


119899Φ

11989411989012057301199031sdotsdotsdot119903119899119905 (21)



(1199111 119911

119899)isin ⨉

119899

119894=1119862+([minus119903

119894 119879]) and 119861 = (119861

1 119861

119899) isin

⨉119899

119894=1119862+([minus119903




⨉119899

119894=1119862+([minus119903


119879= (Z1

119879 Z119899

119879)

where

Z119894

119879(119911) (119905) = int

119905

0

B119894

119879(119911) (119886) 119890

minusint119905

119886120582119894(120591minus119886119911120591)119889120591

119889119886

+ int

infin

0

120593119894(119886 0) 119890

minusint119905

0120582119894(119886+120591119911120591)119889120591

119889119886

(22)


1 119861


119879=

(B1

119879 B119899

119879)



119899

119894=1119862+[minus119903

119894 119879] rarr

⨉119899

119894=1119862+


for any 119879 gt 0



119894119879= sup

119905isin[minus119903119894119879]119890minus119870119905

|119891119894(119905)| for any


119894=1119862[minus119903



10038171003817100381710038171003817Z

119894

119879(119911) minusZ

119894

119879()

10038171003817100381710038171003817119879


119894119879]

119890minus119870119905

times int

119905

0

1003816100381610038161003816100381610038161003816


119886120582119894(120591minus119886119911120591)119889120591


119886120582119894(120591minus119886120591)119889120591

1003816100381610038161003816100381610038161003816

B119894

119879(119911) (119886) 119889119886


119894119879]

119890minus119870119905

times int

119905

0


119886120582119894(120591minus119886120591)119889120591

10038161003816100381610038161003816B

119894

119879(119911) (119886) minusB

119894

119879() (119886)

10038161003816100381610038161003816119889119886


119894119879]

119890minus119870119905

times int

infin

0

1003816100381610038161003816100381610038161003816


0120582119894(119886+120591119911120591)119889120591


0120582119894(119886+120591120591)119889120591

1003816100381610038161003816100381610038161003816

120593119894(119886 0) 119889119886

= 1198681+ 119868

2+ 119868

3

(23)


2 and 119868

3we use the assump-

tions (1198672) (119867

7) and (119867


|119896|119890|119896| Thus

1198681⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

times int

119905

0

1003816100381610038161003816100381610038161003816


119886[120582119894(120591minus119886119911120591)minus120582

119894(120591minus119886120591)]119889120591

1003816100381610038161003816100381610038161003816

B119894

119879(119911) (119886) 119889119886

⩽ 1205821119911 minus 119879


119894119879]

119890minus119870119905

times int

119905

0

int

119905

119886

11989021205820(119905minus119886)

119890119870120591B

119894

119879(119911) (119886) 119889120591 119889119886

⩽

1205821

119870

12057301199031sdot sdot sdot 119903

119899Φ

119894

12057301199031sdot sdot sdot 119903

119899minus 2120582

0


1198683⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

times int

infin

0

int

119905

119886

11989021205820(119905minus119886)

1198901198701205911205821119911 minus 119879

120593119894(119886 0) 119889120591 119889119886

⩽

1205821

119870

Φ11989411989021205820119879

119911 minus 119879

(24)


B119894

119879(119911) (119905) minusB

119894

119879() (119905)

= int

[minus1199030]

int

119905+119904119894

0

120573119894(119905+119904

119894minus119886 119911 (119905+119904)) sdot 119890

minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

times (B119894

119879(119911) (119886) minusB

119894

119879() (119886)) 119889119886 119889119904

(25)

+ int

[minus1199030]

int

119905+119904119894

0

B119894

119879() (119886)

times (120573119894(119905 + 119904

119894minus 119886 119911 (119905 + 119904))

times119890minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

minus120573119894(119905+ 119904

119894minus119886 (119905 + 119904))

times 119890minusint119905+119904119894

119886120582119894(120591minus119886120591)119889120591

)119889119886 119889119904

+ int

[minus1199030]

int

infin

0

120593119894(119886 0)

times (120573119894(119886 + 119905 + 119904

119894 119911 (119905 + 119904))

times119890minusint119905+119904119894

0120582119894(119886+120591119911120591)119889120591

minus120573119894(119886 + 119905 + 119904

119894 (119905 + 119904))

times 119890minusint119905+119904119894

0120582119894(119886+120591120591)119889120591

)119889119886 119889119904

(26)


B119894

119879(119911) (119905) minusB

119894

119879() (119905)

⩽ 12057301199031sdot sdot sdot 119903

119899int

119905

0

(B119894

119879(119911) (119886) minusB

119894

119879() (119886)) 119889119886

+

10038161003816100381610038161003816119891119894(119905)

10038161003816100381610038161003816

(27)


B119894

119879(119911) (119905) minusB

119894

119879() (119905)

⩽

10038161003816100381610038161003816119891119894(119905)

10038161003816100381610038161003816+ 120573


119899int

119905

0

10038161003816100381610038161003816119891119894(119886)


119889119886

(28)

Let us estimate |119891119894(119905)|

10038161003816100381610038161003816119891119894(119905)

10038161003816100381610038161003816

⩽ int

[minus1199030]

int

119905+119904119894

0

1198621B

119894

119879() (119886)

times (120573119894(119905 + 119904

119894minus 119886 119911 (119905 + 119904))

minus120573119894(119905 + 119904

119894minus 119886 (119905 + 119904))) 119889119886 119889119904


+ int

[minus1199030]

int

infin

0

1198622120593119894(119886 0)

times (120573119894(119886 + 119905 + 119904

119894 119911 (119905 + 119904))

minus120573119894(119886 + 119905 + 119904

119894 (119905 + 119904))) 119889119886 119889119904

⩽ 11986212057311199031sdot sdot sdot 119903

119899119890119870119905119911 minus 119879

times (Φ119894int

119905

0

12057301199031sdot sdot sdot 119903


infin

0

120593119894(119886 0) 119889119886)

⩽ 11986212057311199031sdot sdot sdot 119903

119899Φ


(29)

where1198621and119862


1 119862

2

Finally we have

1198682⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

timesint

119905

0

(

10038161003816100381610038161003816119891119894(119886)

10038161003816100381610038161003816+ 120573


119899int

119886

0

10038161003816100381610038161003816119891119894(120585)


119889120585) 119889119886

⩽ 11986212057311199031sdot sdot sdot 119903

119899Φ

119894119890119870119905119911 minus 119879


119894119879]

119890minus119870119905

(int

119905

0

119890(119870+1205730119903

1sdotsdotsdot119903119899)119886119889119886

+12057301199031sdot sdot sdot 119903

119899int

119905

0

int

119886

0

11989011987012058511989012057301199031sdotsdotsdot119903119899119886119889120585 119889119886)

⩽

11986212057311199031sdot sdot sdot 119903

119899Φ

119894

119870 + 12057301199031sdot sdot sdot 119903

119899119911 minus 119879

11989012057301199031sdotsdotsdot119903119899119879(1 +

12057301199031sdot sdot sdot 119903

119899

119870

)

(30)


1 119868

2 and 119868


119879with the






119899(119886)) of the model


(119906119894

0)

1015840

(119886) + 120582119894(119886 119911

0) 119906

119894

0= 0

119911119894

0= int

infin

0

119906119894

0(119886) 119889119886

119906119894

0(0) = 119903


119899int

infin

0

120573119894(119886 119911

0) 119906

119894

0(119886) 119889119886

(31)

for 119894 = 1 119899


1

0 119911

119899

0)


0(0) = (119906

1

0(0) 119906

119899

0(0)) are con-


1(R




120587119894

0(119886) = 119890

minusint119886

0120582119894(1205721199110)119889120572

(32)

The quantity

119877119894(119911

0) = 119903


119899int

infin

0

120573119894(119886 119911

0) 120587

119894

0(119886) 119889119886 (33)




Theorem 3 Let 1199110= (119911

1

0 119911

119899

0) and 119911119894

0gt 0 for 119894 = 1 119899


0(sdot) 120573119894

(sdot 1199110)120587

119894

0(sdot) isin 119871

1(R

+) for each 119894 =

1 119899 Then

119877 (1199110) = (119877

1(119911

0) 119877

119899(119911

0)) (34)

with

119877119894(119911

0) = 1 119894 = 1 119899 (35)


1

0 119906

119899


0= (119911

1

0 119911

119899

0) is given

by

119906119894

0(119886) = 119861

119894

0120587119894

0(119886) 119894 = 1 119899 (36)

where

119861119894

0=

119911119894

0

int

infin

0120587119894

0(119886) 119889119886

(37)



0(0) = 119861

119894

0 By (31)

2we obtain the

formula (37) for 119861119894




119906 (119886 119905) = 1199060(119886) + 120585 (119886 119905)

= (1199061

0(119886) + 120585

1(119886 119905) 119906

2

0(119886)

+1205852(119886 119905) 119906

119899

0(119886) + 120585

119899(119886 119905))

(38)

119906119894(119886 119905) = 119906

119894

0(119886) + 120585

119894(119886 119905) (39)


where 120585119894(119886 119905) = 120593119894(119886 119905) minus 119906

119894


119911 (119905) = 1199110+ 119901 (119905)

= (1199111

0+ 119901

1(119905) 119911

2

0+ 119901

2(119905) 119911

119899

0+ 119901

119899(119905))

119911119905= 119911

0+ 119901

119905= (119911

1

0+ 119901

1

119905 119911

2

0+ 119901

2

119905 119911

119899

0+ 119901

119899

119905)

(40)

119911119894(119905) = 119911

119894

0+ 119901

119894(119905)

119911119894

119905= 119911

119894

0+ 119901

119894

119905

(41)

where119901119894

119905(119904) = 119901




119863120585119894(119886 119905) + 120582

119894

0(119886) 120585

119894(119886 119905) + 120596

119894(119886) 119901

119894

119905= 119909

119894(119886 119905) (42)

for

119909119894(119886 119905) = minus 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905120585119894(119886 119905)

minus Λ119894(119886 119901

119905) [119861

119894

0120587119894

0(119886) + 120585

119894(119886 119905)]

120582119894

0(119886) = 120582

119894(119886 119911

0)

(43)

where 120596119894 R

+times 119862([minus119903


120596119894(119886) 119901

119894

119905= 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905(119861

119894

0120587119894

0(119886))

Λ119894(119886 119901

119905) = 120582

119894(119886 119911

0+ 119901

119905) minus 120582

119894

0(119886) minus 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905

(44)


120585119894(0 119905) = int

infin

0

int

[minus1199030]

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119904 119889119886 + 120581

119894119901119894

119905+ 120595

119894(119905)

(45)

where

120573119894(119886 119911

0) = 120573

119894

0(119886)

120595119894(119905) = int

infin

0

int

[minus1199030]

119863119911119894120573

119894(119886 119911

0) 119901

119894(119905 + 119904

119894) 120585

119894(119886 119905 + 119904

119894) 119889119904 119889119886

+ int

infin

0

int

[minus1199030]

Ω119894(119886 119901 (119905 + 119904))

times [119861119894

0120587119894

0(119886) + 120585

119894(119886 119905 + 119904

119894)] 119889119904 119889119886

120581119894119901119894

119905= 119861

119894

0int

infin

0

int

[minus1199030]

119863119911119894120573

119894(119886 119911

0) 119901

119894

119905120587119894

0(119886) 119889119904 119889119886

(46)


Ω119894(119886 119901 (119905 + 119904)) = 120573

119894(119886 119911

0+ 119901 (119905 + 119904))

minus 120573119894

0(119886) minus 119863

119911119894120573

119894(119886 119911

0) 119901

119894(119905 + 119904

119894)

(47)


119901119894(119905) = int

infin

0

120585119894(119886 119905) 119889119886 (48)


120585119894(119886 0) = 120578

119894(119886) (49)

where

120578119894(119886) = 120593

119894(119886 0) minus 119861

119894

0 (50)


0 119905

0) isin R

+times [0 119879] 120585119894(ℎ) = 120585

119894(119886

0+ ℎ 119905

0+ ℎ) 120582119894

0(ℎ) =

120582119894

0(119886

0+ ℎ) 120596119894

(ℎ) = 120596119894(119886

0+ ℎ) 119901119894

ℎ= 119901

119894

119905+ℎ and 119909119894(ℎ) = 119909119894(119886

0+


119889

119889ℎ

120585119894(ℎ) + 120582

119894

0(ℎ) 120585

119894(ℎ) = 119909

119894(ℎ) minus 120596

119894(ℎ) 119901

119894

ℎ(51)


120585119894(119886

0+ ℎ 119905

0+ ℎ)

= 120585119894(119886

0 119905

0) 119890

minusintℎ

0120582119894

0(1198860+120591)119889120591

+ 119890minusintℎ

0120582119894

0(1198860+120591)119889120591

times int

ℎ

0

[119909119894(119886

0+ 120591 119905

0+ 120591) minus 120596

119894(119886

0+ 120591) 119901

119894

1199050+120591]

times 119890int120591

0120582119894

0(1198860+120572)119889120572

119889120591

(52)


0) = (119886 minus 119905 0) ℎ = 119905 for 119886 ⩾ 119905 and (119886

0 119905

0) =


120585119894(119886 119905) = 120578

119894(119886 minus 119905)

119894(119886 minus 119905 119886)

+ int

119905

0

[119909119894(119886 minus 119905 + 120591 120591) minus 120596

119894(119886 minus 119905 + 120591) 119901

119894

120591]

times 119894

0(119886 minus 119905 + 120591 119886) 119889120591 for 119886 ⩾ 119905

120585119894(119886 119905) = 119887

119894(119905 minus 119886) 120587

119894

0(119886)

+ int

119905

119905minus119886

[119909119894(120591 + 119886 minus 119905 120591) minus 120596

119894(120591 + 119886 minus 119905) 119901

119894

120591]

times 119894

0(120591 + 119886 minus 119905 119886) 119889120591 for 119886 lt 119905

(53)

where

119894

0(119886 minus 119905 119886) =

120587119894

0(119886)

120587119894

0(119886 minus 119905)

(54)

119887119894(119905 minus 119886) = 120585

119894(0 119905 minus 119886) (55)


119901119894(119905) = int

119905

0

120585119894(119886 119905) 119889119886 + int

infin

119905

120585119894(119886 119905) 119889119886 (56)


It follows that

119901119894(119905) + int

119905

0

int

infin

0

120596119894(119886) 119901

119894

120591119894

0(119886 119886 + 119905 minus 120591) 119889119886 119889120591

minus int

119905

0

119887119894(120591) 120587

119894

0(119905 minus 120591) 119889120591

= int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591

(57)


119887119894(119905) = int

[minus1199030]

int

119905+119904119894

0

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ int

[minus1199030]

int

infin

119905+119904119894

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ 120581119894(119905) 119901

119894

119905+ 120595

119894(119905)

(58)

and hence

minus 120581119894119901119894

119905+ 119887

119894(119905)

+ int

119905

0

int

minus[1199030]

int

infin

0

120573119894

0(119886 minus 120591 + 119905 + 119904

119894) 120596

119894(119886) 119901

119894

120591

times 119894

0(119886 119886 minus 120591 + 119905 + 119904

119894) 119889119886 119889119904 119889120591

minus int

119905

0

int

[minus1199030]

120573119894

0(119905 + 119904

119894minus 120591) 119887

119894(120591) 120587

119894

0(119905 + 119904

119894minus 120591) 119889119904 119889120591

= int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(59)


y (119905) = [[[

y1 (119905)

y119899 (119905)

]

]

]

(60)

where

y119894 (119905) = [119901119894(119905)

119887119894(119905)

] (61)

y119894119905(119904) = y119894 (119905 + 119904) for 119904 isin [minus119903119894 0] 119894 = 1 2 119899 (62)


Ay119905+ int

119905

0

K (119905 minus 120591) y120591119889120591 = f (119905) (63)


A =

[

[

[

[

[

A1 O OO A2

O

d

O O A119899

]

]

]

]

]

K =

[

[

[

[

[

K1 O OO K2

O

d

O O K119899

]

]

]

]

]

(64)


1 0

minus1205811198941] A119894y

119905= [

119901119894

119905(0) 0

minus120581119894119901119894

119905119887119894

119905(0)] =

[119901119894(119905) 0

minus120581119894119901119894

119905119887119894(119905)] and

K119894(119905) =

[

[

[

[

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) 119889119886 minus120587

119894

0(119905)

int

0

minus119903119894

int

infin

0

120573119894

0(119886 + 119905 + 119904)

119894

0(119886 119886 + 119905 + 119904) 120596

119894(119886) 119889119886 119889119904 minusint

0

minus119903119894

120573119894

0(119905 + 119904) 120587

119894

0(119905 + 119904) 119889119904

]

]

]

]

(65)


f (119905) = [[[

f1 (119905)

f119899 (119905)

]

]

]

(66)

for

f 119894 (119905) = [119891119894

1(119905)

119891119894

2(119905)


119905

0

K119894(119905 minus 120591) y119894

1120591119889120591

119891119894

1(119905) = int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591


119891119894

2(119905) = int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(67)

and y1198941(119905) = [

119901119894(119905)

0] |

[minus1199031198940] y119894

1119905(119904) = y119894

1(119905 + 119904) for 119904 isin [minus119903119894 0]




119911119894120582

119894(119886 119911

0) and the deriva-

tion 119863119911119894120573

119894(119886 119911



(11986711) (Λ

119894(119886 120588)120588

0) and (Ω

119894(119886 120588)120588

0) tend to zero as


0denotes


12) 120582

lowast= inf

119886⩾0

120582119894(119886 119911

0) 119894 = 1 2 119899 gt 0


120579(119904) = 119890



1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 +

119904119894)119889119904 and let 120587119894

1(119905) = int

[minus1199030]120573119894

0(119905+119904

119894)120587

119894

0(119905+119904

119894)119889119904 Let us assume


1 = int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905

minus int

infin

0

119890minus119905120574

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905

sdot (1 minus int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905)

minus int

infin

0

119890minus119905120574120587119894

0(119905) 119889119905

sdot (int

infin

0

119890minus119905120574

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905 minus 120581

119894(119890

120574))

(68)




0

1003817100381710038171003817= 119874 (119890

minus120583119905) (69)

1003817100381710038171003817119906 (119886 119905) minus 119906

0(119886)1003817100381710038171003817= 119874 (119890

minus120583119905) for each 119886


(70)


119894 0])

lowast


984858 (120579) = int

infin

0

984858 (119905) (119890120579) 119890

minus120579119905119889119905 (71)


(984858 lowast V120591) (119905) = int

119905

0

984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)


984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)


A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)

where

A (120579) =[

[

[

[

[



d


]

]

]

]

]

A119894(120579) = [

1 0

minus120581119894(119890

120579) 1

]

K (120579) =[

[

[

[

[



d


(120579)

]

]

]

]

]

K119894(120579)

=[

[

[

int

infin

0

119890minus119905120579

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

0(120579)

int

infin

0

119890minus119905120579

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

1(120579)

]

]

]

(75)

for 119894

1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 + 119904

119894)119889119904 and

120587119894

1(119905) = int

[minus1199030]120573119894

0(119905 + 119904

119894)120587

119894

0(119905 + 119904



Aminus1(120579) =

[

[

[

[

[

[

[

(A1)

minus1


A2)

minus1


d


minus1

(120579)

]

]

]

]

]

]

]

(A119894)

minus1

(120579) = [

1 0

120581119894(119890

120579) 1

]

(76)




y (119905) = Aminus1f119905+ int

119905

0

J (119905 minus 120591) f120591119889120591 (77)

where f120591(119904) =

[

[

f1(120591+119904)

f119899(120591+119904)

]

]


J (119905) = 1

2120587

119890minus120583119905

int

infin

minusinfin

119890119894120577119905J (minus120583 + 119894120577) 119889120577 (78)


function J and

|J (119905)| ⩽ 1198621119890minus120583119905 (79)


1 119862

2 119862

3 denote


12) (32) and (54)

120587119894

0(119886) ⩽ 119890


119894

0(119886

1015840 119886) ⩽ 119890


(1198861015840⩽ 119886) (80)

For 120582lowastgt 120583 and 120573


f (119905) ⩽ 1198622int

119905

0


119909 (sdot 120591)1198711119889120591

+100381710038171003817100381712057810038171003817100381710038171198711119890

minus120582lowast119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

(81)

1003817100381710038171003817y (119905)100381710038171003817

1003817⩽ 119862

3100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

+int

119905

0

119890minus120583(119905minus120591)

[1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(82)

where y(119905) = (sum119899

119894=1(119901

119894)2+ (119887

119894)2)

12

1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 ⩽ int

119905

0

119890minus120582lowast120591

119887 (119905 minus 120591) 119889120591 +100381710038171003817100381712057810038171003817100381710038171198711119890


+ int

119905

0


(119909 (sdot 120591)1198711 + 120596119871

1

1003817100381710038171003817119901120591

10038171003817100381710038170) 119889120591

⩽ 1198624100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905

+ int

119905

0

119890minus120583(119905minus120591)

times [1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(83)


120575(120576) such that1003817100381710038171003817Λ (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

1003817100381710038171003817Ω (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(84)


119863119911120582(119886 119911

0)

119871infin 119863119911

120573(119886 1199110)


estimate that

119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

1003817100381710038171003817119901119905

10038171003817100381710038170sdot1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 + 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(85)



119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

120576120590 (119905)

120590 (119905) =1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 +

1003817100381710038171003817119901119905

10038171003817100381710038170

(86)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+ 2120576int

119905

0

119890minus120583(119905minus120591)

120590 (120591) 119889120591 (87)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

(minus120583+2119872120576)119905 (88)


01198711 = 120578

1198711 lt 120575 In


1198711 120595(119905)


(41) (86)2imply (69) and (53)

2 (80) and (39) imply (70)


120582119894(119886 119911) = 120582

119894(119886 119911

1

1 119911

1

2 119911

2

1 119911

2

2)

120573119894(119886 119911) = 120573

119894(119886 119911

1 119911

2)

(89)

where

120582119894(119886 119911) = 120582

119894(119911

1(0) 119911

1(minus119903

1) 119911

2(0) 119911

2(minus119903

2))

120573119894(119886 119911) = 120573

119894(119911

1 119911

2) 119890

minus120572119894119886

(90)


120582119894

0= 120582

119894(119911

1

0 119911

1

0 119911

2

0 119911

2

0)

120573119894

0= 120573

119894(119911

1

0 119911

2

0) 119890

minus120572119894119886

(91)


1

0 119911

2


conclude that

120587119894

0(119886) = 119890

minus119886120582119894

0

119877119894(119911

0) =

11990311199032120573119894

0

120572119894+ 120582

119894

0

(92)


0(119886) = 119911

119894

0120582119894

0119890minus120582119894

0119886 where 119861

0= 119911

119894

0120582119894

0if and only if 11990311199032120573119894

0(120572

119894+

120582119894




1

0 119911

2

0) be the solution


119877119894(119911

0) = 1 119894 = 1 2 (93)


120596119894(119890

120574) = (

120597120582119894

120597119911119894

1

+

120597120582119894

120597119911119894

2

119890minus120574119903119894

)120582119894

0119911119894

0119890minus119886120582119894

0 (94)


|120597120582119894120597119911

119894

1+(120597120582

119894120597119911

119894

2)119890

minus120574119903119894


120581119894(119890

120574) =

120582119894

0

120573119894

0

sdot

119911119894

0

119903119894sdot

120597120573119894

120597119911119894(119911

0) int

0

minus119903119894

119890120574119904119889119904 (95)


119894(119890



119860

119861 + 120574

(96)


Re(120574) gt minus120583

10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + 120574

10038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + Re (120574)

100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(97)


100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590 (98)


10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

⩽

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⩽

100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590

(99)



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(1198677) 120593

119894⩾ 0 120582

119894⩾ 0 120573

119894⩾ 0 for 119894 = 1 119899


0and 120573(119886 119911) ⩽ 120573

0where 120582

0and 120573

0



1and 120573

1such that

1003817100381710038171003817120582 (119886 120595

1) minus 120582 (119886 120595

2)1003817100381710038171003817⩽ 120582

1

10038171003817100381710038171205951minus 120595

2

1003817100381710038171003817

(12)

for every 119886 1205951 120595

2and

119899

sum

119894=1

119899

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

120597120573119894(119886 119911)

120597119911119895

100381610038161003816100381610038161003816100381610038161003816

⩽ 1205731 (13)

for every 119886 119911




1

119905

119911119899

119905) and 119861 = (1198611

119861119899) where

119861119894(119905) = int

[minus1199030]

int

119905+119904119894

0

120573119894(119905 + 119904

119894minus 119886 119911 (119905 + 119904))

times 119861119894(119886) 119890

minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

119889119886 119889119904

+ int

[minus1199030]

int

infin

0

120573119894(119886 + 119905 + 119904

119894 119911 (119905 + 119904))

times 120593119894(119886) 119890

minusint119905+119904119894

0120582119894(119886+120591119911120591)119889120591

119889119886 119889119904

(14)

119911119894

119905(119904

119894) = int

119905+119904119894

0

119861119894(119886) 119890

minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

119889119886

+ int

infin

0

120593119894(119886 0) 119890

minusint119905+119904119894

0120582119894(119886+120591119911120591)119889120591

119889119886 for 119905 + 119904119894⩾ 0

119911119894

119905(119904

119894) = 119911

119894

0(119905 + 119904

119894) for 119905 + 119904

119894lt 0

(15)


119906119894(119886 119905) =

120593119894(119886 minus 119905 0) 119890

minusint119905

0120582119894(119886minus119905+120591119911120591)119889120591 for 119886 ⩾ 119905

119861119894(119905 minus 119886) 119890

minusint119886

0120582119894(120572119911119905minus119886+120572)119889120572 for 119905 gt 119886

(16)

where 119861119894(119905) = 119906

119894(0 119905) for each 119894 = 1 119899



119899) be


0+

ℎ 1199050+ ℎ) 120582

119894

(ℎ) = 120582119894(119886

0+ ℎ 119911


the equation 119889119906119894119889ℎ + 120582119894


119906119894(119886

0+ ℎ 119905

0+ ℎ) = 119906

119894(119886

0 119905

0) 119890

minusintℎ

0120582119894

(120578)119889120578 (17)


0) = (119886 minus 119905 0) ℎ = 119905 and (119886

0 119905

0) = (0 119905 minus 119886)



119905⩾ 0 and 119861119894


+times [0 119879]








holds


2) and (119867

8) we get

119861119894(119905)⩽120573


119899int

119905

0


1198940]

119861119894(119886 + 119904) 119889119886+120573


119899Φ

119894 (18)

Denoting

B119894(119905) = sup

120591isin[minus119903119894119905]

119861119894(120591) (19)

we obtain

B119894(119905) ⩽ 120573


119899int

119905

0

B119894(119886) 119889119886 + 120573


119899Φ

119894 (20)


B119894(119905) ⩽ 120573


119899Φ

11989411989012057301199031sdotsdotsdot119903119899119905 (21)



(1199111 119911

119899)isin ⨉

119899

119894=1119862+([minus119903

119894 119879]) and 119861 = (119861

1 119861

119899) isin

⨉119899

119894=1119862+([minus119903




⨉119899

119894=1119862+([minus119903


119879= (Z1

119879 Z119899

119879)

where

Z119894

119879(119911) (119905) = int

119905

0

B119894

119879(119911) (119886) 119890

minusint119905

119886120582119894(120591minus119886119911120591)119889120591

119889119886

+ int

infin

0

120593119894(119886 0) 119890

minusint119905

0120582119894(119886+120591119911120591)119889120591

119889119886

(22)


1 119861


119879=

(B1

119879 B119899

119879)



119899

119894=1119862+[minus119903

119894 119879] rarr

⨉119899

119894=1119862+


for any 119879 gt 0



119894119879= sup

119905isin[minus119903119894119879]119890minus119870119905

|119891119894(119905)| for any


119894=1119862[minus119903



10038171003817100381710038171003817Z

119894

119879(119911) minusZ

119894

119879()

10038171003817100381710038171003817119879


119894119879]

119890minus119870119905

times int

119905

0

1003816100381610038161003816100381610038161003816


119886120582119894(120591minus119886119911120591)119889120591


119886120582119894(120591minus119886120591)119889120591

1003816100381610038161003816100381610038161003816

B119894

119879(119911) (119886) 119889119886


119894119879]

119890minus119870119905

times int

119905

0


119886120582119894(120591minus119886120591)119889120591

10038161003816100381610038161003816B

119894

119879(119911) (119886) minusB

119894

119879() (119886)

10038161003816100381610038161003816119889119886


119894119879]

119890minus119870119905

times int

infin

0

1003816100381610038161003816100381610038161003816


0120582119894(119886+120591119911120591)119889120591


0120582119894(119886+120591120591)119889120591

1003816100381610038161003816100381610038161003816

120593119894(119886 0) 119889119886

= 1198681+ 119868

2+ 119868

3

(23)


2 and 119868

3we use the assump-

tions (1198672) (119867

7) and (119867


|119896|119890|119896| Thus

1198681⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

times int

119905

0

1003816100381610038161003816100381610038161003816


119886[120582119894(120591minus119886119911120591)minus120582

119894(120591minus119886120591)]119889120591

1003816100381610038161003816100381610038161003816

B119894

119879(119911) (119886) 119889119886

⩽ 1205821119911 minus 119879


119894119879]

119890minus119870119905

times int

119905

0

int

119905

119886

11989021205820(119905minus119886)

119890119870120591B

119894

119879(119911) (119886) 119889120591 119889119886

⩽

1205821

119870

12057301199031sdot sdot sdot 119903

119899Φ

119894

12057301199031sdot sdot sdot 119903

119899minus 2120582

0


1198683⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

times int

infin

0

int

119905

119886

11989021205820(119905minus119886)

1198901198701205911205821119911 minus 119879

120593119894(119886 0) 119889120591 119889119886

⩽

1205821

119870

Φ11989411989021205820119879

119911 minus 119879

(24)


B119894

119879(119911) (119905) minusB

119894

119879() (119905)

= int

[minus1199030]

int

119905+119904119894

0

120573119894(119905+119904

119894minus119886 119911 (119905+119904)) sdot 119890

minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

times (B119894

119879(119911) (119886) minusB

119894

119879() (119886)) 119889119886 119889119904

(25)

+ int

[minus1199030]

int

119905+119904119894

0

B119894

119879() (119886)

times (120573119894(119905 + 119904

119894minus 119886 119911 (119905 + 119904))

times119890minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

minus120573119894(119905+ 119904

119894minus119886 (119905 + 119904))

times 119890minusint119905+119904119894

119886120582119894(120591minus119886120591)119889120591

)119889119886 119889119904

+ int

[minus1199030]

int

infin

0

120593119894(119886 0)

times (120573119894(119886 + 119905 + 119904

119894 119911 (119905 + 119904))

times119890minusint119905+119904119894

0120582119894(119886+120591119911120591)119889120591

minus120573119894(119886 + 119905 + 119904

119894 (119905 + 119904))

times 119890minusint119905+119904119894

0120582119894(119886+120591120591)119889120591

)119889119886 119889119904

(26)


B119894

119879(119911) (119905) minusB

119894

119879() (119905)

⩽ 12057301199031sdot sdot sdot 119903

119899int

119905

0

(B119894

119879(119911) (119886) minusB

119894

119879() (119886)) 119889119886

+

10038161003816100381610038161003816119891119894(119905)

10038161003816100381610038161003816

(27)


B119894

119879(119911) (119905) minusB

119894

119879() (119905)

⩽

10038161003816100381610038161003816119891119894(119905)

10038161003816100381610038161003816+ 120573


119899int

119905

0

10038161003816100381610038161003816119891119894(119886)


119889119886

(28)

Let us estimate |119891119894(119905)|

10038161003816100381610038161003816119891119894(119905)

10038161003816100381610038161003816

⩽ int

[minus1199030]

int

119905+119904119894

0

1198621B

119894

119879() (119886)

times (120573119894(119905 + 119904

119894minus 119886 119911 (119905 + 119904))

minus120573119894(119905 + 119904

119894minus 119886 (119905 + 119904))) 119889119886 119889119904


+ int

[minus1199030]

int

infin

0

1198622120593119894(119886 0)

times (120573119894(119886 + 119905 + 119904

119894 119911 (119905 + 119904))

minus120573119894(119886 + 119905 + 119904

119894 (119905 + 119904))) 119889119886 119889119904

⩽ 11986212057311199031sdot sdot sdot 119903

119899119890119870119905119911 minus 119879

times (Φ119894int

119905

0

12057301199031sdot sdot sdot 119903


infin

0

120593119894(119886 0) 119889119886)

⩽ 11986212057311199031sdot sdot sdot 119903

119899Φ


(29)

where1198621and119862


1 119862

2

Finally we have

1198682⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

timesint

119905

0

(

10038161003816100381610038161003816119891119894(119886)

10038161003816100381610038161003816+ 120573


119899int

119886

0

10038161003816100381610038161003816119891119894(120585)


119889120585) 119889119886

⩽ 11986212057311199031sdot sdot sdot 119903

119899Φ

119894119890119870119905119911 minus 119879


119894119879]

119890minus119870119905

(int

119905

0

119890(119870+1205730119903

1sdotsdotsdot119903119899)119886119889119886

+12057301199031sdot sdot sdot 119903

119899int

119905

0

int

119886

0

11989011987012058511989012057301199031sdotsdotsdot119903119899119886119889120585 119889119886)

⩽

11986212057311199031sdot sdot sdot 119903

119899Φ

119894

119870 + 12057301199031sdot sdot sdot 119903

119899119911 minus 119879

11989012057301199031sdotsdotsdot119903119899119879(1 +

12057301199031sdot sdot sdot 119903

119899

119870

)

(30)


1 119868

2 and 119868


119879with the






119899(119886)) of the model


(119906119894

0)

1015840

(119886) + 120582119894(119886 119911

0) 119906

119894

0= 0

119911119894

0= int

infin

0

119906119894

0(119886) 119889119886

119906119894

0(0) = 119903


119899int

infin

0

120573119894(119886 119911

0) 119906

119894

0(119886) 119889119886

(31)

for 119894 = 1 119899


1

0 119911

119899

0)


0(0) = (119906

1

0(0) 119906

119899

0(0)) are con-


1(R




120587119894

0(119886) = 119890

minusint119886

0120582119894(1205721199110)119889120572

(32)

The quantity

119877119894(119911

0) = 119903


119899int

infin

0

120573119894(119886 119911

0) 120587

119894

0(119886) 119889119886 (33)




Theorem 3 Let 1199110= (119911

1

0 119911

119899

0) and 119911119894

0gt 0 for 119894 = 1 119899


0(sdot) 120573119894

(sdot 1199110)120587

119894

0(sdot) isin 119871

1(R

+) for each 119894 =

1 119899 Then

119877 (1199110) = (119877

1(119911

0) 119877

119899(119911

0)) (34)

with

119877119894(119911

0) = 1 119894 = 1 119899 (35)


1

0 119906

119899


0= (119911

1

0 119911

119899

0) is given

by

119906119894

0(119886) = 119861

119894

0120587119894

0(119886) 119894 = 1 119899 (36)

where

119861119894

0=

119911119894

0

int

infin

0120587119894

0(119886) 119889119886

(37)



0(0) = 119861

119894

0 By (31)

2we obtain the

formula (37) for 119861119894




119906 (119886 119905) = 1199060(119886) + 120585 (119886 119905)

= (1199061

0(119886) + 120585

1(119886 119905) 119906

2

0(119886)

+1205852(119886 119905) 119906

119899

0(119886) + 120585

119899(119886 119905))

(38)

119906119894(119886 119905) = 119906

119894

0(119886) + 120585

119894(119886 119905) (39)


where 120585119894(119886 119905) = 120593119894(119886 119905) minus 119906

119894


119911 (119905) = 1199110+ 119901 (119905)

= (1199111

0+ 119901

1(119905) 119911

2

0+ 119901

2(119905) 119911

119899

0+ 119901

119899(119905))

119911119905= 119911

0+ 119901

119905= (119911

1

0+ 119901

1

119905 119911

2

0+ 119901

2

119905 119911

119899

0+ 119901

119899

119905)

(40)

119911119894(119905) = 119911

119894

0+ 119901

119894(119905)

119911119894

119905= 119911

119894

0+ 119901

119894

119905

(41)

where119901119894

119905(119904) = 119901




119863120585119894(119886 119905) + 120582

119894

0(119886) 120585

119894(119886 119905) + 120596

119894(119886) 119901

119894

119905= 119909

119894(119886 119905) (42)

for

119909119894(119886 119905) = minus 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905120585119894(119886 119905)

minus Λ119894(119886 119901

119905) [119861

119894

0120587119894

0(119886) + 120585

119894(119886 119905)]

120582119894

0(119886) = 120582

119894(119886 119911

0)

(43)

where 120596119894 R

+times 119862([minus119903


120596119894(119886) 119901

119894

119905= 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905(119861

119894

0120587119894

0(119886))

Λ119894(119886 119901

119905) = 120582

119894(119886 119911

0+ 119901

119905) minus 120582

119894

0(119886) minus 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905

(44)


120585119894(0 119905) = int

infin

0

int

[minus1199030]

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119904 119889119886 + 120581

119894119901119894

119905+ 120595

119894(119905)

(45)

where

120573119894(119886 119911

0) = 120573

119894

0(119886)

120595119894(119905) = int

infin

0

int

[minus1199030]

119863119911119894120573

119894(119886 119911

0) 119901

119894(119905 + 119904

119894) 120585

119894(119886 119905 + 119904

119894) 119889119904 119889119886

+ int

infin

0

int

[minus1199030]

Ω119894(119886 119901 (119905 + 119904))

times [119861119894

0120587119894

0(119886) + 120585

119894(119886 119905 + 119904

119894)] 119889119904 119889119886

120581119894119901119894

119905= 119861

119894

0int

infin

0

int

[minus1199030]

119863119911119894120573

119894(119886 119911

0) 119901

119894

119905120587119894

0(119886) 119889119904 119889119886

(46)


Ω119894(119886 119901 (119905 + 119904)) = 120573

119894(119886 119911

0+ 119901 (119905 + 119904))

minus 120573119894

0(119886) minus 119863

119911119894120573

119894(119886 119911

0) 119901

119894(119905 + 119904

119894)

(47)


119901119894(119905) = int

infin

0

120585119894(119886 119905) 119889119886 (48)


120585119894(119886 0) = 120578

119894(119886) (49)

where

120578119894(119886) = 120593

119894(119886 0) minus 119861

119894

0 (50)


0 119905

0) isin R

+times [0 119879] 120585119894(ℎ) = 120585

119894(119886

0+ ℎ 119905

0+ ℎ) 120582119894

0(ℎ) =

120582119894

0(119886

0+ ℎ) 120596119894

(ℎ) = 120596119894(119886

0+ ℎ) 119901119894

ℎ= 119901

119894

119905+ℎ and 119909119894(ℎ) = 119909119894(119886

0+


119889

119889ℎ

120585119894(ℎ) + 120582

119894

0(ℎ) 120585

119894(ℎ) = 119909

119894(ℎ) minus 120596

119894(ℎ) 119901

119894

ℎ(51)


120585119894(119886

0+ ℎ 119905

0+ ℎ)

= 120585119894(119886

0 119905

0) 119890

minusintℎ

0120582119894

0(1198860+120591)119889120591

+ 119890minusintℎ

0120582119894

0(1198860+120591)119889120591

times int

ℎ

0

[119909119894(119886

0+ 120591 119905

0+ 120591) minus 120596

119894(119886

0+ 120591) 119901

119894

1199050+120591]

times 119890int120591

0120582119894

0(1198860+120572)119889120572

119889120591

(52)


0) = (119886 minus 119905 0) ℎ = 119905 for 119886 ⩾ 119905 and (119886

0 119905

0) =


120585119894(119886 119905) = 120578

119894(119886 minus 119905)

119894(119886 minus 119905 119886)

+ int

119905

0

[119909119894(119886 minus 119905 + 120591 120591) minus 120596

119894(119886 minus 119905 + 120591) 119901

119894

120591]

times 119894

0(119886 minus 119905 + 120591 119886) 119889120591 for 119886 ⩾ 119905

120585119894(119886 119905) = 119887

119894(119905 minus 119886) 120587

119894

0(119886)

+ int

119905

119905minus119886

[119909119894(120591 + 119886 minus 119905 120591) minus 120596

119894(120591 + 119886 minus 119905) 119901

119894

120591]

times 119894

0(120591 + 119886 minus 119905 119886) 119889120591 for 119886 lt 119905

(53)

where

119894

0(119886 minus 119905 119886) =

120587119894

0(119886)

120587119894

0(119886 minus 119905)

(54)

119887119894(119905 minus 119886) = 120585

119894(0 119905 minus 119886) (55)


119901119894(119905) = int

119905

0

120585119894(119886 119905) 119889119886 + int

infin

119905

120585119894(119886 119905) 119889119886 (56)


It follows that

119901119894(119905) + int

119905

0

int

infin

0

120596119894(119886) 119901

119894

120591119894

0(119886 119886 + 119905 minus 120591) 119889119886 119889120591

minus int

119905

0

119887119894(120591) 120587

119894

0(119905 minus 120591) 119889120591

= int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591

(57)


119887119894(119905) = int

[minus1199030]

int

119905+119904119894

0

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ int

[minus1199030]

int

infin

119905+119904119894

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ 120581119894(119905) 119901

119894

119905+ 120595

119894(119905)

(58)

and hence

minus 120581119894119901119894

119905+ 119887

119894(119905)

+ int

119905

0

int

minus[1199030]

int

infin

0

120573119894

0(119886 minus 120591 + 119905 + 119904

119894) 120596

119894(119886) 119901

119894

120591

times 119894

0(119886 119886 minus 120591 + 119905 + 119904

119894) 119889119886 119889119904 119889120591

minus int

119905

0

int

[minus1199030]

120573119894

0(119905 + 119904

119894minus 120591) 119887

119894(120591) 120587

119894

0(119905 + 119904

119894minus 120591) 119889119904 119889120591

= int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(59)


y (119905) = [[[

y1 (119905)

y119899 (119905)

]

]

]

(60)

where

y119894 (119905) = [119901119894(119905)

119887119894(119905)

] (61)

y119894119905(119904) = y119894 (119905 + 119904) for 119904 isin [minus119903119894 0] 119894 = 1 2 119899 (62)


Ay119905+ int

119905

0

K (119905 minus 120591) y120591119889120591 = f (119905) (63)


A =

[

[

[

[

[

A1 O OO A2

O

d

O O A119899

]

]

]

]

]

K =

[

[

[

[

[

K1 O OO K2

O

d

O O K119899

]

]

]

]

]

(64)


1 0

minus1205811198941] A119894y

119905= [

119901119894

119905(0) 0

minus120581119894119901119894

119905119887119894

119905(0)] =

[119901119894(119905) 0

minus120581119894119901119894

119905119887119894(119905)] and

K119894(119905) =

[

[

[

[

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) 119889119886 minus120587

119894

0(119905)

int

0

minus119903119894

int

infin

0

120573119894

0(119886 + 119905 + 119904)

119894

0(119886 119886 + 119905 + 119904) 120596

119894(119886) 119889119886 119889119904 minusint

0

minus119903119894

120573119894

0(119905 + 119904) 120587

119894

0(119905 + 119904) 119889119904

]

]

]

]

(65)


f (119905) = [[[

f1 (119905)

f119899 (119905)

]

]

]

(66)

for

f 119894 (119905) = [119891119894

1(119905)

119891119894

2(119905)


119905

0

K119894(119905 minus 120591) y119894

1120591119889120591

119891119894

1(119905) = int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591


119891119894

2(119905) = int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(67)

and y1198941(119905) = [

119901119894(119905)

0] |

[minus1199031198940] y119894

1119905(119904) = y119894

1(119905 + 119904) for 119904 isin [minus119903119894 0]




119911119894120582

119894(119886 119911

0) and the deriva-

tion 119863119911119894120573

119894(119886 119911



(11986711) (Λ

119894(119886 120588)120588

0) and (Ω

119894(119886 120588)120588

0) tend to zero as


0denotes


12) 120582

lowast= inf

119886⩾0

120582119894(119886 119911

0) 119894 = 1 2 119899 gt 0


120579(119904) = 119890



1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 +

119904119894)119889119904 and let 120587119894

1(119905) = int

[minus1199030]120573119894

0(119905+119904

119894)120587

119894

0(119905+119904

119894)119889119904 Let us assume


1 = int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905

minus int

infin

0

119890minus119905120574

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905

sdot (1 minus int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905)

minus int

infin

0

119890minus119905120574120587119894

0(119905) 119889119905

sdot (int

infin

0

119890minus119905120574

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905 minus 120581

119894(119890

120574))

(68)




0

1003817100381710038171003817= 119874 (119890

minus120583119905) (69)

1003817100381710038171003817119906 (119886 119905) minus 119906

0(119886)1003817100381710038171003817= 119874 (119890

minus120583119905) for each 119886


(70)


119894 0])

lowast


984858 (120579) = int

infin

0

984858 (119905) (119890120579) 119890

minus120579119905119889119905 (71)


(984858 lowast V120591) (119905) = int

119905

0

984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)


984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)


A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)

where

A (120579) =[

[

[

[

[



d


]

]

]

]

]

A119894(120579) = [

1 0

minus120581119894(119890

120579) 1

]

K (120579) =[

[

[

[

[



d


(120579)

]

]

]

]

]

K119894(120579)

=[

[

[

int

infin

0

119890minus119905120579

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

0(120579)

int

infin

0

119890minus119905120579

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

1(120579)

]

]

]

(75)

for 119894

1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 + 119904

119894)119889119904 and

120587119894

1(119905) = int

[minus1199030]120573119894

0(119905 + 119904

119894)120587

119894

0(119905 + 119904



Aminus1(120579) =

[

[

[

[

[

[

[

(A1)

minus1


A2)

minus1


d


minus1

(120579)

]

]

]

]

]

]

]

(A119894)

minus1

(120579) = [

1 0

120581119894(119890

120579) 1

]

(76)




y (119905) = Aminus1f119905+ int

119905

0

J (119905 minus 120591) f120591119889120591 (77)

where f120591(119904) =

[

[

f1(120591+119904)

f119899(120591+119904)

]

]


J (119905) = 1

2120587

119890minus120583119905

int

infin

minusinfin

119890119894120577119905J (minus120583 + 119894120577) 119889120577 (78)


function J and

|J (119905)| ⩽ 1198621119890minus120583119905 (79)


1 119862

2 119862

3 denote


12) (32) and (54)

120587119894

0(119886) ⩽ 119890


119894

0(119886

1015840 119886) ⩽ 119890


(1198861015840⩽ 119886) (80)

For 120582lowastgt 120583 and 120573


f (119905) ⩽ 1198622int

119905

0


119909 (sdot 120591)1198711119889120591

+100381710038171003817100381712057810038171003817100381710038171198711119890

minus120582lowast119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

(81)

1003817100381710038171003817y (119905)100381710038171003817

1003817⩽ 119862

3100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

+int

119905

0

119890minus120583(119905minus120591)

[1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(82)

where y(119905) = (sum119899

119894=1(119901

119894)2+ (119887

119894)2)

12

1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 ⩽ int

119905

0

119890minus120582lowast120591

119887 (119905 minus 120591) 119889120591 +100381710038171003817100381712057810038171003817100381710038171198711119890


+ int

119905

0


(119909 (sdot 120591)1198711 + 120596119871

1

1003817100381710038171003817119901120591

10038171003817100381710038170) 119889120591

⩽ 1198624100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905

+ int

119905

0

119890minus120583(119905minus120591)

times [1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(83)


120575(120576) such that1003817100381710038171003817Λ (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

1003817100381710038171003817Ω (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(84)


119863119911120582(119886 119911

0)

119871infin 119863119911

120573(119886 1199110)


estimate that

119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

1003817100381710038171003817119901119905

10038171003817100381710038170sdot1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 + 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(85)



119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

120576120590 (119905)

120590 (119905) =1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 +

1003817100381710038171003817119901119905

10038171003817100381710038170

(86)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+ 2120576int

119905

0

119890minus120583(119905minus120591)

120590 (120591) 119889120591 (87)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

(minus120583+2119872120576)119905 (88)


01198711 = 120578

1198711 lt 120575 In


1198711 120595(119905)


(41) (86)2imply (69) and (53)

2 (80) and (39) imply (70)


120582119894(119886 119911) = 120582

119894(119886 119911

1

1 119911

1

2 119911

2

1 119911

2

2)

120573119894(119886 119911) = 120573

119894(119886 119911

1 119911

2)

(89)

where

120582119894(119886 119911) = 120582

119894(119911

1(0) 119911

1(minus119903

1) 119911

2(0) 119911

2(minus119903

2))

120573119894(119886 119911) = 120573

119894(119911

1 119911

2) 119890

minus120572119894119886

(90)


120582119894

0= 120582

119894(119911

1

0 119911

1

0 119911

2

0 119911

2

0)

120573119894

0= 120573

119894(119911

1

0 119911

2

0) 119890

minus120572119894119886

(91)


1

0 119911

2


conclude that

120587119894

0(119886) = 119890

minus119886120582119894

0

119877119894(119911

0) =

11990311199032120573119894

0

120572119894+ 120582

119894

0

(92)


0(119886) = 119911

119894

0120582119894

0119890minus120582119894

0119886 where 119861

0= 119911

119894

0120582119894

0if and only if 11990311199032120573119894

0(120572

119894+

120582119894




1

0 119911

2

0) be the solution


119877119894(119911

0) = 1 119894 = 1 2 (93)


120596119894(119890

120574) = (

120597120582119894

120597119911119894

1

+

120597120582119894

120597119911119894

2

119890minus120574119903119894

)120582119894

0119911119894

0119890minus119886120582119894

0 (94)


|120597120582119894120597119911

119894

1+(120597120582

119894120597119911

119894

2)119890

minus120574119903119894


120581119894(119890

120574) =

120582119894

0

120573119894

0

sdot

119911119894

0

119903119894sdot

120597120573119894

120597119911119894(119911

0) int

0

minus119903119894

119890120574119904119889119904 (95)


119894(119890



119860

119861 + 120574

(96)


Re(120574) gt minus120583

10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + 120574

10038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + Re (120574)

100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(97)


100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590 (98)


10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

⩽

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⩽

100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590

(99)



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899

119894=1119862+[minus119903

119894 119879] rarr

⨉119899

119894=1119862+


for any 119879 gt 0



119894119879= sup

119905isin[minus119903119894119879]119890minus119870119905

|119891119894(119905)| for any


119894=1119862[minus119903



10038171003817100381710038171003817Z

119894

119879(119911) minusZ

119894

119879()

10038171003817100381710038171003817119879


119894119879]

119890minus119870119905

times int

119905

0

1003816100381610038161003816100381610038161003816


119886120582119894(120591minus119886119911120591)119889120591


119886120582119894(120591minus119886120591)119889120591

1003816100381610038161003816100381610038161003816

B119894

119879(119911) (119886) 119889119886


119894119879]

119890minus119870119905

times int

119905

0


119886120582119894(120591minus119886120591)119889120591

10038161003816100381610038161003816B

119894

119879(119911) (119886) minusB

119894

119879() (119886)

10038161003816100381610038161003816119889119886


119894119879]

119890minus119870119905

times int

infin

0

1003816100381610038161003816100381610038161003816


0120582119894(119886+120591119911120591)119889120591


0120582119894(119886+120591120591)119889120591

1003816100381610038161003816100381610038161003816

120593119894(119886 0) 119889119886

= 1198681+ 119868

2+ 119868

3

(23)


2 and 119868

3we use the assump-

tions (1198672) (119867

7) and (119867


|119896|119890|119896| Thus

1198681⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

times int

119905

0

1003816100381610038161003816100381610038161003816


119886[120582119894(120591minus119886119911120591)minus120582

119894(120591minus119886120591)]119889120591

1003816100381610038161003816100381610038161003816

B119894

119879(119911) (119886) 119889119886

⩽ 1205821119911 minus 119879


119894119879]

119890minus119870119905

times int

119905

0

int

119905

119886

11989021205820(119905minus119886)

119890119870120591B

119894

119879(119911) (119886) 119889120591 119889119886

⩽

1205821

119870

12057301199031sdot sdot sdot 119903

119899Φ

119894

12057301199031sdot sdot sdot 119903

119899minus 2120582

0


1198683⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

times int

infin

0

int

119905

119886

11989021205820(119905minus119886)

1198901198701205911205821119911 minus 119879

120593119894(119886 0) 119889120591 119889119886

⩽

1205821

119870

Φ11989411989021205820119879

119911 minus 119879

(24)


B119894

119879(119911) (119905) minusB

119894

119879() (119905)

= int

[minus1199030]

int

119905+119904119894

0

120573119894(119905+119904

119894minus119886 119911 (119905+119904)) sdot 119890

minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

times (B119894

119879(119911) (119886) minusB

119894

119879() (119886)) 119889119886 119889119904

(25)

+ int

[minus1199030]

int

119905+119904119894

0

B119894

119879() (119886)

times (120573119894(119905 + 119904

119894minus 119886 119911 (119905 + 119904))

times119890minusint119905+119904119894

119886120582119894(120591minus119886119911120591)119889120591

minus120573119894(119905+ 119904

119894minus119886 (119905 + 119904))

times 119890minusint119905+119904119894

119886120582119894(120591minus119886120591)119889120591

)119889119886 119889119904

+ int

[minus1199030]

int

infin

0

120593119894(119886 0)

times (120573119894(119886 + 119905 + 119904

119894 119911 (119905 + 119904))

times119890minusint119905+119904119894

0120582119894(119886+120591119911120591)119889120591

minus120573119894(119886 + 119905 + 119904

119894 (119905 + 119904))

times 119890minusint119905+119904119894

0120582119894(119886+120591120591)119889120591

)119889119886 119889119904

(26)


B119894

119879(119911) (119905) minusB

119894

119879() (119905)

⩽ 12057301199031sdot sdot sdot 119903

119899int

119905

0

(B119894

119879(119911) (119886) minusB

119894

119879() (119886)) 119889119886

+

10038161003816100381610038161003816119891119894(119905)

10038161003816100381610038161003816

(27)


B119894

119879(119911) (119905) minusB

119894

119879() (119905)

⩽

10038161003816100381610038161003816119891119894(119905)

10038161003816100381610038161003816+ 120573


119899int

119905

0

10038161003816100381610038161003816119891119894(119886)


119889119886

(28)

Let us estimate |119891119894(119905)|

10038161003816100381610038161003816119891119894(119905)

10038161003816100381610038161003816

⩽ int

[minus1199030]

int

119905+119904119894

0

1198621B

119894

119879() (119886)

times (120573119894(119905 + 119904

119894minus 119886 119911 (119905 + 119904))

minus120573119894(119905 + 119904

119894minus 119886 (119905 + 119904))) 119889119886 119889119904


+ int

[minus1199030]

int

infin

0

1198622120593119894(119886 0)

times (120573119894(119886 + 119905 + 119904

119894 119911 (119905 + 119904))

minus120573119894(119886 + 119905 + 119904

119894 (119905 + 119904))) 119889119886 119889119904

⩽ 11986212057311199031sdot sdot sdot 119903

119899119890119870119905119911 minus 119879

times (Φ119894int

119905

0

12057301199031sdot sdot sdot 119903


infin

0

120593119894(119886 0) 119889119886)

⩽ 11986212057311199031sdot sdot sdot 119903

119899Φ


(29)

where1198621and119862


1 119862

2

Finally we have

1198682⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

timesint

119905

0

(

10038161003816100381610038161003816119891119894(119886)

10038161003816100381610038161003816+ 120573


119899int

119886

0

10038161003816100381610038161003816119891119894(120585)


119889120585) 119889119886

⩽ 11986212057311199031sdot sdot sdot 119903

119899Φ

119894119890119870119905119911 minus 119879


119894119879]

119890minus119870119905

(int

119905

0

119890(119870+1205730119903

1sdotsdotsdot119903119899)119886119889119886

+12057301199031sdot sdot sdot 119903

119899int

119905

0

int

119886

0

11989011987012058511989012057301199031sdotsdotsdot119903119899119886119889120585 119889119886)

⩽

11986212057311199031sdot sdot sdot 119903

119899Φ

119894

119870 + 12057301199031sdot sdot sdot 119903

119899119911 minus 119879

11989012057301199031sdotsdotsdot119903119899119879(1 +

12057301199031sdot sdot sdot 119903

119899

119870

)

(30)


1 119868

2 and 119868


119879with the






119899(119886)) of the model


(119906119894

0)

1015840

(119886) + 120582119894(119886 119911

0) 119906

119894

0= 0

119911119894

0= int

infin

0

119906119894

0(119886) 119889119886

119906119894

0(0) = 119903


119899int

infin

0

120573119894(119886 119911

0) 119906

119894

0(119886) 119889119886

(31)

for 119894 = 1 119899


1

0 119911

119899

0)


0(0) = (119906

1

0(0) 119906

119899

0(0)) are con-


1(R




120587119894

0(119886) = 119890

minusint119886

0120582119894(1205721199110)119889120572

(32)

The quantity

119877119894(119911

0) = 119903


119899int

infin

0

120573119894(119886 119911

0) 120587

119894

0(119886) 119889119886 (33)




Theorem 3 Let 1199110= (119911

1

0 119911

119899

0) and 119911119894

0gt 0 for 119894 = 1 119899


0(sdot) 120573119894

(sdot 1199110)120587

119894

0(sdot) isin 119871

1(R

+) for each 119894 =

1 119899 Then

119877 (1199110) = (119877

1(119911

0) 119877

119899(119911

0)) (34)

with

119877119894(119911

0) = 1 119894 = 1 119899 (35)


1

0 119906

119899


0= (119911

1

0 119911

119899

0) is given

by

119906119894

0(119886) = 119861

119894

0120587119894

0(119886) 119894 = 1 119899 (36)

where

119861119894

0=

119911119894

0

int

infin

0120587119894

0(119886) 119889119886

(37)



0(0) = 119861

119894

0 By (31)

2we obtain the

formula (37) for 119861119894




119906 (119886 119905) = 1199060(119886) + 120585 (119886 119905)

= (1199061

0(119886) + 120585

1(119886 119905) 119906

2

0(119886)

+1205852(119886 119905) 119906

119899

0(119886) + 120585

119899(119886 119905))

(38)

119906119894(119886 119905) = 119906

119894

0(119886) + 120585

119894(119886 119905) (39)


where 120585119894(119886 119905) = 120593119894(119886 119905) minus 119906

119894


119911 (119905) = 1199110+ 119901 (119905)

= (1199111

0+ 119901

1(119905) 119911

2

0+ 119901

2(119905) 119911

119899

0+ 119901

119899(119905))

119911119905= 119911

0+ 119901

119905= (119911

1

0+ 119901

1

119905 119911

2

0+ 119901

2

119905 119911

119899

0+ 119901

119899

119905)

(40)

119911119894(119905) = 119911

119894

0+ 119901

119894(119905)

119911119894

119905= 119911

119894

0+ 119901

119894

119905

(41)

where119901119894

119905(119904) = 119901




119863120585119894(119886 119905) + 120582

119894

0(119886) 120585

119894(119886 119905) + 120596

119894(119886) 119901

119894

119905= 119909

119894(119886 119905) (42)

for

119909119894(119886 119905) = minus 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905120585119894(119886 119905)

minus Λ119894(119886 119901

119905) [119861

119894

0120587119894

0(119886) + 120585

119894(119886 119905)]

120582119894

0(119886) = 120582

119894(119886 119911

0)

(43)

where 120596119894 R

+times 119862([minus119903


120596119894(119886) 119901

119894

119905= 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905(119861

119894

0120587119894

0(119886))

Λ119894(119886 119901

119905) = 120582

119894(119886 119911

0+ 119901

119905) minus 120582

119894

0(119886) minus 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905

(44)


120585119894(0 119905) = int

infin

0

int

[minus1199030]

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119904 119889119886 + 120581

119894119901119894

119905+ 120595

119894(119905)

(45)

where

120573119894(119886 119911

0) = 120573

119894

0(119886)

120595119894(119905) = int

infin

0

int

[minus1199030]

119863119911119894120573

119894(119886 119911

0) 119901

119894(119905 + 119904

119894) 120585

119894(119886 119905 + 119904

119894) 119889119904 119889119886

+ int

infin

0

int

[minus1199030]

Ω119894(119886 119901 (119905 + 119904))

times [119861119894

0120587119894

0(119886) + 120585

119894(119886 119905 + 119904

119894)] 119889119904 119889119886

120581119894119901119894

119905= 119861

119894

0int

infin

0

int

[minus1199030]

119863119911119894120573

119894(119886 119911

0) 119901

119894

119905120587119894

0(119886) 119889119904 119889119886

(46)


Ω119894(119886 119901 (119905 + 119904)) = 120573

119894(119886 119911

0+ 119901 (119905 + 119904))

minus 120573119894

0(119886) minus 119863

119911119894120573

119894(119886 119911

0) 119901

119894(119905 + 119904

119894)

(47)


119901119894(119905) = int

infin

0

120585119894(119886 119905) 119889119886 (48)


120585119894(119886 0) = 120578

119894(119886) (49)

where

120578119894(119886) = 120593

119894(119886 0) minus 119861

119894

0 (50)


0 119905

0) isin R

+times [0 119879] 120585119894(ℎ) = 120585

119894(119886

0+ ℎ 119905

0+ ℎ) 120582119894

0(ℎ) =

120582119894

0(119886

0+ ℎ) 120596119894

(ℎ) = 120596119894(119886

0+ ℎ) 119901119894

ℎ= 119901

119894

119905+ℎ and 119909119894(ℎ) = 119909119894(119886

0+


119889

119889ℎ

120585119894(ℎ) + 120582

119894

0(ℎ) 120585

119894(ℎ) = 119909

119894(ℎ) minus 120596

119894(ℎ) 119901

119894

ℎ(51)


120585119894(119886

0+ ℎ 119905

0+ ℎ)

= 120585119894(119886

0 119905

0) 119890

minusintℎ

0120582119894

0(1198860+120591)119889120591

+ 119890minusintℎ

0120582119894

0(1198860+120591)119889120591

times int

ℎ

0

[119909119894(119886

0+ 120591 119905

0+ 120591) minus 120596

119894(119886

0+ 120591) 119901

119894

1199050+120591]

times 119890int120591

0120582119894

0(1198860+120572)119889120572

119889120591

(52)


0) = (119886 minus 119905 0) ℎ = 119905 for 119886 ⩾ 119905 and (119886

0 119905

0) =


120585119894(119886 119905) = 120578

119894(119886 minus 119905)

119894(119886 minus 119905 119886)

+ int

119905

0

[119909119894(119886 minus 119905 + 120591 120591) minus 120596

119894(119886 minus 119905 + 120591) 119901

119894

120591]

times 119894

0(119886 minus 119905 + 120591 119886) 119889120591 for 119886 ⩾ 119905

120585119894(119886 119905) = 119887

119894(119905 minus 119886) 120587

119894

0(119886)

+ int

119905

119905minus119886

[119909119894(120591 + 119886 minus 119905 120591) minus 120596

119894(120591 + 119886 minus 119905) 119901

119894

120591]

times 119894

0(120591 + 119886 minus 119905 119886) 119889120591 for 119886 lt 119905

(53)

where

119894

0(119886 minus 119905 119886) =

120587119894

0(119886)

120587119894

0(119886 minus 119905)

(54)

119887119894(119905 minus 119886) = 120585

119894(0 119905 minus 119886) (55)


119901119894(119905) = int

119905

0

120585119894(119886 119905) 119889119886 + int

infin

119905

120585119894(119886 119905) 119889119886 (56)


It follows that

119901119894(119905) + int

119905

0

int

infin

0

120596119894(119886) 119901

119894

120591119894

0(119886 119886 + 119905 minus 120591) 119889119886 119889120591

minus int

119905

0

119887119894(120591) 120587

119894

0(119905 minus 120591) 119889120591

= int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591

(57)


119887119894(119905) = int

[minus1199030]

int

119905+119904119894

0

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ int

[minus1199030]

int

infin

119905+119904119894

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ 120581119894(119905) 119901

119894

119905+ 120595

119894(119905)

(58)

and hence

minus 120581119894119901119894

119905+ 119887

119894(119905)

+ int

119905

0

int

minus[1199030]

int

infin

0

120573119894

0(119886 minus 120591 + 119905 + 119904

119894) 120596

119894(119886) 119901

119894

120591

times 119894

0(119886 119886 minus 120591 + 119905 + 119904

119894) 119889119886 119889119904 119889120591

minus int

119905

0

int

[minus1199030]

120573119894

0(119905 + 119904

119894minus 120591) 119887

119894(120591) 120587

119894

0(119905 + 119904

119894minus 120591) 119889119904 119889120591

= int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(59)


y (119905) = [[[

y1 (119905)

y119899 (119905)

]

]

]

(60)

where

y119894 (119905) = [119901119894(119905)

119887119894(119905)

] (61)

y119894119905(119904) = y119894 (119905 + 119904) for 119904 isin [minus119903119894 0] 119894 = 1 2 119899 (62)


Ay119905+ int

119905

0

K (119905 minus 120591) y120591119889120591 = f (119905) (63)


A =

[

[

[

[

[

A1 O OO A2

O

d

O O A119899

]

]

]

]

]

K =

[

[

[

[

[

K1 O OO K2

O

d

O O K119899

]

]

]

]

]

(64)


1 0

minus1205811198941] A119894y

119905= [

119901119894

119905(0) 0

minus120581119894119901119894

119905119887119894

119905(0)] =

[119901119894(119905) 0

minus120581119894119901119894

119905119887119894(119905)] and

K119894(119905) =

[

[

[

[

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) 119889119886 minus120587

119894

0(119905)

int

0

minus119903119894

int

infin

0

120573119894

0(119886 + 119905 + 119904)

119894

0(119886 119886 + 119905 + 119904) 120596

119894(119886) 119889119886 119889119904 minusint

0

minus119903119894

120573119894

0(119905 + 119904) 120587

119894

0(119905 + 119904) 119889119904

]

]

]

]

(65)


f (119905) = [[[

f1 (119905)

f119899 (119905)

]

]

]

(66)

for

f 119894 (119905) = [119891119894

1(119905)

119891119894

2(119905)


119905

0

K119894(119905 minus 120591) y119894

1120591119889120591

119891119894

1(119905) = int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591


119891119894

2(119905) = int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(67)

and y1198941(119905) = [

119901119894(119905)

0] |

[minus1199031198940] y119894

1119905(119904) = y119894

1(119905 + 119904) for 119904 isin [minus119903119894 0]




119911119894120582

119894(119886 119911

0) and the deriva-

tion 119863119911119894120573

119894(119886 119911



(11986711) (Λ

119894(119886 120588)120588

0) and (Ω

119894(119886 120588)120588

0) tend to zero as


0denotes


12) 120582

lowast= inf

119886⩾0

120582119894(119886 119911

0) 119894 = 1 2 119899 gt 0


120579(119904) = 119890



1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 +

119904119894)119889119904 and let 120587119894

1(119905) = int

[minus1199030]120573119894

0(119905+119904

119894)120587

119894

0(119905+119904

119894)119889119904 Let us assume


1 = int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905

minus int

infin

0

119890minus119905120574

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905

sdot (1 minus int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905)

minus int

infin

0

119890minus119905120574120587119894

0(119905) 119889119905

sdot (int

infin

0

119890minus119905120574

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905 minus 120581

119894(119890

120574))

(68)




0

1003817100381710038171003817= 119874 (119890

minus120583119905) (69)

1003817100381710038171003817119906 (119886 119905) minus 119906

0(119886)1003817100381710038171003817= 119874 (119890

minus120583119905) for each 119886


(70)


119894 0])

lowast


984858 (120579) = int

infin

0

984858 (119905) (119890120579) 119890

minus120579119905119889119905 (71)


(984858 lowast V120591) (119905) = int

119905

0

984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)


984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)


A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)

where

A (120579) =[

[

[

[

[



d


]

]

]

]

]

A119894(120579) = [

1 0

minus120581119894(119890

120579) 1

]

K (120579) =[

[

[

[

[



d


(120579)

]

]

]

]

]

K119894(120579)

=[

[

[

int

infin

0

119890minus119905120579

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

0(120579)

int

infin

0

119890minus119905120579

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

1(120579)

]

]

]

(75)

for 119894

1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 + 119904

119894)119889119904 and

120587119894

1(119905) = int

[minus1199030]120573119894

0(119905 + 119904

119894)120587

119894

0(119905 + 119904



Aminus1(120579) =

[

[

[

[

[

[

[

(A1)

minus1


A2)

minus1


d


minus1

(120579)

]

]

]

]

]

]

]

(A119894)

minus1

(120579) = [

1 0

120581119894(119890

120579) 1

]

(76)




y (119905) = Aminus1f119905+ int

119905

0

J (119905 minus 120591) f120591119889120591 (77)

where f120591(119904) =

[

[

f1(120591+119904)

f119899(120591+119904)

]

]


J (119905) = 1

2120587

119890minus120583119905

int

infin

minusinfin

119890119894120577119905J (minus120583 + 119894120577) 119889120577 (78)


function J and

|J (119905)| ⩽ 1198621119890minus120583119905 (79)


1 119862

2 119862

3 denote


12) (32) and (54)

120587119894

0(119886) ⩽ 119890


119894

0(119886

1015840 119886) ⩽ 119890


(1198861015840⩽ 119886) (80)

For 120582lowastgt 120583 and 120573


f (119905) ⩽ 1198622int

119905

0


119909 (sdot 120591)1198711119889120591

+100381710038171003817100381712057810038171003817100381710038171198711119890

minus120582lowast119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

(81)

1003817100381710038171003817y (119905)100381710038171003817

1003817⩽ 119862

3100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

+int

119905

0

119890minus120583(119905minus120591)

[1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(82)

where y(119905) = (sum119899

119894=1(119901

119894)2+ (119887

119894)2)

12

1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 ⩽ int

119905

0

119890minus120582lowast120591

119887 (119905 minus 120591) 119889120591 +100381710038171003817100381712057810038171003817100381710038171198711119890


+ int

119905

0


(119909 (sdot 120591)1198711 + 120596119871

1

1003817100381710038171003817119901120591

10038171003817100381710038170) 119889120591

⩽ 1198624100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905

+ int

119905

0

119890minus120583(119905minus120591)

times [1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(83)


120575(120576) such that1003817100381710038171003817Λ (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

1003817100381710038171003817Ω (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(84)


119863119911120582(119886 119911

0)

119871infin 119863119911

120573(119886 1199110)


estimate that

119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

1003817100381710038171003817119901119905

10038171003817100381710038170sdot1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 + 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(85)



119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

120576120590 (119905)

120590 (119905) =1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 +

1003817100381710038171003817119901119905

10038171003817100381710038170

(86)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+ 2120576int

119905

0

119890minus120583(119905minus120591)

120590 (120591) 119889120591 (87)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

(minus120583+2119872120576)119905 (88)


01198711 = 120578

1198711 lt 120575 In


1198711 120595(119905)


(41) (86)2imply (69) and (53)

2 (80) and (39) imply (70)


120582119894(119886 119911) = 120582

119894(119886 119911

1

1 119911

1

2 119911

2

1 119911

2

2)

120573119894(119886 119911) = 120573

119894(119886 119911

1 119911

2)

(89)

where

120582119894(119886 119911) = 120582

119894(119911

1(0) 119911

1(minus119903

1) 119911

2(0) 119911

2(minus119903

2))

120573119894(119886 119911) = 120573

119894(119911

1 119911

2) 119890

minus120572119894119886

(90)


120582119894

0= 120582

119894(119911

1

0 119911

1

0 119911

2

0 119911

2

0)

120573119894

0= 120573

119894(119911

1

0 119911

2

0) 119890

minus120572119894119886

(91)


1

0 119911

2


conclude that

120587119894

0(119886) = 119890

minus119886120582119894

0

119877119894(119911

0) =

11990311199032120573119894

0

120572119894+ 120582

119894

0

(92)


0(119886) = 119911

119894

0120582119894

0119890minus120582119894

0119886 where 119861

0= 119911

119894

0120582119894

0if and only if 11990311199032120573119894

0(120572

119894+

120582119894




1

0 119911

2

0) be the solution


119877119894(119911

0) = 1 119894 = 1 2 (93)


120596119894(119890

120574) = (

120597120582119894

120597119911119894

1

+

120597120582119894

120597119911119894

2

119890minus120574119903119894

)120582119894

0119911119894

0119890minus119886120582119894

0 (94)


|120597120582119894120597119911

119894

1+(120597120582

119894120597119911

119894

2)119890

minus120574119903119894


120581119894(119890

120574) =

120582119894

0

120573119894

0

sdot

119911119894

0

119903119894sdot

120597120573119894

120597119911119894(119911

0) int

0

minus119903119894

119890120574119904119889119904 (95)


119894(119890



119860

119861 + 120574

(96)


Re(120574) gt minus120583

10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + 120574

10038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + Re (120574)

100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(97)


100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590 (98)


10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

⩽

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⩽

100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590

(99)



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ int

[minus1199030]

int

infin

0

1198622120593119894(119886 0)

times (120573119894(119886 + 119905 + 119904

119894 119911 (119905 + 119904))

minus120573119894(119886 + 119905 + 119904

119894 (119905 + 119904))) 119889119886 119889119904

⩽ 11986212057311199031sdot sdot sdot 119903

119899119890119870119905119911 minus 119879

times (Φ119894int

119905

0

12057301199031sdot sdot sdot 119903


infin

0

120593119894(119886 0) 119889119886)

⩽ 11986212057311199031sdot sdot sdot 119903

119899Φ


(29)

where1198621and119862


1 119862

2

Finally we have

1198682⩽ sup

119905isin[minus119903119894119879]

119890minus119870119905

timesint

119905

0

(

10038161003816100381610038161003816119891119894(119886)

10038161003816100381610038161003816+ 120573


119899int

119886

0

10038161003816100381610038161003816119891119894(120585)


119889120585) 119889119886

⩽ 11986212057311199031sdot sdot sdot 119903

119899Φ

119894119890119870119905119911 minus 119879


119894119879]

119890minus119870119905

(int

119905

0

119890(119870+1205730119903

1sdotsdotsdot119903119899)119886119889119886

+12057301199031sdot sdot sdot 119903

119899int

119905

0

int

119886

0

11989011987012058511989012057301199031sdotsdotsdot119903119899119886119889120585 119889119886)

⩽

11986212057311199031sdot sdot sdot 119903

119899Φ

119894

119870 + 12057301199031sdot sdot sdot 119903

119899119911 minus 119879

11989012057301199031sdotsdotsdot119903119899119879(1 +

12057301199031sdot sdot sdot 119903

119899

119870

)

(30)


1 119868

2 and 119868


119879with the






119899(119886)) of the model


(119906119894

0)

1015840

(119886) + 120582119894(119886 119911

0) 119906

119894

0= 0

119911119894

0= int

infin

0

119906119894

0(119886) 119889119886

119906119894

0(0) = 119903


119899int

infin

0

120573119894(119886 119911

0) 119906

119894

0(119886) 119889119886

(31)

for 119894 = 1 119899


1

0 119911

119899

0)


0(0) = (119906

1

0(0) 119906

119899

0(0)) are con-


1(R




120587119894

0(119886) = 119890

minusint119886

0120582119894(1205721199110)119889120572

(32)

The quantity

119877119894(119911

0) = 119903


119899int

infin

0

120573119894(119886 119911

0) 120587

119894

0(119886) 119889119886 (33)




Theorem 3 Let 1199110= (119911

1

0 119911

119899

0) and 119911119894

0gt 0 for 119894 = 1 119899


0(sdot) 120573119894

(sdot 1199110)120587

119894

0(sdot) isin 119871

1(R

+) for each 119894 =

1 119899 Then

119877 (1199110) = (119877

1(119911

0) 119877

119899(119911

0)) (34)

with

119877119894(119911

0) = 1 119894 = 1 119899 (35)


1

0 119906

119899


0= (119911

1

0 119911

119899

0) is given

by

119906119894

0(119886) = 119861

119894

0120587119894

0(119886) 119894 = 1 119899 (36)

where

119861119894

0=

119911119894

0

int

infin

0120587119894

0(119886) 119889119886

(37)



0(0) = 119861

119894

0 By (31)

2we obtain the

formula (37) for 119861119894




119906 (119886 119905) = 1199060(119886) + 120585 (119886 119905)

= (1199061

0(119886) + 120585

1(119886 119905) 119906

2

0(119886)

+1205852(119886 119905) 119906

119899

0(119886) + 120585

119899(119886 119905))

(38)

119906119894(119886 119905) = 119906

119894

0(119886) + 120585

119894(119886 119905) (39)


where 120585119894(119886 119905) = 120593119894(119886 119905) minus 119906

119894


119911 (119905) = 1199110+ 119901 (119905)

= (1199111

0+ 119901

1(119905) 119911

2

0+ 119901

2(119905) 119911

119899

0+ 119901

119899(119905))

119911119905= 119911

0+ 119901

119905= (119911

1

0+ 119901

1

119905 119911

2

0+ 119901

2

119905 119911

119899

0+ 119901

119899

119905)

(40)

119911119894(119905) = 119911

119894

0+ 119901

119894(119905)

119911119894

119905= 119911

119894

0+ 119901

119894

119905

(41)

where119901119894

119905(119904) = 119901




119863120585119894(119886 119905) + 120582

119894

0(119886) 120585

119894(119886 119905) + 120596

119894(119886) 119901

119894

119905= 119909

119894(119886 119905) (42)

for

119909119894(119886 119905) = minus 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905120585119894(119886 119905)

minus Λ119894(119886 119901

119905) [119861

119894

0120587119894

0(119886) + 120585

119894(119886 119905)]

120582119894

0(119886) = 120582

119894(119886 119911

0)

(43)

where 120596119894 R

+times 119862([minus119903


120596119894(119886) 119901

119894

119905= 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905(119861

119894

0120587119894

0(119886))

Λ119894(119886 119901

119905) = 120582

119894(119886 119911

0+ 119901

119905) minus 120582

119894

0(119886) minus 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905

(44)


120585119894(0 119905) = int

infin

0

int

[minus1199030]

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119904 119889119886 + 120581

119894119901119894

119905+ 120595

119894(119905)

(45)

where

120573119894(119886 119911

0) = 120573

119894

0(119886)

120595119894(119905) = int

infin

0

int

[minus1199030]

119863119911119894120573

119894(119886 119911

0) 119901

119894(119905 + 119904

119894) 120585

119894(119886 119905 + 119904

119894) 119889119904 119889119886

+ int

infin

0

int

[minus1199030]

Ω119894(119886 119901 (119905 + 119904))

times [119861119894

0120587119894

0(119886) + 120585

119894(119886 119905 + 119904

119894)] 119889119904 119889119886

120581119894119901119894

119905= 119861

119894

0int

infin

0

int

[minus1199030]

119863119911119894120573

119894(119886 119911

0) 119901

119894

119905120587119894

0(119886) 119889119904 119889119886

(46)


Ω119894(119886 119901 (119905 + 119904)) = 120573

119894(119886 119911

0+ 119901 (119905 + 119904))

minus 120573119894

0(119886) minus 119863

119911119894120573

119894(119886 119911

0) 119901

119894(119905 + 119904

119894)

(47)


119901119894(119905) = int

infin

0

120585119894(119886 119905) 119889119886 (48)


120585119894(119886 0) = 120578

119894(119886) (49)

where

120578119894(119886) = 120593

119894(119886 0) minus 119861

119894

0 (50)


0 119905

0) isin R

+times [0 119879] 120585119894(ℎ) = 120585

119894(119886

0+ ℎ 119905

0+ ℎ) 120582119894

0(ℎ) =

120582119894

0(119886

0+ ℎ) 120596119894

(ℎ) = 120596119894(119886

0+ ℎ) 119901119894

ℎ= 119901

119894

119905+ℎ and 119909119894(ℎ) = 119909119894(119886

0+


119889

119889ℎ

120585119894(ℎ) + 120582

119894

0(ℎ) 120585

119894(ℎ) = 119909

119894(ℎ) minus 120596

119894(ℎ) 119901

119894

ℎ(51)


120585119894(119886

0+ ℎ 119905

0+ ℎ)

= 120585119894(119886

0 119905

0) 119890

minusintℎ

0120582119894

0(1198860+120591)119889120591

+ 119890minusintℎ

0120582119894

0(1198860+120591)119889120591

times int

ℎ

0

[119909119894(119886

0+ 120591 119905

0+ 120591) minus 120596

119894(119886

0+ 120591) 119901

119894

1199050+120591]

times 119890int120591

0120582119894

0(1198860+120572)119889120572

119889120591

(52)


0) = (119886 minus 119905 0) ℎ = 119905 for 119886 ⩾ 119905 and (119886

0 119905

0) =


120585119894(119886 119905) = 120578

119894(119886 minus 119905)

119894(119886 minus 119905 119886)

+ int

119905

0

[119909119894(119886 minus 119905 + 120591 120591) minus 120596

119894(119886 minus 119905 + 120591) 119901

119894

120591]

times 119894

0(119886 minus 119905 + 120591 119886) 119889120591 for 119886 ⩾ 119905

120585119894(119886 119905) = 119887

119894(119905 minus 119886) 120587

119894

0(119886)

+ int

119905

119905minus119886

[119909119894(120591 + 119886 minus 119905 120591) minus 120596

119894(120591 + 119886 minus 119905) 119901

119894

120591]

times 119894

0(120591 + 119886 minus 119905 119886) 119889120591 for 119886 lt 119905

(53)

where

119894

0(119886 minus 119905 119886) =

120587119894

0(119886)

120587119894

0(119886 minus 119905)

(54)

119887119894(119905 minus 119886) = 120585

119894(0 119905 minus 119886) (55)


119901119894(119905) = int

119905

0

120585119894(119886 119905) 119889119886 + int

infin

119905

120585119894(119886 119905) 119889119886 (56)


It follows that

119901119894(119905) + int

119905

0

int

infin

0

120596119894(119886) 119901

119894

120591119894

0(119886 119886 + 119905 minus 120591) 119889119886 119889120591

minus int

119905

0

119887119894(120591) 120587

119894

0(119905 minus 120591) 119889120591

= int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591

(57)


119887119894(119905) = int

[minus1199030]

int

119905+119904119894

0

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ int

[minus1199030]

int

infin

119905+119904119894

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ 120581119894(119905) 119901

119894

119905+ 120595

119894(119905)

(58)

and hence

minus 120581119894119901119894

119905+ 119887

119894(119905)

+ int

119905

0

int

minus[1199030]

int

infin

0

120573119894

0(119886 minus 120591 + 119905 + 119904

119894) 120596

119894(119886) 119901

119894

120591

times 119894

0(119886 119886 minus 120591 + 119905 + 119904

119894) 119889119886 119889119904 119889120591

minus int

119905

0

int

[minus1199030]

120573119894

0(119905 + 119904

119894minus 120591) 119887

119894(120591) 120587

119894

0(119905 + 119904

119894minus 120591) 119889119904 119889120591

= int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(59)


y (119905) = [[[

y1 (119905)

y119899 (119905)

]

]

]

(60)

where

y119894 (119905) = [119901119894(119905)

119887119894(119905)

] (61)

y119894119905(119904) = y119894 (119905 + 119904) for 119904 isin [minus119903119894 0] 119894 = 1 2 119899 (62)


Ay119905+ int

119905

0

K (119905 minus 120591) y120591119889120591 = f (119905) (63)


A =

[

[

[

[

[

A1 O OO A2

O

d

O O A119899

]

]

]

]

]

K =

[

[

[

[

[

K1 O OO K2

O

d

O O K119899

]

]

]

]

]

(64)


1 0

minus1205811198941] A119894y

119905= [

119901119894

119905(0) 0

minus120581119894119901119894

119905119887119894

119905(0)] =

[119901119894(119905) 0

minus120581119894119901119894

119905119887119894(119905)] and

K119894(119905) =

[

[

[

[

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) 119889119886 minus120587

119894

0(119905)

int

0

minus119903119894

int

infin

0

120573119894

0(119886 + 119905 + 119904)

119894

0(119886 119886 + 119905 + 119904) 120596

119894(119886) 119889119886 119889119904 minusint

0

minus119903119894

120573119894

0(119905 + 119904) 120587

119894

0(119905 + 119904) 119889119904

]

]

]

]

(65)


f (119905) = [[[

f1 (119905)

f119899 (119905)

]

]

]

(66)

for

f 119894 (119905) = [119891119894

1(119905)

119891119894

2(119905)


119905

0

K119894(119905 minus 120591) y119894

1120591119889120591

119891119894

1(119905) = int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591


119891119894

2(119905) = int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(67)

and y1198941(119905) = [

119901119894(119905)

0] |

[minus1199031198940] y119894

1119905(119904) = y119894

1(119905 + 119904) for 119904 isin [minus119903119894 0]




119911119894120582

119894(119886 119911

0) and the deriva-

tion 119863119911119894120573

119894(119886 119911



(11986711) (Λ

119894(119886 120588)120588

0) and (Ω

119894(119886 120588)120588

0) tend to zero as


0denotes


12) 120582

lowast= inf

119886⩾0

120582119894(119886 119911

0) 119894 = 1 2 119899 gt 0


120579(119904) = 119890



1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 +

119904119894)119889119904 and let 120587119894

1(119905) = int

[minus1199030]120573119894

0(119905+119904

119894)120587

119894

0(119905+119904

119894)119889119904 Let us assume


1 = int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905

minus int

infin

0

119890minus119905120574

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905

sdot (1 minus int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905)

minus int

infin

0

119890minus119905120574120587119894

0(119905) 119889119905

sdot (int

infin

0

119890minus119905120574

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905 minus 120581

119894(119890

120574))

(68)




0

1003817100381710038171003817= 119874 (119890

minus120583119905) (69)

1003817100381710038171003817119906 (119886 119905) minus 119906

0(119886)1003817100381710038171003817= 119874 (119890

minus120583119905) for each 119886


(70)


119894 0])

lowast


984858 (120579) = int

infin

0

984858 (119905) (119890120579) 119890

minus120579119905119889119905 (71)


(984858 lowast V120591) (119905) = int

119905

0

984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)


984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)


A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)

where

A (120579) =[

[

[

[

[



d


]

]

]

]

]

A119894(120579) = [

1 0

minus120581119894(119890

120579) 1

]

K (120579) =[

[

[

[

[



d


(120579)

]

]

]

]

]

K119894(120579)

=[

[

[

int

infin

0

119890minus119905120579

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

0(120579)

int

infin

0

119890minus119905120579

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

1(120579)

]

]

]

(75)

for 119894

1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 + 119904

119894)119889119904 and

120587119894

1(119905) = int

[minus1199030]120573119894

0(119905 + 119904

119894)120587

119894

0(119905 + 119904



Aminus1(120579) =

[

[

[

[

[

[

[

(A1)

minus1


A2)

minus1


d


minus1

(120579)

]

]

]

]

]

]

]

(A119894)

minus1

(120579) = [

1 0

120581119894(119890

120579) 1

]

(76)




y (119905) = Aminus1f119905+ int

119905

0

J (119905 minus 120591) f120591119889120591 (77)

where f120591(119904) =

[

[

f1(120591+119904)

f119899(120591+119904)

]

]


J (119905) = 1

2120587

119890minus120583119905

int

infin

minusinfin

119890119894120577119905J (minus120583 + 119894120577) 119889120577 (78)


function J and

|J (119905)| ⩽ 1198621119890minus120583119905 (79)


1 119862

2 119862

3 denote


12) (32) and (54)

120587119894

0(119886) ⩽ 119890


119894

0(119886

1015840 119886) ⩽ 119890


(1198861015840⩽ 119886) (80)

For 120582lowastgt 120583 and 120573


f (119905) ⩽ 1198622int

119905

0


119909 (sdot 120591)1198711119889120591

+100381710038171003817100381712057810038171003817100381710038171198711119890

minus120582lowast119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

(81)

1003817100381710038171003817y (119905)100381710038171003817

1003817⩽ 119862

3100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

+int

119905

0

119890minus120583(119905minus120591)

[1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(82)

where y(119905) = (sum119899

119894=1(119901

119894)2+ (119887

119894)2)

12

1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 ⩽ int

119905

0

119890minus120582lowast120591

119887 (119905 minus 120591) 119889120591 +100381710038171003817100381712057810038171003817100381710038171198711119890


+ int

119905

0


(119909 (sdot 120591)1198711 + 120596119871

1

1003817100381710038171003817119901120591

10038171003817100381710038170) 119889120591

⩽ 1198624100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905

+ int

119905

0

119890minus120583(119905minus120591)

times [1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(83)


120575(120576) such that1003817100381710038171003817Λ (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

1003817100381710038171003817Ω (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(84)


119863119911120582(119886 119911

0)

119871infin 119863119911

120573(119886 1199110)


estimate that

119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

1003817100381710038171003817119901119905

10038171003817100381710038170sdot1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 + 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(85)



119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

120576120590 (119905)

120590 (119905) =1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 +

1003817100381710038171003817119901119905

10038171003817100381710038170

(86)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+ 2120576int

119905

0

119890minus120583(119905minus120591)

120590 (120591) 119889120591 (87)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

(minus120583+2119872120576)119905 (88)


01198711 = 120578

1198711 lt 120575 In


1198711 120595(119905)


(41) (86)2imply (69) and (53)

2 (80) and (39) imply (70)


120582119894(119886 119911) = 120582

119894(119886 119911

1

1 119911

1

2 119911

2

1 119911

2

2)

120573119894(119886 119911) = 120573

119894(119886 119911

1 119911

2)

(89)

where

120582119894(119886 119911) = 120582

119894(119911

1(0) 119911

1(minus119903

1) 119911

2(0) 119911

2(minus119903

2))

120573119894(119886 119911) = 120573

119894(119911

1 119911

2) 119890

minus120572119894119886

(90)


120582119894

0= 120582

119894(119911

1

0 119911

1

0 119911

2

0 119911

2

0)

120573119894

0= 120573

119894(119911

1

0 119911

2

0) 119890

minus120572119894119886

(91)


1

0 119911

2


conclude that

120587119894

0(119886) = 119890

minus119886120582119894

0

119877119894(119911

0) =

11990311199032120573119894

0

120572119894+ 120582

119894

0

(92)


0(119886) = 119911

119894

0120582119894

0119890minus120582119894

0119886 where 119861

0= 119911

119894

0120582119894

0if and only if 11990311199032120573119894

0(120572

119894+

120582119894




1

0 119911

2

0) be the solution


119877119894(119911

0) = 1 119894 = 1 2 (93)


120596119894(119890

120574) = (

120597120582119894

120597119911119894

1

+

120597120582119894

120597119911119894

2

119890minus120574119903119894

)120582119894

0119911119894

0119890minus119886120582119894

0 (94)


|120597120582119894120597119911

119894

1+(120597120582

119894120597119911

119894

2)119890

minus120574119903119894


120581119894(119890

120574) =

120582119894

0

120573119894

0

sdot

119911119894

0

119903119894sdot

120597120573119894

120597119911119894(119911

0) int

0

minus119903119894

119890120574119904119889119904 (95)


119894(119890



119860

119861 + 120574

(96)


Re(120574) gt minus120583

10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + 120574

10038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + Re (120574)

100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(97)


100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590 (98)


10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

⩽

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⩽

100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590

(99)



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 120585119894(119886 119905) = 120593119894(119886 119905) minus 119906

119894


119911 (119905) = 1199110+ 119901 (119905)

= (1199111

0+ 119901

1(119905) 119911

2

0+ 119901

2(119905) 119911

119899

0+ 119901

119899(119905))

119911119905= 119911

0+ 119901

119905= (119911

1

0+ 119901

1

119905 119911

2

0+ 119901

2

119905 119911

119899

0+ 119901

119899

119905)

(40)

119911119894(119905) = 119911

119894

0+ 119901

119894(119905)

119911119894

119905= 119911

119894

0+ 119901

119894

119905

(41)

where119901119894

119905(119904) = 119901




119863120585119894(119886 119905) + 120582

119894

0(119886) 120585

119894(119886 119905) + 120596

119894(119886) 119901

119894

119905= 119909

119894(119886 119905) (42)

for

119909119894(119886 119905) = minus 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905120585119894(119886 119905)

minus Λ119894(119886 119901

119905) [119861

119894

0120587119894

0(119886) + 120585

119894(119886 119905)]

120582119894

0(119886) = 120582

119894(119886 119911

0)

(43)

where 120596119894 R

+times 119862([minus119903


120596119894(119886) 119901

119894

119905= 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905(119861

119894

0120587119894

0(119886))

Λ119894(119886 119901

119905) = 120582

119894(119886 119911

0+ 119901

119905) minus 120582

119894

0(119886) minus 119863

119911119894120582

119894(119886 119911

0) 119901

119894

119905

(44)


120585119894(0 119905) = int

infin

0

int

[minus1199030]

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119904 119889119886 + 120581

119894119901119894

119905+ 120595

119894(119905)

(45)

where

120573119894(119886 119911

0) = 120573

119894

0(119886)

120595119894(119905) = int

infin

0

int

[minus1199030]

119863119911119894120573

119894(119886 119911

0) 119901

119894(119905 + 119904

119894) 120585

119894(119886 119905 + 119904

119894) 119889119904 119889119886

+ int

infin

0

int

[minus1199030]

Ω119894(119886 119901 (119905 + 119904))

times [119861119894

0120587119894

0(119886) + 120585

119894(119886 119905 + 119904

119894)] 119889119904 119889119886

120581119894119901119894

119905= 119861

119894

0int

infin

0

int

[minus1199030]

119863119911119894120573

119894(119886 119911

0) 119901

119894

119905120587119894

0(119886) 119889119904 119889119886

(46)


Ω119894(119886 119901 (119905 + 119904)) = 120573

119894(119886 119911

0+ 119901 (119905 + 119904))

minus 120573119894

0(119886) minus 119863

119911119894120573

119894(119886 119911

0) 119901

119894(119905 + 119904

119894)

(47)


119901119894(119905) = int

infin

0

120585119894(119886 119905) 119889119886 (48)


120585119894(119886 0) = 120578

119894(119886) (49)

where

120578119894(119886) = 120593

119894(119886 0) minus 119861

119894

0 (50)


0 119905

0) isin R

+times [0 119879] 120585119894(ℎ) = 120585

119894(119886

0+ ℎ 119905

0+ ℎ) 120582119894

0(ℎ) =

120582119894

0(119886

0+ ℎ) 120596119894

(ℎ) = 120596119894(119886

0+ ℎ) 119901119894

ℎ= 119901

119894

119905+ℎ and 119909119894(ℎ) = 119909119894(119886

0+


119889

119889ℎ

120585119894(ℎ) + 120582

119894

0(ℎ) 120585

119894(ℎ) = 119909

119894(ℎ) minus 120596

119894(ℎ) 119901

119894

ℎ(51)


120585119894(119886

0+ ℎ 119905

0+ ℎ)

= 120585119894(119886

0 119905

0) 119890

minusintℎ

0120582119894

0(1198860+120591)119889120591

+ 119890minusintℎ

0120582119894

0(1198860+120591)119889120591

times int

ℎ

0

[119909119894(119886

0+ 120591 119905

0+ 120591) minus 120596

119894(119886

0+ 120591) 119901

119894

1199050+120591]

times 119890int120591

0120582119894

0(1198860+120572)119889120572

119889120591

(52)


0) = (119886 minus 119905 0) ℎ = 119905 for 119886 ⩾ 119905 and (119886

0 119905

0) =


120585119894(119886 119905) = 120578

119894(119886 minus 119905)

119894(119886 minus 119905 119886)

+ int

119905

0

[119909119894(119886 minus 119905 + 120591 120591) minus 120596

119894(119886 minus 119905 + 120591) 119901

119894

120591]

times 119894

0(119886 minus 119905 + 120591 119886) 119889120591 for 119886 ⩾ 119905

120585119894(119886 119905) = 119887

119894(119905 minus 119886) 120587

119894

0(119886)

+ int

119905

119905minus119886

[119909119894(120591 + 119886 minus 119905 120591) minus 120596

119894(120591 + 119886 minus 119905) 119901

119894

120591]

times 119894

0(120591 + 119886 minus 119905 119886) 119889120591 for 119886 lt 119905

(53)

where

119894

0(119886 minus 119905 119886) =

120587119894

0(119886)

120587119894

0(119886 minus 119905)

(54)

119887119894(119905 minus 119886) = 120585

119894(0 119905 minus 119886) (55)


119901119894(119905) = int

119905

0

120585119894(119886 119905) 119889119886 + int

infin

119905

120585119894(119886 119905) 119889119886 (56)


It follows that

119901119894(119905) + int

119905

0

int

infin

0

120596119894(119886) 119901

119894

120591119894

0(119886 119886 + 119905 minus 120591) 119889119886 119889120591

minus int

119905

0

119887119894(120591) 120587

119894

0(119905 minus 120591) 119889120591

= int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591

(57)


119887119894(119905) = int

[minus1199030]

int

119905+119904119894

0

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ int

[minus1199030]

int

infin

119905+119904119894

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ 120581119894(119905) 119901

119894

119905+ 120595

119894(119905)

(58)

and hence

minus 120581119894119901119894

119905+ 119887

119894(119905)

+ int

119905

0

int

minus[1199030]

int

infin

0

120573119894

0(119886 minus 120591 + 119905 + 119904

119894) 120596

119894(119886) 119901

119894

120591

times 119894

0(119886 119886 minus 120591 + 119905 + 119904

119894) 119889119886 119889119904 119889120591

minus int

119905

0

int

[minus1199030]

120573119894

0(119905 + 119904

119894minus 120591) 119887

119894(120591) 120587

119894

0(119905 + 119904

119894minus 120591) 119889119904 119889120591

= int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(59)


y (119905) = [[[

y1 (119905)

y119899 (119905)

]

]

]

(60)

where

y119894 (119905) = [119901119894(119905)

119887119894(119905)

] (61)

y119894119905(119904) = y119894 (119905 + 119904) for 119904 isin [minus119903119894 0] 119894 = 1 2 119899 (62)


Ay119905+ int

119905

0

K (119905 minus 120591) y120591119889120591 = f (119905) (63)


A =

[

[

[

[

[

A1 O OO A2

O

d

O O A119899

]

]

]

]

]

K =

[

[

[

[

[

K1 O OO K2

O

d

O O K119899

]

]

]

]

]

(64)


1 0

minus1205811198941] A119894y

119905= [

119901119894

119905(0) 0

minus120581119894119901119894

119905119887119894

119905(0)] =

[119901119894(119905) 0

minus120581119894119901119894

119905119887119894(119905)] and

K119894(119905) =

[

[

[

[

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) 119889119886 minus120587

119894

0(119905)

int

0

minus119903119894

int

infin

0

120573119894

0(119886 + 119905 + 119904)

119894

0(119886 119886 + 119905 + 119904) 120596

119894(119886) 119889119886 119889119904 minusint

0

minus119903119894

120573119894

0(119905 + 119904) 120587

119894

0(119905 + 119904) 119889119904

]

]

]

]

(65)


f (119905) = [[[

f1 (119905)

f119899 (119905)

]

]

]

(66)

for

f 119894 (119905) = [119891119894

1(119905)

119891119894

2(119905)


119905

0

K119894(119905 minus 120591) y119894

1120591119889120591

119891119894

1(119905) = int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591


119891119894

2(119905) = int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(67)

and y1198941(119905) = [

119901119894(119905)

0] |

[minus1199031198940] y119894

1119905(119904) = y119894

1(119905 + 119904) for 119904 isin [minus119903119894 0]




119911119894120582

119894(119886 119911

0) and the deriva-

tion 119863119911119894120573

119894(119886 119911



(11986711) (Λ

119894(119886 120588)120588

0) and (Ω

119894(119886 120588)120588

0) tend to zero as


0denotes


12) 120582

lowast= inf

119886⩾0

120582119894(119886 119911

0) 119894 = 1 2 119899 gt 0


120579(119904) = 119890



1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 +

119904119894)119889119904 and let 120587119894

1(119905) = int

[minus1199030]120573119894

0(119905+119904

119894)120587

119894

0(119905+119904

119894)119889119904 Let us assume


1 = int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905

minus int

infin

0

119890minus119905120574

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905

sdot (1 minus int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905)

minus int

infin

0

119890minus119905120574120587119894

0(119905) 119889119905

sdot (int

infin

0

119890minus119905120574

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905 minus 120581

119894(119890

120574))

(68)




0

1003817100381710038171003817= 119874 (119890

minus120583119905) (69)

1003817100381710038171003817119906 (119886 119905) minus 119906

0(119886)1003817100381710038171003817= 119874 (119890

minus120583119905) for each 119886


(70)


119894 0])

lowast


984858 (120579) = int

infin

0

984858 (119905) (119890120579) 119890

minus120579119905119889119905 (71)


(984858 lowast V120591) (119905) = int

119905

0

984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)


984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)


A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)

where

A (120579) =[

[

[

[

[



d


]

]

]

]

]

A119894(120579) = [

1 0

minus120581119894(119890

120579) 1

]

K (120579) =[

[

[

[

[



d


(120579)

]

]

]

]

]

K119894(120579)

=[

[

[

int

infin

0

119890minus119905120579

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

0(120579)

int

infin

0

119890minus119905120579

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

1(120579)

]

]

]

(75)

for 119894

1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 + 119904

119894)119889119904 and

120587119894

1(119905) = int

[minus1199030]120573119894

0(119905 + 119904

119894)120587

119894

0(119905 + 119904



Aminus1(120579) =

[

[

[

[

[

[

[

(A1)

minus1


A2)

minus1


d


minus1

(120579)

]

]

]

]

]

]

]

(A119894)

minus1

(120579) = [

1 0

120581119894(119890

120579) 1

]

(76)




y (119905) = Aminus1f119905+ int

119905

0

J (119905 minus 120591) f120591119889120591 (77)

where f120591(119904) =

[

[

f1(120591+119904)

f119899(120591+119904)

]

]


J (119905) = 1

2120587

119890minus120583119905

int

infin

minusinfin

119890119894120577119905J (minus120583 + 119894120577) 119889120577 (78)


function J and

|J (119905)| ⩽ 1198621119890minus120583119905 (79)


1 119862

2 119862

3 denote


12) (32) and (54)

120587119894

0(119886) ⩽ 119890


119894

0(119886

1015840 119886) ⩽ 119890


(1198861015840⩽ 119886) (80)

For 120582lowastgt 120583 and 120573


f (119905) ⩽ 1198622int

119905

0


119909 (sdot 120591)1198711119889120591

+100381710038171003817100381712057810038171003817100381710038171198711119890

minus120582lowast119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

(81)

1003817100381710038171003817y (119905)100381710038171003817

1003817⩽ 119862

3100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

+int

119905

0

119890minus120583(119905minus120591)

[1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(82)

where y(119905) = (sum119899

119894=1(119901

119894)2+ (119887

119894)2)

12

1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 ⩽ int

119905

0

119890minus120582lowast120591

119887 (119905 minus 120591) 119889120591 +100381710038171003817100381712057810038171003817100381710038171198711119890


+ int

119905

0


(119909 (sdot 120591)1198711 + 120596119871

1

1003817100381710038171003817119901120591

10038171003817100381710038170) 119889120591

⩽ 1198624100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905

+ int

119905

0

119890minus120583(119905minus120591)

times [1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(83)


120575(120576) such that1003817100381710038171003817Λ (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

1003817100381710038171003817Ω (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(84)


119863119911120582(119886 119911

0)

119871infin 119863119911

120573(119886 1199110)


estimate that

119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

1003817100381710038171003817119901119905

10038171003817100381710038170sdot1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 + 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(85)



119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

120576120590 (119905)

120590 (119905) =1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 +

1003817100381710038171003817119901119905

10038171003817100381710038170

(86)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+ 2120576int

119905

0

119890minus120583(119905minus120591)

120590 (120591) 119889120591 (87)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

(minus120583+2119872120576)119905 (88)


01198711 = 120578

1198711 lt 120575 In


1198711 120595(119905)


(41) (86)2imply (69) and (53)

2 (80) and (39) imply (70)


120582119894(119886 119911) = 120582

119894(119886 119911

1

1 119911

1

2 119911

2

1 119911

2

2)

120573119894(119886 119911) = 120573

119894(119886 119911

1 119911

2)

(89)

where

120582119894(119886 119911) = 120582

119894(119911

1(0) 119911

1(minus119903

1) 119911

2(0) 119911

2(minus119903

2))

120573119894(119886 119911) = 120573

119894(119911

1 119911

2) 119890

minus120572119894119886

(90)


120582119894

0= 120582

119894(119911

1

0 119911

1

0 119911

2

0 119911

2

0)

120573119894

0= 120573

119894(119911

1

0 119911

2

0) 119890

minus120572119894119886

(91)


1

0 119911

2


conclude that

120587119894

0(119886) = 119890

minus119886120582119894

0

119877119894(119911

0) =

11990311199032120573119894

0

120572119894+ 120582

119894

0

(92)


0(119886) = 119911

119894

0120582119894

0119890minus120582119894

0119886 where 119861

0= 119911

119894

0120582119894

0if and only if 11990311199032120573119894

0(120572

119894+

120582119894




1

0 119911

2

0) be the solution


119877119894(119911

0) = 1 119894 = 1 2 (93)


120596119894(119890

120574) = (

120597120582119894

120597119911119894

1

+

120597120582119894

120597119911119894

2

119890minus120574119903119894

)120582119894

0119911119894

0119890minus119886120582119894

0 (94)


|120597120582119894120597119911

119894

1+(120597120582

119894120597119911

119894

2)119890

minus120574119903119894


120581119894(119890

120574) =

120582119894

0

120573119894

0

sdot

119911119894

0

119903119894sdot

120597120573119894

120597119911119894(119911

0) int

0

minus119903119894

119890120574119904119889119904 (95)


119894(119890



119860

119861 + 120574

(96)


Re(120574) gt minus120583

10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + 120574

10038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + Re (120574)

100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(97)


100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590 (98)


10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

⩽

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⩽

100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590

(99)



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










It follows that

119901119894(119905) + int

119905

0

int

infin

0

120596119894(119886) 119901

119894

120591119894

0(119886 119886 + 119905 minus 120591) 119889119886 119889120591

minus int

119905

0

119887119894(120591) 120587

119894

0(119905 minus 120591) 119889120591

= int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591

(57)


119887119894(119905) = int

[minus1199030]

int

119905+119904119894

0

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ int

[minus1199030]

int

infin

119905+119904119894

120573119894

0(119886) 120585

119894(119886 119905 + 119904

119894) 119889119886 119889119904

+ 120581119894(119905) 119901

119894

119905+ 120595

119894(119905)

(58)

and hence

minus 120581119894119901119894

119905+ 119887

119894(119905)

+ int

119905

0

int

minus[1199030]

int

infin

0

120573119894

0(119886 minus 120591 + 119905 + 119904

119894) 120596

119894(119886) 119901

119894

120591

times 119894

0(119886 119886 minus 120591 + 119905 + 119904

119894) 119889119886 119889119904 119889120591

minus int

119905

0

int

[minus1199030]

120573119894

0(119905 + 119904

119894minus 120591) 119887

119894(120591) 120587

119894

0(119905 + 119904

119894minus 120591) 119889119904 119889120591

= int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(59)


y (119905) = [[[

y1 (119905)

y119899 (119905)

]

]

]

(60)

where

y119894 (119905) = [119901119894(119905)

119887119894(119905)

] (61)

y119894119905(119904) = y119894 (119905 + 119904) for 119904 isin [minus119903119894 0] 119894 = 1 2 119899 (62)


Ay119905+ int

119905

0

K (119905 minus 120591) y120591119889120591 = f (119905) (63)


A =

[

[

[

[

[

A1 O OO A2

O

d

O O A119899

]

]

]

]

]

K =

[

[

[

[

[

K1 O OO K2

O

d

O O K119899

]

]

]

]

]

(64)


1 0

minus1205811198941] A119894y

119905= [

119901119894

119905(0) 0

minus120581119894119901119894

119905119887119894

119905(0)] =

[119901119894(119905) 0

minus120581119894119901119894

119905119887119894(119905)] and

K119894(119905) =

[

[

[

[

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) 119889119886 minus120587

119894

0(119905)

int

0

minus119903119894

int

infin

0

120573119894

0(119886 + 119905 + 119904)

119894

0(119886 119886 + 119905 + 119904) 120596

119894(119886) 119889119886 119889119904 minusint

0

minus119903119894

120573119894

0(119905 + 119904) 120587

119894

0(119905 + 119904) 119889119904

]

]

]

]

(65)


f (119905) = [[[

f1 (119905)

f119899 (119905)

]

]

]

(66)

for

f 119894 (119905) = [119891119894

1(119905)

119891119894

2(119905)


119905

0

K119894(119905 minus 120591) y119894

1120591119889120591

119891119894

1(119905) = int

119905

0

int

infin

119905minus120591

119909119894(120591 + 119886 minus 119905 120591)

119894

0(120591 + 119886 minus 119905 119886) 119889119886 119889120591

+ int

infin

0

120578119894(120591)

119894

0(120591 120591 + 119905) 119889120591


119891119894

2(119905) = int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(67)

and y1198941(119905) = [

119901119894(119905)

0] |

[minus1199031198940] y119894

1119905(119904) = y119894

1(119905 + 119904) for 119904 isin [minus119903119894 0]




119911119894120582

119894(119886 119911

0) and the deriva-

tion 119863119911119894120573

119894(119886 119911



(11986711) (Λ

119894(119886 120588)120588

0) and (Ω

119894(119886 120588)120588

0) tend to zero as


0denotes


12) 120582

lowast= inf

119886⩾0

120582119894(119886 119911

0) 119894 = 1 2 119899 gt 0


120579(119904) = 119890



1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 +

119904119894)119889119904 and let 120587119894

1(119905) = int

[minus1199030]120573119894

0(119905+119904

119894)120587

119894

0(119905+119904

119894)119889119904 Let us assume


1 = int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905

minus int

infin

0

119890minus119905120574

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905

sdot (1 minus int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905)

minus int

infin

0

119890minus119905120574120587119894

0(119905) 119889119905

sdot (int

infin

0

119890minus119905120574

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905 minus 120581

119894(119890

120574))

(68)




0

1003817100381710038171003817= 119874 (119890

minus120583119905) (69)

1003817100381710038171003817119906 (119886 119905) minus 119906

0(119886)1003817100381710038171003817= 119874 (119890

minus120583119905) for each 119886


(70)


119894 0])

lowast


984858 (120579) = int

infin

0

984858 (119905) (119890120579) 119890

minus120579119905119889119905 (71)


(984858 lowast V120591) (119905) = int

119905

0

984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)


984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)


A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)

where

A (120579) =[

[

[

[

[



d


]

]

]

]

]

A119894(120579) = [

1 0

minus120581119894(119890

120579) 1

]

K (120579) =[

[

[

[

[



d


(120579)

]

]

]

]

]

K119894(120579)

=[

[

[

int

infin

0

119890minus119905120579

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

0(120579)

int

infin

0

119890minus119905120579

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

1(120579)

]

]

]

(75)

for 119894

1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 + 119904

119894)119889119904 and

120587119894

1(119905) = int

[minus1199030]120573119894

0(119905 + 119904

119894)120587

119894

0(119905 + 119904



Aminus1(120579) =

[

[

[

[

[

[

[

(A1)

minus1


A2)

minus1


d


minus1

(120579)

]

]

]

]

]

]

]

(A119894)

minus1

(120579) = [

1 0

120581119894(119890

120579) 1

]

(76)




y (119905) = Aminus1f119905+ int

119905

0

J (119905 minus 120591) f120591119889120591 (77)

where f120591(119904) =

[

[

f1(120591+119904)

f119899(120591+119904)

]

]


J (119905) = 1

2120587

119890minus120583119905

int

infin

minusinfin

119890119894120577119905J (minus120583 + 119894120577) 119889120577 (78)


function J and

|J (119905)| ⩽ 1198621119890minus120583119905 (79)


1 119862

2 119862

3 denote


12) (32) and (54)

120587119894

0(119886) ⩽ 119890


119894

0(119886

1015840 119886) ⩽ 119890


(1198861015840⩽ 119886) (80)

For 120582lowastgt 120583 and 120573


f (119905) ⩽ 1198622int

119905

0


119909 (sdot 120591)1198711119889120591

+100381710038171003817100381712057810038171003817100381710038171198711119890

minus120582lowast119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

(81)

1003817100381710038171003817y (119905)100381710038171003817

1003817⩽ 119862

3100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

+int

119905

0

119890minus120583(119905minus120591)

[1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(82)

where y(119905) = (sum119899

119894=1(119901

119894)2+ (119887

119894)2)

12

1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 ⩽ int

119905

0

119890minus120582lowast120591

119887 (119905 minus 120591) 119889120591 +100381710038171003817100381712057810038171003817100381710038171198711119890


+ int

119905

0


(119909 (sdot 120591)1198711 + 120596119871

1

1003817100381710038171003817119901120591

10038171003817100381710038170) 119889120591

⩽ 1198624100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905

+ int

119905

0

119890minus120583(119905minus120591)

times [1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(83)


120575(120576) such that1003817100381710038171003817Λ (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

1003817100381710038171003817Ω (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(84)


119863119911120582(119886 119911

0)

119871infin 119863119911

120573(119886 1199110)


estimate that

119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

1003817100381710038171003817119901119905

10038171003817100381710038170sdot1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 + 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(85)



119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

120576120590 (119905)

120590 (119905) =1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 +

1003817100381710038171003817119901119905

10038171003817100381710038170

(86)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+ 2120576int

119905

0

119890minus120583(119905minus120591)

120590 (120591) 119889120591 (87)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

(minus120583+2119872120576)119905 (88)


01198711 = 120578

1198711 lt 120575 In


1198711 120595(119905)


(41) (86)2imply (69) and (53)

2 (80) and (39) imply (70)


120582119894(119886 119911) = 120582

119894(119886 119911

1

1 119911

1

2 119911

2

1 119911

2

2)

120573119894(119886 119911) = 120573

119894(119886 119911

1 119911

2)

(89)

where

120582119894(119886 119911) = 120582

119894(119911

1(0) 119911

1(minus119903

1) 119911

2(0) 119911

2(minus119903

2))

120573119894(119886 119911) = 120573

119894(119911

1 119911

2) 119890

minus120572119894119886

(90)


120582119894

0= 120582

119894(119911

1

0 119911

1

0 119911

2

0 119911

2

0)

120573119894

0= 120573

119894(119911

1

0 119911

2

0) 119890

minus120572119894119886

(91)


1

0 119911

2


conclude that

120587119894

0(119886) = 119890

minus119886120582119894

0

119877119894(119911

0) =

11990311199032120573119894

0

120572119894+ 120582

119894

0

(92)


0(119886) = 119911

119894

0120582119894

0119890minus120582119894

0119886 where 119861

0= 119911

119894

0120582119894

0if and only if 11990311199032120573119894

0(120572

119894+

120582119894




1

0 119911

2

0) be the solution


119877119894(119911

0) = 1 119894 = 1 2 (93)


120596119894(119890

120574) = (

120597120582119894

120597119911119894

1

+

120597120582119894

120597119911119894

2

119890minus120574119903119894

)120582119894

0119911119894

0119890minus119886120582119894

0 (94)


|120597120582119894120597119911

119894

1+(120597120582

119894120597119911

119894

2)119890

minus120574119903119894


120581119894(119890

120574) =

120582119894

0

120573119894

0

sdot

119911119894

0

119903119894sdot

120597120573119894

120597119911119894(119911

0) int

0

minus119903119894

119890120574119904119889119904 (95)


119894(119890



119860

119861 + 120574

(96)


Re(120574) gt minus120583

10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + 120574

10038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + Re (120574)

100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(97)


100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590 (98)


10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

⩽

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⩽

100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590

(99)



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119891119894

2(119905) = int

[minus1199030]

int

119905+119904119894

0

int

infin

119905+119904119894minus120591

120573119894

0(119886) 119909

119894(120591 + 119886 minus 119905 minus 119904

119894 120591)

times 119894

0(120591 + 119886 minus 119905 minus 119904

119894 119886) 119889119886 119889120591 119889119904

+ int

[minus1199030]

int

infin

0

120573119894

0(120591 + 119905 + 119904

119894) 120578

119894(120591)

times 119894

0(120591 120591 + 119905 + 119904

119894) 119889120591 119889119904 + 120595

119894(119905)

(67)

and y1198941(119905) = [

119901119894(119905)

0] |

[minus1199031198940] y119894

1119905(119904) = y119894

1(119905 + 119904) for 119904 isin [minus119903119894 0]




119911119894120582

119894(119886 119911

0) and the deriva-

tion 119863119911119894120573

119894(119886 119911



(11986711) (Λ

119894(119886 120588)120588

0) and (Ω

119894(119886 120588)120588

0) tend to zero as


0denotes


12) 120582

lowast= inf

119886⩾0

120582119894(119886 119911

0) 119894 = 1 2 119899 gt 0


120579(119904) = 119890



1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 +

119904119894)119889119904 and let 120587119894

1(119905) = int

[minus1199030]120573119894

0(119905+119904

119894)120587

119894

0(119905+119904

119894)119889119904 Let us assume


1 = int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905

minus int

infin

0

119890minus119905120574

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905

sdot (1 minus int

infin

0

119890minus119905120574120587119894

1(119905) 119889119905)

minus int

infin

0

119890minus119905120574120587119894

0(119905) 119889119905

sdot (int

infin

0

119890minus119905120574

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120574) 119889119886 119889119905 minus 120581

119894(119890

120574))

(68)




0

1003817100381710038171003817= 119874 (119890

minus120583119905) (69)

1003817100381710038171003817119906 (119886 119905) minus 119906

0(119886)1003817100381710038171003817= 119874 (119890

minus120583119905) for each 119886


(70)


119894 0])

lowast


984858 (120579) = int

infin

0

984858 (119905) (119890120579) 119890

minus120579119905119889119905 (71)


(984858 lowast V120591) (119905) = int

119905

0

984858 (119905 minus 120591) V (120591 + sdot) 119889120591 (72)


984858 lowast V120591(120579) = 984858 (120579) V (120579) (73)


A (120579) y (120579) + K (120579) y (120579) = f (120579) (74)

where

A (120579) =[

[

[

[

[



d


]

]

]

]

]

A119894(120579) = [

1 0

minus120581119894(119890

120579) 1

]

K (120579) =[

[

[

[

[



d


(120579)

]

]

]

]

]

K119894(120579)

=[

[

[

int

infin

0

119890minus119905120579

int

infin

0

119894

0(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

0(120579)

int

infin

0

119890minus119905120579

int

infin

0

119894

1(119886 119886 + 119905) 120596

119894(119886) (119890

120579) 119889119886 119889119905 minus

119894

1(120579)

]

]

]

(75)

for 119894

1(119886 119886 + 119905) = int

[minus1199030]120573119894

0(119886 + 119905 + 119904

119894)

119894

0(119886 119886 + 119905 + 119904

119894)119889119904 and

120587119894

1(119905) = int

[minus1199030]120573119894

0(119905 + 119904

119894)120587

119894

0(119905 + 119904



Aminus1(120579) =

[

[

[

[

[

[

[

(A1)

minus1


A2)

minus1


d


minus1

(120579)

]

]

]

]

]

]

]

(A119894)

minus1

(120579) = [

1 0

120581119894(119890

120579) 1

]

(76)




y (119905) = Aminus1f119905+ int

119905

0

J (119905 minus 120591) f120591119889120591 (77)

where f120591(119904) =

[

[

f1(120591+119904)

f119899(120591+119904)

]

]


J (119905) = 1

2120587

119890minus120583119905

int

infin

minusinfin

119890119894120577119905J (minus120583 + 119894120577) 119889120577 (78)


function J and

|J (119905)| ⩽ 1198621119890minus120583119905 (79)


1 119862

2 119862

3 denote


12) (32) and (54)

120587119894

0(119886) ⩽ 119890


119894

0(119886

1015840 119886) ⩽ 119890


(1198861015840⩽ 119886) (80)

For 120582lowastgt 120583 and 120573


f (119905) ⩽ 1198622int

119905

0


119909 (sdot 120591)1198711119889120591

+100381710038171003817100381712057810038171003817100381710038171198711119890

minus120582lowast119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

(81)

1003817100381710038171003817y (119905)100381710038171003817

1003817⩽ 119862

3100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

+int

119905

0

119890minus120583(119905minus120591)

[1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(82)

where y(119905) = (sum119899

119894=1(119901

119894)2+ (119887

119894)2)

12

1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 ⩽ int

119905

0

119890minus120582lowast120591

119887 (119905 minus 120591) 119889120591 +100381710038171003817100381712057810038171003817100381710038171198711119890


+ int

119905

0


(119909 (sdot 120591)1198711 + 120596119871

1

1003817100381710038171003817119901120591

10038171003817100381710038170) 119889120591

⩽ 1198624100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905

+ int

119905

0

119890minus120583(119905minus120591)

times [1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(83)


120575(120576) such that1003817100381710038171003817Λ (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

1003817100381710038171003817Ω (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(84)


119863119911120582(119886 119911

0)

119871infin 119863119911

120573(119886 1199110)


estimate that

119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

1003817100381710038171003817119901119905

10038171003817100381710038170sdot1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 + 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(85)



119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

120576120590 (119905)

120590 (119905) =1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 +

1003817100381710038171003817119901119905

10038171003817100381710038170

(86)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+ 2120576int

119905

0

119890minus120583(119905minus120591)

120590 (120591) 119889120591 (87)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

(minus120583+2119872120576)119905 (88)


01198711 = 120578

1198711 lt 120575 In


1198711 120595(119905)


(41) (86)2imply (69) and (53)

2 (80) and (39) imply (70)


120582119894(119886 119911) = 120582

119894(119886 119911

1

1 119911

1

2 119911

2

1 119911

2

2)

120573119894(119886 119911) = 120573

119894(119886 119911

1 119911

2)

(89)

where

120582119894(119886 119911) = 120582

119894(119911

1(0) 119911

1(minus119903

1) 119911

2(0) 119911

2(minus119903

2))

120573119894(119886 119911) = 120573

119894(119911

1 119911

2) 119890

minus120572119894119886

(90)


120582119894

0= 120582

119894(119911

1

0 119911

1

0 119911

2

0 119911

2

0)

120573119894

0= 120573

119894(119911

1

0 119911

2

0) 119890

minus120572119894119886

(91)


1

0 119911

2


conclude that

120587119894

0(119886) = 119890

minus119886120582119894

0

119877119894(119911

0) =

11990311199032120573119894

0

120572119894+ 120582

119894

0

(92)


0(119886) = 119911

119894

0120582119894

0119890minus120582119894

0119886 where 119861

0= 119911

119894

0120582119894

0if and only if 11990311199032120573119894

0(120572

119894+

120582119894




1

0 119911

2

0) be the solution


119877119894(119911

0) = 1 119894 = 1 2 (93)


120596119894(119890

120574) = (

120597120582119894

120597119911119894

1

+

120597120582119894

120597119911119894

2

119890minus120574119903119894

)120582119894

0119911119894

0119890minus119886120582119894

0 (94)


|120597120582119894120597119911

119894

1+(120597120582

119894120597119911

119894

2)119890

minus120574119903119894


120581119894(119890

120574) =

120582119894

0

120573119894

0

sdot

119911119894

0

119903119894sdot

120597120573119894

120597119911119894(119911

0) int

0

minus119903119894

119890120574119904119889119904 (95)


119894(119890



119860

119861 + 120574

(96)


Re(120574) gt minus120583

10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + 120574

10038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + Re (120574)

100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(97)


100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590 (98)


10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

⩽

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⩽

100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590

(99)



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











y (119905) = Aminus1f119905+ int

119905

0

J (119905 minus 120591) f120591119889120591 (77)

where f120591(119904) =

[

[

f1(120591+119904)

f119899(120591+119904)

]

]


J (119905) = 1

2120587

119890minus120583119905

int

infin

minusinfin

119890119894120577119905J (minus120583 + 119894120577) 119889120577 (78)


function J and

|J (119905)| ⩽ 1198621119890minus120583119905 (79)


1 119862

2 119862

3 denote


12) (32) and (54)

120587119894

0(119886) ⩽ 119890


119894

0(119886

1015840 119886) ⩽ 119890


(1198861015840⩽ 119886) (80)

For 120582lowastgt 120583 and 120573


f (119905) ⩽ 1198622int

119905

0


119909 (sdot 120591)1198711119889120591

+100381710038171003817100381712057810038171003817100381710038171198711119890

minus120582lowast119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

(81)

1003817100381710038171003817y (119905)100381710038171003817

1003817⩽ 119862

3100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711

+int

119905

0

119890minus120583(119905minus120591)

[1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(82)

where y(119905) = (sum119899

119894=1(119901

119894)2+ (119887

119894)2)

12

1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 ⩽ int

119905

0

119890minus120582lowast120591

119887 (119905 minus 120591) 119889120591 +100381710038171003817100381712057810038171003817100381710038171198711119890


+ int

119905

0


(119909 (sdot 120591)1198711 + 120596119871

1

1003817100381710038171003817119901120591

10038171003817100381710038170) 119889120591

⩽ 1198624100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905

+ int

119905

0

119890minus120583(119905minus120591)

times [1003817100381710038171003817120595 (120591)

10038171003817100381710038171198711 + 119909 (sdot 120591)119871

1] 119889120591

(83)


120575(120576) such that1003817100381710038171003817Λ (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

1003817100381710038171003817Ω (119886 119901

119905)1003817100381710038171003817⩽ 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(84)


119863119911120582(119886 119911

0)

119871infin 119863119911

120573(119886 1199110)


estimate that

119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

1003817100381710038171003817119901119905

10038171003817100381710038170sdot1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 + 120576

1003817100381710038171003817119901119905

10038171003817100381710038170

(85)



119909 (sdot 119905)1198711 1003817100381710038171003817120595 (119905)

10038171003817100381710038171198711 ⩽ 1198625

120576120590 (119905)

120590 (119905) =1003817100381710038171003817120585 (sdot 119905)

10038171003817100381710038171198711 +

1003817100381710038171003817119901119905

10038171003817100381710038170

(86)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

minus120583119905+ 2120576int

119905

0

119890minus120583(119905minus120591)

120590 (120591) 119889120591 (87)


120590 (119905) ⩽ 119872100381710038171003817100381712057810038171003817100381710038171198711119890

(minus120583+2119872120576)119905 (88)


01198711 = 120578

1198711 lt 120575 In


1198711 120595(119905)


(41) (86)2imply (69) and (53)

2 (80) and (39) imply (70)


120582119894(119886 119911) = 120582

119894(119886 119911

1

1 119911

1

2 119911

2

1 119911

2

2)

120573119894(119886 119911) = 120573

119894(119886 119911

1 119911

2)

(89)

where

120582119894(119886 119911) = 120582

119894(119911

1(0) 119911

1(minus119903

1) 119911

2(0) 119911

2(minus119903

2))

120573119894(119886 119911) = 120573

119894(119911

1 119911

2) 119890

minus120572119894119886

(90)


120582119894

0= 120582

119894(119911

1

0 119911

1

0 119911

2

0 119911

2

0)

120573119894

0= 120573

119894(119911

1

0 119911

2

0) 119890

minus120572119894119886

(91)


1

0 119911

2


conclude that

120587119894

0(119886) = 119890

minus119886120582119894

0

119877119894(119911

0) =

11990311199032120573119894

0

120572119894+ 120582

119894

0

(92)


0(119886) = 119911

119894

0120582119894

0119890minus120582119894

0119886 where 119861

0= 119911

119894

0120582119894

0if and only if 11990311199032120573119894

0(120572

119894+

120582119894




1

0 119911

2

0) be the solution


119877119894(119911

0) = 1 119894 = 1 2 (93)


120596119894(119890

120574) = (

120597120582119894

120597119911119894

1

+

120597120582119894

120597119911119894

2

119890minus120574119903119894

)120582119894

0119911119894

0119890minus119886120582119894

0 (94)


|120597120582119894120597119911

119894

1+(120597120582

119894120597119911

119894

2)119890

minus120574119903119894


120581119894(119890

120574) =

120582119894

0

120573119894

0

sdot

119911119894

0

119903119894sdot

120597120573119894

120597119911119894(119911

0) int

0

minus119903119894

119890120574119904119889119904 (95)


119894(119890



119860

119861 + 120574

(96)


Re(120574) gt minus120583

10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + 120574

10038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + Re (120574)

100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(97)


100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590 (98)


10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

⩽

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⩽

100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590

(99)



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1

0 119911

2

0) be the solution


119877119894(119911

0) = 1 119894 = 1 2 (93)


120596119894(119890

120574) = (

120597120582119894

120597119911119894

1

+

120597120582119894

120597119911119894

2

119890minus120574119903119894

)120582119894

0119911119894

0119890minus119886120582119894

0 (94)


|120597120582119894120597119911

119894

1+(120597120582

119894120597119911

119894

2)119890

minus120574119903119894


120581119894(119890

120574) =

120582119894

0

120573119894

0

sdot

119911119894

0

119903119894sdot

120597120573119894

120597119911119894(119911

0) int

0

minus119903119894

119890120574119904119889119904 (95)


119894(119890



119860

119861 + 120574

(96)


Re(120574) gt minus120583

10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + 120574

10038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119860

119861 minus 120583 + 120583 + Re (120574)

100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(97)


100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590 (98)


10038161003816100381610038161003816100381610038161003816

119860

119861 + 120574

10038161003816100381610038161003816100381610038161003816

⩽

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119860


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⩽

100381610038161003816100381610038161003816100381610038161003816

119860


100381610038161003816100381610038161003816100381610038161003816

lt 120590

(99)



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article On the Multispecies Delayed Gurtin...

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