1

A STUDY ON THE ASYMPTOTIC BEHAVIOR OF

SOME DIFFERENCE EQUATIONS

Thesis submitted in Partial fulfilment for the award of

Degree of Doctor of Philosophy in Mathematics

By

V. ANANTHAN

Research Supervisor/Guide

Prof. Dr. S. KANDASAMY M.Sc., M.Phil., Ph.D.,

Professor, Department of Mathematics,

VMKV Engineering College, Salem - 636 308.

VINAYAKA MISSIONS UNIVERSITY

SALEM, TAMILNADU, INDIA

JULY - 2017

2

VINAYAKA MISSIONS UNIVERSITY

DECLARATION

I, V. ANANTHAN declare that the thesis entitled “A STUDY

ON THE ASYMPTOTIC BEHAVIOR OF SOME DIFFERENCE

EQUATIONS” submitted for the Degree of Doctor of Philosophy in

Mathematics is the record of work carried out by me during the period

from January 2011 to July 2017 under the guidance of

Dr. S. KANDASAMY M.Sc., M.Phil., Ph.D., Professor, Department of

Mathematics, VMKV Engineering College, Salem and that has not

formed the basis for the award of any other degree, diploma,

associatesship, fellowship or other titles in this university or any other

university or other similar institutions of higher learning.

Signature of the Candidate

Place: Salem

Date:

3

VINAYAKA MISSIONS UNIVERSITY

CERTIFICATE BY THE GUIDE

I, Dr. S. KANDASAMY M.Sc., M.Phil., Ph.D., certify that the

thesis entitled “A STUDY ON THE ASYMPTOTIC BEHAVIOR OF

SOME DIFFERENCE EQUATIONS” submitted for the Degree of

Doctor of Philosophy in Mathematics by V. ANANTHAN is the record

of research work carried out by him during the period from January

2011 to July 2017 under my guidance and supervision and that this work

has not formed the basis for the award of any other degree, diploma,

associate-ship, fellowship or other titles in this University or any other

University or Institutions of higher learning.

Signature of the Supervisor

Place: Salem

Date:

4

ACKNOWLEDGEMENT

At first, I would like to express my deepest gratitude to my

research supervisor and guide Dr. S. KANDASAMY, without whose

guidance none of this would have been possible. His encouragement,

support, freedom and keen insight have been invaluable and I have indeed

been fortunate to have such an ideal advisor. He played a very important

role in leading me towards scientific maturity. I am indebted to his

support through the years on both scientific and personal matters.

I am grateful to the founders of Vinayaka Missions University,

especially I thank our respected Madam Founder Chancellor, Vinyaka

Missions University Mrs. Annapoorani Shanmugasundaram,

Dato Dr. S. Sharavanan, Vice Chairman, Vinayaka Missions and

respected Chancellor Dr. S. Ganesan, for their valuable support. I thank

the Vice-Chancellor Prof. Dr. V. R. Rajendran and Registrar

Prof. Dr. Y. Abraham, Vinayaka Missions University and I express my

sincere thanks to Dr. S. Prabhavathi, Dean (Research) Vinayaka

Missions University Salem, Tamilnadu, India.

I express my profound thanks to Dr. P. Mohankumar and

Dr. J. Pandurangan, Retired Professors, Aarupadai Veedu Institute of

Technology, Paiyanoor, Chennai. They played a vital role in leading me

5

towards scientific maturity. I am grateful to their support through the

years on scientific matters.

I warmly thank Dr. A. Selvaraj, Professor and Head of the

Department of Mathematics, Nehru Institute of Technology, Coimbatore

for his support throughout my research work. I also thank to my best

friend Dr. A. Ramesh, Principal Incharge, District Institute of Education

and Training, Uthamacholapuram, Salem-636010 for his kind co-

operation.

I thank our Principal Dr. Vemmuri Lakshmi Narayana, Vice

Principal Dr. Vijendrababu, Head of the Department of Science and

Humanities Dr. Jennifer G. Joseph, Head of the Mathematics Division

Dr. L. Tamilselvi and my Department colleagues, Aarupadai Veedu

Institute of Technology Paiyanoor, Kanchipuram District and Vinayaka

Missions Kirupananda Variyar Engineering College, Salem.

Last but not least, I am indebted to my parents, wife, my kid, my

sister, brother, and my friends for their endless support throughout my

studies.

V. ANANTHAN

6

CONTENTS

Chapter No Title Page No

Chapter 1 Introduction …. 1

1.1 Introduction of Difference Equations …. 1

1.2 Asymptotic Behavior of Difference Equation …. 3

1.3 Examples of Difference Equations …. 4

1.4 Examples of Asymptotic Behavior of Difference Equation

…. 6

1.5 Review of Literature …. 7

1.6 Need for the Study …. 34

1.7 Objectives of the Thesis …. 35

1.8 Methodology …. 35

1.9 Plan of the Thesis …. 37

Chapter 2 Asymptotic and Oscillatory Behavior of Third Order Nonlinear Neutral Delay Difference Equation

…. 40

2.1 Introduction …. 40

2.2 Main Results …. 41

2.3 Examples …. 42

Chapter 3 Asymptotic Properties of Third Order Nonlinear Neutral Delay Equation

…. 48

3.1 Introduction …. 48

3.2 Main Results …. 49

3.3 Examples …. 52

7

Chapter 4 Asymptotic Properties Third Order

Nonlinear Neutral Delay Difference

Equation

…. 53

4.1 Introduction …. 53

4.2 Main results …. 53

Chapter 5

Asymptotic of Oscillation Solutions of

Third Order Nonlinear Difference Equations with Delay

…. 57

5.1 Introduction …. 57

5.2 Main Results …. 58

5.3 Boundedness and Oscillation …. 60

Chapter 6 Uncountably Many Positive Solutions of

First Order Nonlinear Neutral Delay Difference Equations

…. 66

6.1 Introduction …. 66

6.2 Krasnoselskii’s Fixed Point Theorem …. 67

6.3 Example with Diagrammatic Representation …. 72

Chapter 7 Asymptotic of Non Oscillatory Behaviour of Third Order Neutral Delay Difference Equations

…. 74

7.1 Introduction …. 74

7.2 Asymptotic and Non-Oscillation Theorem …. 75

Chapter 8 Oscillatory and Non Oscillatory Properties

of Fourth Order Difference Equation

…. 79

8.1 Introduction …. 79

8.2 Main Results …. 82

8.3 Oscillation Theorem by Using Monotonic Property

…. 88

Chapter 9 Conclusion …. 91

Bibliography …. 92

List of Publications …. 104

1

1

CHAPTER - 1 INTRODUCTION

1.1 INTRODUCTION OF DIFFERENCE EQUATION

Difference equations are of interest both as approximations of

Differential equations and as methods for describing fundamentally

discrete systems in biology, economic, statistics and other problems in

science where the system is described by the discrete variable. For

example in economics the price changes are considered from year to year

or month to month or week to week or from day to day; in every case the

time variable is discrete. In genetics, the genetic Characteristics changes

from generation to generation and the variable representing a generation

is discrete variable.

In population dynamics, we consider the changes in population

from one age group to another and the variable representing the age group

is discrete variable. In finding the probability of ‘n’ success in certain

number of trials, the independent variable is discrete.

A detailed study of difference equations with many examples from

diverse fields can be found as well as in chapter.

In the theory of difference equations oscillatory and non-oscillatory

Behavior of solutions play important role. A nontrivial solution of

2

Difference equation is said to be oscillatory if it is neither eventually

positive or eventually negative. Otherwise it is called non-oscillatory. If

all the solutions of a difference equation are Oscillatory then the

difference equation itself is said to be oscillatory.

Although several results regarding oscillatory theory in the discrete

case are similar to those of already known in the continuous case, The

adaption from the continuous to the discrete is not direct, but it Requires

some special devices. Further it has been show in that there exists some

properties of differential equations which do not carry over directly to

corresponding difference equations. Therefore, it is useful study the

oscillatory and non-oscillatory behavior of solutions of difference

equations. The problem of oscillation and non- oscillation for second and

high order nonlinear difference equations has received more attention in

the Last few years. It is also interesting to studying asymptotic behavior

of difference equation because they are analogues of differential

equations.

In mathematics, delay difference equations are a type of difference

equations in which the variables of the unknown function at a certain

Interval are given in terms of the values of the function at previous

Intervals. A difference equation of the form

( ) 0( , ) 0, 0 .m

n n ky f n y k N

3

1.2 ASYMPTOTIC BEHAVIOR OF DIFFERENCE EQUATION

If

lim 1,t

y t

z t then we say that “ y t is asymptotic to z t as ' 't

tends to infinity” and write ~ ,y t z t .t

Suppose we wish to describe the asymptotic behavior of the

function 3

2 324 ~ 8 , t t t t some other elementary examples are

2 2

1 1~ , 

3 2 3t

t t t

and sin  ~ ,  .

2

teht t the famous nontrivial

example is the prime number theorem.

Let t the number of primes less than t.

~ , .log

tt

tt

We describe the nature and behavior of solutions of difference

systems, without actually constructing or approximating them. Since in

contrast with differential equations, the existence and uniqueness of

solution of discrete initial value problems is already guaranteed we shall

begin with the continuous dependence on the initial conditions and

parameters. This is followed by the as asymptotic behavior of solutions of

linear as well as nonlinear difference systems. In particular, easily

verifiable sufficient conditions are obtained so that the solutions of

perturbed systems remain bounded or eventually tend to zero, provided

the solutions of the unperturbed systems have the same property.

4

For a given difference system one of the pioneer problems is the

study of ultimate behavior of its solutions. In particular, for linear

systems we shall provide sufficient conditions on the known quantities so

that all their solutions remain bounded or tend to zero as k . Thus

from the practical point of view the results we shall discuss are very

important because an explicit form of the solutions is not needed.

1.3 EXAMPLES OF DIFFERENCE EQUATIONS

Example 1.3.1

The difference equation 1 ,  0 1

nn

n k

yy n

y

where 1,  ,

  0, and 0k N is known as delay logistic equation and was

proposed by pie low as model equation of dynamic of single species.

Example 1.3.2

The difference equation 2

1 0,  0n n ny q y n

where α is

quotient of all positive integers 0, nq with nq not eventually equal

to zero is the discrete analogue of the well-known Emden fowler

differential equation, which appear in astrophysics, nuclear physics and

chemical reaction etc.

Example 1.3.3

It is observed that the decrease in the mass of a radioactive

substance over a fixed time period is proportional to the mass that was

5

present at the beginning of the time period. If the half life of radium is

1600 years. Let m t represent the mass of the radium after t years. Then

1m t m t km t , where k is a positive constant. Then

1m t 1 .k m t

Example 1.3.4

Let G(government expenditure), D(government deficit), Y(national

income) be functions of time t related in such a way that

4t t tY G D Make the following inference from this relation:

“Then, an extra dollar of deficit would allow four dollar reduction in tax-

financed government spending without affecting the magnitude of

national income…”

Example 1.3.5

In 1626, Peter Minuit purchased Manhattan Island for goods worth

$24. If the $24 could have been invested at an annual interest rate of 7%

compounded quarterly, what would it have been worth in 1998?.

Solution

Let y t be the value of the investment after t quarters of a year.

Then 0 24,y since the interest rate is 1.75% per quarter, y t satisfies

the difference equation 1 1.075 .y t y t

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1.4 EXAMPLES OF ASYMPTOTIC BEHAVIOR OF

DIFFERENCE EQUATION

Example 1.4.1

The asymptotic behavior of 1

.n

k

k

k

We begin by factoring out the

largest term:

1

nk

k

k

1 21 2 1

1  

n n

n

n n n

n nn

n n n

The sum in braces is less than

2 1

1 1 1nn n n

2

0

1 1kn

kn n

11

11

11

n

n

n

n

Since the expression

11

1

11

n

n

n

is bounded, we have

1

nk

k

k

1

1 ,  nn nn

Then the asymptotic value of 1

nk

k

k

is given by the largest term with

a relative error that approaches zero like 1

n as .n

In a similar way, it can be shown that

1

nk

k

k

1

2

1 1 11 ,    .

n

n nn n

n n n

7

Consequently, the two largest terms of the series yield an 2

1

n

asymptotic estimate.

Example 1.4.2

The asymptotic behavior of 1

1

2 log .n

k

k

k

We begin by applying

summation by parts.

1

2 logn

k

k

k

1

1

(2 )logn

k

k

k

11

1

2  log 2 .logn

n k

k

kn

By the mean value theorem, 1

log  kk

,

11

1

2 logn

k

k

k

1

1

1

12

nk

k k

2

2 1 1 1 1 1 11 . .

1 2 2 3 4 1 2

n

n

n n n

n n n

.

1.5 REVIEW OF LITERATURE

Berry (1967) believed that, “The view of related literature is a must

for scientific approach in all areas of scientific research. One cannot

develop an insight into the problem to be investigating unless and until

one has learnt one has done in a particular area of his own interest. Thus

the related literature forms the foundation upon which all work can be

built.

8

In recent years, there has been considerable interest in the study of

Oscillatory and asymptotic behavior of difference equations. An intensive

survey of the literature related to the present work have been made by the

researcher by referring to a large number of Journals, Books,

Encyclopedias, International Dissertation Abstracts and National Level

publications, etc.

Josef Diblik and Irena Hlavickova [22] discussed the asymptotic

behavior of solutions of systems of difference equations with an

application to delayed discrete equation for k of the form

u k ,F k u k 1.5.1

Where k is the independent variable assuming value from the set

: ,  1,  . N a a a with a fixed

,a N 1 2,  ,   . , mu u u u 1 ,u k u k u k

and : ,m mF N a R R 1 2,  ,  ..  .mF f f f

William F. Trench [63] discussed asymptotic behavior of

solutions of Poincare difference equations of the form

1 1 1 . 0n ny n m a p m y n m a p m y m 1.5.2

Where 0,na the polynomial 1

1 ..q a a n n

nhas

distinct zeros 1 2,  ,  .. ,n and 0,1  .kp m k n

9

M. Maria susai manuel, G.Britto Antony Xavier, D.S.Dillip and

G.Dominic Babu [29] discussed the asymptotic behavior of solutions of

the generalized nonlinear difference equation of the form

0l lp k u k f k F u k g k ,  ,k a 1.5.3

Where the function , ,p f F and g are defined in their domain of

definition and l is a positive real.

Further,

0uF u for 0,u 0p k for all , k a

for some 0, a and for all 0 ,j i , ,a j kR

Where

, 

0

1,  , 

1

k l i j

l

i j k

i

R l ap j rl

and 1lk N j l .

Laurens de HAAN, Holger ROOTZEN and Casper G. de. VRJES

[27] discussed extremal behavior of solutions to a stochastic difference

equation with applications to arch processes ,    1nY n which satisfies

the stochastic difference equation

1 ,n n n nY A Y B 1,n 0 0Y 1.5.4

Where ,  ,   1n nA B n are i.i.d, 2R valued random pairs. We

study the extremal behavior of nY under rather mild assumptions.

10

William F. Trench [62] discussed asymptotic behavior of solutions

of a linear second order difference equation of the form

2

1ny ,n np y   1, 2, n 1.5.5

Where ∆ is the forward difference operator with unit spacing

nu1n nu u equation of the form (1.5.5) arise in discretizing a second

order linear differential equation to solve it numerically.

E. Thandapani. R. Arul and P.S. Raja [52] discussed the asymptotic

behavior of non-oscillatory solutions of nonlinear neutral type difference

equations of the form

2 ,  0,n n k n ly py f n y n N 1.5.6

2 ,  ,  0,n n k n l n ly py f n y y n N 1.5.7

Using some difference inequalities. We establish conditions under

all non-oscillatory solutions are asymptotic to an b as nwith

,  .a b R

Vadivel sadhasivam, Pon sundar and Annamalai santhi [60]

discussed on the asymptotic behavior of second order quasilinear

difference equations of the form

1y n y n

1

,p n y n y n

1.5.8

Where 0n N 0 0 0,  1,  2,  . ,n n n 0 .n N

11

We classified the solutions into six types by means of their

asymptotic behavior. We establish the necessary and/or sufficient

conditions for such equations to possess a solution of each of these six

types.

Hajnalka Peics and Andrea Roznjik [14] discussed in the

asymptotic behavior of solution of a scalar delay difference equations

with continuous time of the form

x t 1t x t b t x p t 1.5.9

Where , ,a b p are given real functions such that p t t and ‘p’ is

monotone increasing, and the special case of the above equation for

1 .b t a t

E. Thandapani and S. Pandian [48] discussed about the asymptotic

and oscillatory behavior of solutions of some general second order

nonlinear difference equations of the form

1 1 10,n n n n n n n

a h y y p y q f y

n Z 1.5.10

Here ∆ is the forward difference operator defined by

ny1 ,n ny y   0,1  , 2, Z and the real sequences ,  ,n n np a q ,

  n and ,f h are the functions.

12

Eithiraju Thandapani, Zhaoshuang Liu, Ramalingam Arul and

Palanisamy S. Raja [13] discussed the oscillation and asymptotic

behavior certain second order Non-linear neutral difference equations of

the form

1 0n n n k n n la y py q f y

, 0 0n n 1.5.11

Where p is a real number, 0, 0k l are integers, α is a ratio of

odd positive integers, ∆ is the forward difference operator defined by

ny 1n ny y . na is a positive sequence with 0

1, n

n n n

qa

is a non

negative real sequence with a positive subsequence and :f R R is

continuous and non-decreasing with 0uf u for 0.u

Hulima, Hui Feng, Jiaofeng wang and Wandi Ding [17] discussed

the bounded and asymptotic behavior of positive solutions for difference

equations of

1nx 1nx

na bx e 1.5.12

Where a, b are positive constants and the initial values, 1 0,x x are

positive numbers.

J.S.Yu and Z. C. Wang [65] discussed the oscillation of neutral

delay difference equations of the form

0,n n k n n ly py q y 0,1,2,n 1.5.13

13

Whose oscillation and asymptotic behavior have been investigated,

where ,   0,1  , 2, np q n real numbers, k and l are positive

constants, the forward difference operator is defined by 1 .n n nx x x

H. Sedaghat and W. Wang [38] discussed the asymptotic behavior

of non linear delay difference equation of

nx 1

1

1 ( ) ,m

p

n i n i

i

x g f x

0, ,p   1, 2, 3,...n 1.5.14

Where g and each if are continuous real functions with g

decreasing and if increasing.

R. P. Agarwal and J. V. Manjlovic [1] discussed the asymptotic

behavior of positive solutions of fourth order nonlinear difference

equations of the form

2 2

3 0,n n n np y q y

n N 1.5.15

Where , are ratios of odd positive integers and ,np nq are

positive real sequences detailed for all 0 .n N n We establish necessary

and sufficient conditions for the existence of non-oscillatory solutions

with specific asymptotic behavior under suitable combinations of

convergence or divergence conditions for the sum

0

2n n

n

n

p

and 0

1

n n n

n

p

1.5.16

14

John R. Graef, Agnes miciano, Paul w.Spikas, P.Sundaram and

E.Thandapani [20] discussed oscillatory and asymptotic behavior of

solutions of non-linear neutral difference equation of

1 1 1 , 0m

n m n m n m k n ly p y F n y 1.5.17

Here k and l are non negative integers, m is a positive integer, p is a

constant and nq is a sequence of real numbers.

G. Papaschinopoulos, M. Radin and C. J. Schinas [32] discussed

the boundedness, the asymptotic behavior, the periodicity and the

stability of the positive solutions of the difference equation of the form

1ny

1

ny

n

e

y

1.5.18

where ,  ,  positive constants and the initial values 1 0, x x are

positive numbers.

G.Ladas and C.Qian [26] described the oscillatory behavior of

difference equations with positive and negative coefficients of

1 0,n n n n k n n ly y p y q y 0,1  , 2, n 1.5.19

Where k and l are non-negative integers and the coefficients np

and nq are sequences of nonnegative real numbers and also studied

asymptotic behavior.

15

B. G. Zhang [67] discussed the Oscillation and asymptotic

behavior of second order difference equations of the form

1 0,n n n nC y p y

0,1  , 2, n 1.5.20

where 1n n ny y y γ is a quotient of odd positive integers. Some

necessary and sufficient conditions for oscillation of (1.5.20) with

1 and 1 are obtained. Asymptotic behavior of non-oscillatory

solutions of difference equations with forced term

2

1 ,n n n ny p y f 1, 2, n is considered also.

John R. Grafe and E. Thandapani [21] discussed the

Oscillatory and Asymptotic Behavior of Solutions Of Third

order delay difference equation of the form

1 ,n n n n n m na b y q f y h 0 0,1  , 2,  .n N 1.5.21

Where ,   ,   ,n n na b q and  nh are real sequences, :f R R is

continuous, 0,na 0,nb and   0nq for all 0 0,n n N   0uf u for all

0,u and m is a positive integer. A solution of (1.5.1) is a real sequence

ny defined for all 0 1n n m and satisfying (1.5.1) for all 0n n . In

what follows, we assume that equation (1.5.1) has solutions which are

nontrivial and defined for all large n. A nontrivial solution ny of

equation (1.5.1) is said to be oscillatory if any 0N n there exists

16

n N such that 1 0.n ny y otherwise, the solution is said to be non-

oscillatory. Equation (1.5.1) is said to be oscillatory if every solution of

(1.5.1) is oscillatory, and it is said to be almost oscillatory if every

solution ny is either oscillatory or satisfies lim 0i

nn

y

for 0,1  , 2,...i

J.S. Yu and Z. C. Wang [66] discussed the asymptotic behavior and

oscillation in neutral delay difference equation of the form

1 0,n n k n ny py q y   0,1  , 2, n 1.5.22

Whose oscillation and asymptotic behavior have been investigated,

where ,  0,1  , 2, ...np q n are real numbers, k and l are non-negative

integers, and ∆ denotes the forward difference operator

1 .n n nx x x

Jerzy Popenda and Ewa Schmeidel [18] discussed on the

asymptotic behavior of solutions of linear difference equations of the

form

0

,r

i

n n n i

i

x a x

 n N 1.5.23

Here by N, R we denote the set of positive integers and reals

respectively. For any function :y N R the difference operator is

defined as follows 1 ,n n ny y y n N and 1 ,i i

n ny y for 1.i

Andrzej Drozdowicz and Jerzy Popenda [3] discussed on the

asymptotic behavior of solutions of difference equations of second order

of the form

17

2 ,n n ny p y n N 1.5.24

We would like to present some elements of qualitative theory of

difference equations. Asymptotic behavior of solutions of second order

difference equations will be investigated.

Let N denote the set of positive integers, R the set of all real

numbers and R the set of non negative reals.

For a function :a N R we introduce the difference operator

by 1 ,n n na a a 2 ,n na a where ,  .na a n n N

Moreover, let 1

0k

j

j k

a

and 1

1.k

j

j k

a

The sequence 1n n

x

is

called oscillatory if for every n N there exists , m m n such that

1 0.n nx x Otherwise the sequence is called non oscillatory.

Toshiki Natto, Pham Huu Anh Ngoc and Jong Son Shin [16]

discussed representations and asymptotic behavior of solutions to

periodic linear difference equations of the form

1 ,x n Bx n b n 0 px C 1.5.25

Where 0 : 0 ,n N N B is a complex p p matrix and

pb n C is p-periodic, that is .b n b n p

18

E.Thandapani, K. Mahalingam and John R.Graef [50] discussed

the oscillatory and asymptotic behavior of second order neutral type

difference equations of the form

2

1n n k nx ax bx ,n n m n n rq x p x n N 1.5.26

and 2

1 0,   n n k n n rx px Q x n N 1.5.27

Where 1, 2, 3, N , is the forward difference operator defined by

1 .n n nx x x

John W. Hooker and William T. Patula [19] discussed a second

order nonlinear and asymptotic behavior of the form

1 12 0, n n n n ny y y q y

1, 2, 3, n 1.5.28

Where is a quotient of odd positive integers. It is interesting to

study second order nonlinear difference equations because they are

discrete analogues of differential equations, where the forward difference

operator ∆ is defined by the equation

1n n ny y y

and 2

1 1 1 1 12n n n n n n ny y y y y y y

In (1.5.28) ,    1, 2, 3, nq q n is a given infinite sequence of real

numbers. By a solution of (1.5.28) we mean a real sequence

,   0,1  , 2, 3, ny y n satisfying (1.5.28). It is clear from (1.5.28) that a

solution of (1.5.1) is uniquely determined if any two successive values

19

1, k ky y are given. Also, it is clear that any solution can be defined for all

0,1  , 2, 3, n equation (1.5.28) is a discrete analogue of the generalized

Emden-Fowler differential equation

'' 0,  0,  0y q t y t

Kenneth S. Berenhaut and Stevo Stevic [23] discussed the behavior

of the positive solutions of the difference equation of the form

2

1

, 0,1,...

p

nn

n

xx A n

x

1.5.29

With ,    0, ,  1p A p and 2 1,  0, .x x It is shown that:

(a) All solutions converges in the unique equilibrium, 1,x A

whenever min 1,  1 / 2 :p A

(b) All solutions converge to period two solutions whenever

11:

2

Ap

and

(c) There exist unbounded solutions whenever 1.p These results

complement those for the case 1p in A.M.

E. Thandapani, S. Pandian and R. K. Balasubramanian [53]

discussed the oscillatory behavior of second order unstable type neutral

difference equation of the form

( ) 0n c n n k n na y py g f y

1.5.30

20

Where p is a real, α is a ratio of odd positive integer and n is

a sequence of integers.

P. Mohankumar and A. Ramesh [31] discussed the oscillatory

behavior of the solution of the third order nonlinear neutral difference

equation of the form

2

0, 0,n n n n ka x p x f n n n N n 1.5.31

By a solution of equation (1.5.1) we mean real sequence nx

satisfying (1.5.1) 0 0 1 0 2, , ,n n n n a solution nx is said to be

oscillatory if it is neither eventually positive nor eventually negative.

Otherwise it is called non oscillatory. The forward difference operator ∆

is defined by

1 .n n nX X X

Said R. Grace, Ravi P. Agarwal and Sandra Pinelas [37] discussed

on the oscillatory and asymptotic behavior of certain fourth order

difference equations of the form

2 2 0a k x k q k f x g k

1.5.32

With the property that

0x k

k as k are established. Where

∆ is the forward difference operator defined by x k 1x k x k

and is the ratio of positive odd integers. We assume that

21

0, :g a N n R 0, for some 0 0,1,2,...n N and

0 0 0, 1,... ,N n n n 0: :g G g N n N for some 0 :n N g k

,k g k is non-decreasing and limk

g k

. and :f R R is

continuous satisfying 0xf x for 0x and f is non-decreasing.

X. H. Tang [47] discussed the asymptotic behavior of solutions for

neutral difference equations of the form

0,    0,1,2,n n n k n n lx p x q x n 1.5.33

Where is the forward difference operator defined by

1 , ,n n nx x x k l are positive integers, np is a sequence of real

numbers and nq is a sequence of nonnegative real numbers.

Q. Din [11] investigated the qualitative behavior of following

second order fuzzy rational difference equation of the form

11

1

n nn

n n

x xx

p x x

1.5.34

Where ‘p’ is positive fuzzy number and initial conditions 1 0, x x are

positive fuzzy numbers.

A. Aghajani and M. Ejehadi [2] discussed the asymptotic behavior

of a nonautonomous difference equation of the form

1

1

,   0,1  , n nn

n

xx n

x

1.5.35

22

With nonzero initial conditions 1 0, x x where 0n n

is a sequence

of nonzero real numbers. In the case that 0n n

is a constant sequence,

it is easy to see that every solution of equation (1.5.3) is periodic with

period six, and in particular every solution are bounded.

Ravi P. Agarwal, Said R. Grace and Patricia J. Y. Wong [36]

discussed the oscillation of fourth order nonlinear difference equations of

the form

2 21

x ka k

q k f x g k p k h x k 1.5.36

Where is the ratio of two positive odd integers.

By a solution of equation (1.5.1), we mean a real sequence x k

satisfying equation (1.5.1) for all large 0 0.n N A nontrivial solution

x k of (1.5.1) is said to be non-oscillatory if it is either eventually

positive or eventually negative, and it is oscillatory otherwise. The

equation (1.5.1) is said to be oscillatory if all its solutions are oscillatory.

R. Arul and T. J. Raghupathi [6] discussed the some new criteria

are established for the second order neutral difference equation of the

form

0, 0.a n z n a n x n n 1.5.37

23

Where z n x n p n x n and established the extent the

result of the oscillation theorems for second order half-linear neutral

difference equation (1.5.37).

Srinivasam selvarangam, Eithiraju Thandapani and Sandra Peinlas

[43] discussed the oscillation of second order neutral difference equations

of the form

0n n nn n

a x x q x

1.5.38

And established the oscillation of all solutions of this equation

(1.5.38) via comparison theorems.

Vildan kutay and Huseyin Bereketoglu [61] discussed the

oscillation of a class of second order neutral difference equations with

delays of the form

,p n x n q n x n f n x n

, , ,  0g n x n x n n 1.5.39

By using Riccati transformation techniques and establish some

oscillation criteria for the second order neutral delay difference equations

(1.5.39).

R. Arul and G. Ayyappan [5] studied the oscillatory behavior of

solution of the second order neutral difference equation of the form

24

00,  n n n k n n l n n mr x p q x V x n N

1.5.40

00,  n n n k n n l n n mr x p q x V x n N

1.5.41

And established sufficient conditions for the oscillation of all

solutions of equations (1.5.40) and (1.5.41).

S. Lourdu Marian [28] discussed the existence of positive solutions

of nonlinear neutral delay difference equations with positive and negative

co-efficient of the form

1 1[r n x n p n x n q n x n

2 2q n x n e n 1.5.42

Under the following assumption that

0

1,    1, 2,  .i

s j s

q j ir s

1.5.43

For various ranges of p n and authors used Banach’s contraction

mapping principle, some sufficient condition are established for the

existence of non-oscillations solutions of the equations (1.5.42).

B. Selvaraj and S. Kaleeswari [42] discussed the sufficient

condition for the oscillation of second order neutral delay difference

equations of the form

2 0,n ny f y 0,1  , 2, n 1.5.44

25

Ethiraju Thandapani, Ramalingam Arul and Palanisamy [12]

discussed the neutral delay difference equation of the form

0,n n n k n n ly h y q y 0,1  , 2,...n 1.5.45

Where 1, a ratio of odd positive integers, is the forward

difference operator defined by 1 .n n ny y y By a solution of equation

(1.5.8) we mean a real sequence ny which satisfies equation (1.5.45) is

said to be oscillatory if it is neither eventually positive nor eventually

negative and non-oscillatory otherwise.

B. Selvaraj, G. Gomathi jawahar [41] studied the sufficient

condition for the oscillation of second order neutral delay difference

equations of the form

2

( ) ( ) 0( ) ( ) 0, ( )n n k n n ly Py q f x n N n 1.5.46

E. Thandapani and S. Selvarangam [59] discussed some new

oscillation results of the difference equations of the form

0n n n nn na x p x q x

1.5.47

And established oscillatory solution of the equation (1.5.47) via

comparison theorem.

B. Selvaraj and J. Daphy Louis Lovenia [40] studied oscillatory

properties of certain first and second order linear difference equations of

the form

26

0n n n na x q x 1.5.48

1 0n n n na x q x 1.5.49

2

1 0n n n n n na x p x q x 1.5.50

And established the some sufficient conditions for the oscillation of

all solutions of difference equations (1.5.12), (1.5.13) and (1.5.14) proved

by using the results on difference inequalities.

E. Thandapani and M. Vijaya [58] classified all solutions of second

order nonlinear neutral delay difference equations with positive and

negative coefficients of the form

1 0n n n n k n n n n ma x c x p f x q g x

1.5.51

And obtained conditions for the existence and non-existence of

solutions of the equations (1.5.51).

L. K. Kikina and I. P. Stavroulakis [24] described the oscillation

criteria for second order delay difference and functional equations of the

form

2 0,x n p n x n for all 0n 1.5.52

E. Thandapani, K. Thangavelu and Chandrasekarn [57] discussed

the oscillatory behavior of second order neutral difference equations with

positive and negative coefficients of the form

0n n n n k n n l n n ma x c x p f x q f x 1.5.53

27

0n n n n k n n l n n ma x c x p f x q f x 1.5.54

Where 0 0 0 1 0,  ,  , n N n n n n is a nonnegative integer, ,  , k l m

are positive integers ,  ,  , n n n na c p q are real sequences and

:f R R is continuous and non-decreasing with 0uf u for 0.u

J. Cheng and Y. Chu [8] discussed the sufficient and necessary

conditions and established for the oscillation theorem for second order

difference equations of the form

1 1 0,    1, 2, n n n nr x p x n

1.5.55

Where is the quotient of odd positive integers.

E. Thandapani and P. Mohankumar [55] described the oscillation

of difference systems of the neutral type of the form

0,  , n n n n n nn nx a x p g y y q f x n N n

1.5.56

Where 1,  , n na p and nq are real sequences n and

n are real non-negative sequences of integers and ,  :f g R R are

continuous with 0uf u and 0ug u for 0.u

E. Thandapani and P. Mohankumar [56] discussed the sufficient

conditions for oscillation and non-oscillation of second order nonlinear

neutral delay difference equation of the form

28

2

00, n n n k n n lu p u q f u n n 1.5.57

Where ,n np q are non-negative sequences with 0 1,np and

k and l are positive integers.

Y.G. Sun and S. H. Saker [45] studied the oscillation criteria for

the second order perturbed nonlinear difference equations of the form

1 1 1,  ,  ,  1n n n n na x F n x G n x x n

1.5.58

and established the some new oscillation criteria for the second

order perturbed non-linear difference equation (1.5.58) using Riccati

transformation.

A. Murugesan and K. Venkataramanan [53] are discussed

asymptotic behavior of first order delay difference equation with a

forcing term of the form

0,x n p n f x n r n 0n 1.5.59

Where ∆ is the forward operator defined by

1 ,x n x n x n p n is a sequence of positive real numbers,

r n is a sequence of real numbers, is a positive integer and

:f R R is an increasing function.

J. Migda and M. Migda [30] studied the second order difference

equation of the form

29

2 , n n n ku a u 1, 2, n , 0,1  ,2, k 1.5.60

And established asymptotic behavior solution of the second order

difference equation (1.5.60) oscillatory.

J. Deng [9] gave a note on oscillation of second-order nonlinear

difference equation with continuous variable of the form

0,n n n k n n ly h y q y   0,1  , 2, n 1.5.61

And established sufficient conditions for the oscillation of all

solutions of first order neutral delay difference equations. Their approach

is to reduce the oscillation of neutral delay difference equation to the non-

existence of positive solutions of delay difference inequalities (1.5.61).

E. Thandapani and K. Mahalingam [51] discussed the second order

difference equation of the form

2

1 1 0, n n k n n ly py q f y 1, 2, 3, n 1.5.62

Where nq is a non-negative real sequences, :f R R is

continuous such that 0uf u for 0,u  0 1,p k and l are positive

integers and author established the necessary and sufficient condition for

the oscillation of all solutions of the equations (1.5.62).

I. Kubiaczyk [25] studied some new oscillation criteria for second

order nonlinear difference equations of the form

30

0,  0,1  , 2, n n n np x q x n

1.5.63

Where 0 1 is a quotient of odd positive integers and

established Kamenek-type oscillation criteria for sub-linear delay

difference equations of (1.5.63) using Riccati transformation techniques.

S.H. Sakar [37] is studied the difference equations of the form

,  0,  0,1  , 2, 3, n n n n r na y p x f n x n

1.5.64

Where 0 is a quotient of odd positive integers and established

some new oscillatory for the second order nonlinear neutral delay

difference equation (1.5.64) using Riccati transformation techniques.

X. H. Tang and J. S. Yu [46] concerned with oscillations of delay

difference equations in a critical state and the equivalence of the

oscillation of the following two difference equations of the form

0n n n kx p x

and

2

1 11

2 10

1

k k

n n nkk

k ky p y

k k

1.5.65

Under the critical state of the form

1inl m fi

1

k

n kn

kp

k

and

1

1

k

n k

kp

k

1.5.66

Where 0np and k is a positive integer, and they obtain some

sharp oscillation and non-oscillation criteria for equation (1.5.66).

31

R. Arul and E. Thandapani [4] considered the difference equations

1,  0,  0,1, 2, n n np x f n x n 1.5.67

When 0

1

n sp

1.5.68

And gave some sufficient conditions for the existence of positive

solutions of equations (1.5.67).

E. Thandapani and K. Ravi [49] discussed the oscillation of second

order half linear difference equations of the form

1 0,  0,1  , 2, n n n np x q x n

1.5.69

Α is a ratio of odd positive integers.

Ravi. P. Agarwal, M. M. S. Maneul and E. Thandapani [35]

discussed about the oscillatory and non-oscillatory behavior of second

order neutral delay difference equations of the form

1 1 0, n n n n k n n lp y h y q f y 0,1  , 2,  . n Z 1.5.70

Ravi. P. Agarwal and P. J. Y. Wong [34] discussed about discrete

inequalities and used to offer sufficient conditions for the oscillation

theorems for certain second order non-linear difference equations of the

form

1 1 ,  0n n n n na y q f y r n

1.5.71

32

Where 0p

q with p, q are odd positive integers.

Sui-sun Cheng, [44] discussed boundedness and monotoncity

properties of a second order difference equations

1 1 ,  1, 2, k k k kp x q x k 1.5.72

And established the necessary and sufficient condition for all

solutions of the equations (1.5.36) bounded via comparison theorem.

J. Popenda [33] discussed about the oscillatory and non-oscillatory

behavior of the solutions of some second order difference equations of

the form

2 ,  ,  ,a n n b ny F n y y n N 1.5.73

Where a and b are real constants, considering only nontrivial

solutions,

: 0nsup y n i for every .i N

W. J. Hooker and W.T.Patula [15] studied Atkinson’s oscillation of

the difference equation of the form

2

1 0n n ny q y

1.5.74

And established known results on oscillation growth and

asymptotic behavior of solution of the equation (1.5.74).

33

Blazej Szmanda [7] discussed the oscillatory behavior of solutions

of second order nonlinear difference equation of the form

2

1

  ,  0,  0,1  , 2,   m

n in i n n

i

y a f y y n

1.5.75

Where is the forward difference operator, defined by

1 ,n n ny y y 2

1,  ...n n m mny y a a are the real sequence.

J. Diblik and I. Hlavickova [10] discussed the asymptotic behavior

of solutions of delayed difference equations of the form

, , 1 ,  ..v n f n v n v n v n k 1.5.76

Where n is the independent variable assuming values from the set

aZ with a fixed .a N

Yoshihiro Hamaya and Alexandra Rodkina [64] discussed on

global asymptotic stability of nonlinear stochastic difference equations

with delays of the form

1 1 1 0

1

,  ,  ,  .. , k

n n l n l n n n k n

l

x aF x b x g n x x x n N

1.5.77

With arbitrary initial conditions 0 1,  ,  .  ,kx x x R non-linear

continuous functions F and g, and independent zero mean random

variables .n Equation (1.5.41) describes the dynamics of a neutral

network under stochastic perturbations.

34

Ramalingam Arul and Manvel Angayarkanni [66] are discussed

Oscillatory and asymptotic behavior of solutions of second order Neutral

Delay Difference equations with maxima of the form

,

max 0,n n n n n in

a x p x q x

0n N n 1.5.78

Where ∆ is the forward operator defined by

x n 1x x x n , 0 0 0, 1,...N n n n and 0n is a non-negative

integer.

Charles V. Coffman [54] discussed asymptotic behavior of solution

of ordinary difference equations with “almost constant coefficients”

equations having the form

1y n ,Jy n f n y n 1.5.79

Where y is a d-vector, J is a constant d d matrix and ,f n y is

a vector-valued function which is continuous in y for fixed n and

becomes ‘small’ in some sense as , ,0 .n y

1.6 NEED FOR THE STUDY

Most of the difference equations governing various systems are

non linear in nature and it is well known that these equations cannot be

solved with the aid of computers, but a large number of values of the

solution can be calculated. However, if one is interested in the asymptotic

behavior of the solutions, then more computer time will be required and

this often can be very expensive. Hence in the absence of solutions

known explicitly, it is important to know qualitatively how solutions

35

behave, to obtain approximate solutions. Of particular interest is the

oscillation and asymptotic behavior of solutions of difference equations.

By an oscillatory solution of a difference equation we mean a solution

that is neither eventually positive nor eventually negative and non

oscillatory otherwise. These types of solutions occur in many physical

phenomena such as, vibrating mechanical systems and electrical circuits.

In recent years, there has been considerable interest in the study of

oscillatory and asymptotic behavior of solutions of difference equations.

Hence, there is a need to study the asymptotic behavior of some

difference equations.

1.7 OBJECTIVES OF THE THESIS

The objectives are:

1. To find the asymptotic and non oscillatory behavior of third order

non-linear neutral delay difference equation.

2. To find the asymptotic properties of third order non-linear neutral

delay difference equation.

3. To find the asymptotic and oscillation solution of first order

difference equation with delay.

4. To evaluate the asymptotic and oscillatory behavior of fourth

order nonlinear delay difference equation.

1.8 METHODOLOGY

The following methodology used for this thesis.

36

1.8.1 Double sequences

Double sequences are of the form , | 0H m n m n such that

i ,  0H m n for 0  m

ii ,  0H m n for 0m n

iii 2 ,  ) ,  ,   :L H m n h m n H m n for   0m n

1.8.2 Schwarz inequality

Let 1 2,  ,    na a a and 1 2,  ,    nb b b be two sequences of real

numbers, then

2

2 2

1 1 1

n n n

i i i i

i i i

a b a b

.

1.8.3 Riccati Transformation

Non-linear difference equations that can be transformed into

equivalent linear difference equations by a change of dependent variables.

1.8.4 Equicontinuous Functions

A set of function 𝔉 on a metric space X is called an Equi-

continuous family if for every 0 there exists a 0 such that

f x f y for all ,x y X such that ,d x y and all .f F

1.8.5 Lipchitz Functions

A function f such that f x f y C x y for all x and y

Where, C is a constant independent of x and y called a Lipchitz function.

37

1.8.6 Krasnoselskii’s Fixed Point Theroem

Let X be a Banach Space. Let Ω be a bounded closed convex subset

of X and let 1 2,s s be maps of Ω into X such that 1s 2 x s y for every

, x y .

1.9 PLAN OF THE THESIS

This thesis presents the results obtained on the oscillatory behavior

of third and fourth order delay difference equations. This thesis contains

seven chapters.

In Chapter – 1– Introduction

Section 1.1 deals with the difference equations’s introduction,

asymptotic behavior, examples, review of literature, objectives, need for

the study, methodology, the plan of the thesis and contribution of authors.

Chapter – 2 – Results and Discussion

Section 2.1 deals with the oscillatory and non oscillatory behavior

of neutral delay difference equations of the form

1 1 0n n n n n k n n la b x c x q f x 0,1,2,...n Z 2.1.1

Section 2.2 deals with main result of (2.1.1). Section 2.3 deals with

the examples. In Section 2.4 sufficient condition for asymptotic behavior

of solution of the equation 2.1.1 were established.

Chapter – 3

Section 3.1 deals the asymptotic properties of third order nonlinear

neutral delay difference equation of the form

38

2 0n n n n k n n la y p y q f y

3.1.1

Section 3.2 finds the asymptotic and oscillatory behavior of all

solutions of a third order nonlinear neutral delay difference equation

(3.1.1). Section 3.3 deals with suitable examples.

Chapter – 4

Section 4.1 concerned with the asymptotic properties of all the

solutions of a third order nonlinear neutral delay difference equation of

the form

2

1 1 0n n n n k n n lp y h y q f y 4.1.1

Section 4.2 deals with the Main results on the asymptotic and

oscillatory properties of all solutions of a third order nonlinear neutral

delay difference equation (1.5.74). Section 4.3 deals with the examples.

Chapter - 5

Section 5.1 deals with the asymptotic of the oscillation solutions of

third order nonlinear difference equations with delay of the form

2 0,  0,1,2,nn n n nr x q f x n 5.1.1

Section 5.2 deals with results and discussions of asymptotic and

oscillation solutions of third order nonlinear difference equation with

delay. Section 5.3 deals the bounded and oscillation of equation of

(5.1.1). Section 5.4 deals with oscillation behavior for third order delay

difference equation using Schwarz’s Inequality.

39

Chapter -6

Section 6.1 deals with the uncountable many positive solutions of

first order neutral delay nonlinear difference equations of the form

0n n n nnx p x q f x 6.1.1

Section 6.2 deals with the Krasnoselskii’s fixed point theorem.

Section 6.3 deals examples with Diagrammatic representation.

Chapter-7

Section 7.1 deals with the asymptotic and non-oscillatory behavior

of third order neutral delay difference equation of the form

2

1 0n n n n k n np y h y q f y 7.1.1

Section 7.2 deals with non-oscillation.

Chapter- 8

Section 8.1 deals with the oscillatory and non-oscillatory properties

of fourth order difference equation of the form

2 2 2

1 1 2 2 0n n n n n np y q y r y 8.1.1

Section 8.2 deals with the main theorem on oscillatory and non-

oscillatory properties of fourth order difference equation. Section 8.3

deals with the oscillation theorem by using monotonic property.

40

40

CHAPTER - 2

ASYMPTOTIC AND OSCILLATORY BEHAVIOR OF

THIRD ORDER NONLINEAR NEUTRAL

DELAY DIFFERENCE EQUATION

2.1. INTRODUCTION

In this chapter, we concern with the oscillatory behavior of the

solutions of third order Non-linear neutral delay difference equation of

the form

1 1 0,n n n n n k n n la b x c x q f x 0,1,2,...n Z 2.1.1

Subject to the following conditions:

1( ) ,n nH a b and 0nc 0nq for infinitely many values of .n

2( ) :H f R R is continuous and 0xf x for all 0x

3H There exists a real valued function g such that

f u f v , ,g u v u v for all 0u , 0v and

, 0 .g u v L R

4( ) max ,H m k l and 0n be a fixed non-negative integer.

5 nH R x Z there exists an integer N Z such that

0,n nx x for all n N

By a solution of equation (2.1.1), we mean a real sequence ny

satisfying (2.1.1) for 0.n n A solution ny is said to be oscillatory if it is

41

neither eventually positive nor eventually negative. Otherwise it is called

non-oscillatory. The forward difference operator ∆ is defined by

1 .n n ny y y

2.2 MAIN RESULTS

Theorem 2.2.1

With respect to the difference equation (2.1.1), assume that the

following hold

1 nC C is non-negative and non-decreasing for all .n Z

0

2 1( ) limsup ;sn

s n

C q

0 .n Z

0 0

3

1 1( ) .

s n s ns s

Ca b

Then .R

Then the equation (2.1.1) is Oscillatory.

Proof

Suppose that the equation (2.1.1) has a solution .nx R since

0,n nx x for all .n N implies that nx is non-oscillatory without loss

of generality we can assume that there exist an integer 1 0n n such that

0,nx 0,nx 0,n mx 0,n mx for all 1.n n In fact, for 0,nx

0,n mx for all large n Z the proof is similar. Set n n n n kz x c x then

42

the view of 1 , 0nC z and 0nz for all 1.n n Dividing equation

(2.1.1) by 1n lf x and summing from 1n to 1n we have

1 1 1

11 1

n n nn n n

n l n l

a b za b z

f x f x

1

11 1 1 2 1 1

2 1

,ns s s s l s l s l

s n s l s l

a b z g x x x

f x f x

1

1

1

n

s

s n

q

.

and hence

1 1 1

11 1

n n nn n n

n l n l

a b za b z

f x f x

1

1

1

n

s

s n

q

Thus, from 2 ,C we find n n na b z as .n

This implies that 1 1, 0n n na b z k k

Summing the last inequality from 2n to 1,n to obtain

2 2

2

1

1

1;

n

n n n n

s n s

b z k b za

2 1n n

Thus from 3 ,C n nb z as ,n which contradicts the

assumption that 0,nz for all large .n

2.3 EXAMPLES

Example 2.3.1

Consider the difference equation

1

1 11 2 3 2 0,n n n

nn y y n f y

n n

1n 2.3.1

43

and f x

3

2.

1 2

x

x x

Here

1 1

1,

n nn

na

1 1

1 1,

1n nnb n

1

n

nc

n

and 1

1 1

2 3 5 .n

n n

q n

Hence all the assumption of theorem 2.2.1 holds. Hence the

equation (2.3.1) has a solution 1

,ny Rn

since 0.ny

Theorem 2.3.1

With respect to the difference equation (2.1.1), assume that in

addition 1 3 ,C C the following hold:

4 1C k and 1 0,nc

5 0nC q for all 0,n n

0

1

6 1lim .n

sn

s n

C q

Then, .R

Proof

As in theorem 2.2.1. Let the equation (2.1.1) has a solution

nx R such that 0,nx 0,nx 0n mx and 0n mx for all

1 0.n n n Again , set ,n n n n kz x c x then in view of 4C and the fact

that nx R , we have 0n n n n kz x c x for all 1.n n Since equation

(2.1.1) is the same as n n na b z 1 1 1,n n lq f x n n from

44

condition 5C it follows that n n na b z is increasing, for all 1.n n

Now suppose that 0,n n na b z for all 1;n n

2 2; 0.n n na b z k k Summing the last inequalities from 3n to

1 ,n we have

n nb z3 3

3

1

2

1;

n

n n

s n s

k b za

3n n

Letting n and because of 3 ,C we see that ,n nb z a

contradiction. Thus 0.n n na b z

Now following as in theorem 2.2.1, and using the condition

5 ,C we obtain

1

lim ,n n n

nn l

a b z

f x

which is the required

contradiction.

Example 2.3.2

Consider the difference equation

2

1 2

11 1( 2 0,

3 1n n n

n ny y f y

n n

1.n 2.3.1

and .f x x

Here 1

,nan

2 1

,3

n

n nb

2nc and

2

1,

1nq

n

1

1 1

1.

2n

n n

qn n

45

For this difference equation, assumption 3C and 5C hold: but

4C and 6C are violated. Hence the equation (2.3.1) has a

solution 1

,n

ny R

n

Since 0.n ny y

Theorem 2.3.2

Let 1n na b and f be non-decreasing.

If 0

2

n

n n

n q

Then equation (2.1.1) has a non-oscillatory

solution that approaches a nonzero real number as .n

Proof

Let 0C be given and choose N so that

2

2 2n

n N

cn q

f c

Let N be the banach space of all real sequences ,nX x n N

with norm || || sup | |nn N

X x

Let : 2 ,N nX c x c n N and define : NT by

1 1

3 11 2 ,

2 2s mN

s n

cTX s n s n q f x n N

Clearly is a bounded closed and convex subset of .N

First we will show that T maps into itself. For any ,X we

have

3

2N

cTX 21

2 ,2 2

s

s n

cS q f c n N

46

Thus .T Next we let nX x and for each 1,2,...i

Let i i

nx x be a sequence in such that lim 0.i

nx x

Then a straight forward argument using the continuity of f shows

that lim 0.i

nnnTX TX

and so T is continuous, Finally, in order to

apply Schauder’s fixed point theorem, we need to show that Ty is

relatively compact. In view of recent result Ty is uniformly Cauchy. To

this end let nX x y and observe that for any k n N we have

2 | | 2n k s

s n

TX TX s q f c

from the hypothesis, it is clear that for a

given 0 there exist an integer 1N such that for all k n N we

have| | .n kTX TX .

Thus Ty is uniformly Cauchy, and so Ty is relatively compact.

Therefore, by Schauder’s a fixed point theorem, there is a fixed point

.X It is clear that nX X is a non-oscillatory solution of (2.1.1)

for n N and has the required properties.

Theorem 2.3.3

Let f u be non-decreasing and 0c be a constant such that na c for

all 0.n n

47

Suppose that 0

1 1 1| |n n n

s n

D A B

0

1 1 1| || |n n n n

s n

D A B q

Then equation (2.1.1) has a bounded non-oscillatory solution that

approaches a nonzero limit.

Proof

Let 0c and let N be so large that

0

1 1 1| |4

n n n

s n

cD A B

0

1 1 1| || |4 2

n n n n

s n

cD A B q

f c

.

Let the Banach space N and the set N be the same as in

above theorem 2.3.2 and define the operator : NT by

Tx 1 1

3,

2s s

s n

ck s n q f y n N

Where ,k s n1 1 1 1s s s n s sD D A B A B

Similar to the proof of above theorem, we show that the mapping T

satisfies the hypothesis of schauder’s fixed point X and it is clear that

nX x is a non-oscillatory solution of equation (2.1.1) for n N and

has the desired properties.

40

48

CHAPTER - 3

ASYMPTOTIC PROPERTIES OF THIRD ORDER

NONLINEAR NEUTRAL DELAY DIFFERENCE EQUATION

3.1 INTRODUCTION

In the previous chapter, we discussed the oscillatory and non-

oscillatory behavior of neutral delay difference equations. In this chapter,

we concerned with the oscillatory behavior of third order nonlinear

neutral delay difference equation of the form

2 0n n n n k n n la y p y q f y

3.1.1

Where, ∆ is the forward difference operator defined by

1 , ,n n ny y y k l are fixed nonnegative integers and , ,n n na p q are

real sequences with respect to the difference equation (3.1.1) throughout.

It is assumed that the following conditions hold.

1( ) , ,n n nH a p q and 0,nq for infinitely many value of n.

2( ) :H f R R is continuous and 0,uf u for 0.u

3H There exists a real valued function g such that f u f v

, ,g u v u v for all 0,u 0v and , 0 .g u v M R

4 max ,H m k l is a fixed non-negative integer.

5( ) { :nH R x S there exists an integer N Z such that 0,n nx x

,n N Where S is the set of all nontrivial solutions of (3.1.1).

49

By a solution of equation (3.1.1), we mean a real sequence nx

satisfying (3.1.1) for 0.n n A solution nx is said to be oscillatory if it

is neither eventually positive nor eventually negative. Otherwise it is

called non-oscillatory.

3.2 MAIN RESULTS

Theorem 3.2.1

With respect to the difference equation (3.1.1) assumes that the

following hold:

1C np is nonnegative and non decreasing for all .n Z

0

2( ) limsup ,sn

s n

C q

0n Z

0

3

1( ) ,

s n s

Ca

0n Z

Then .R

Proof

Suppose that the equation (3.1.1) has a solution .ny R Since

0,n ny y n N implies that ny is non-oscillatory. Without loss of

generality we can assume that there exists an integer 1 0n n such that

0,ny 0,ny 0,n my 0,n my for all 1.n n Set ' ,n n n n kz y p y

then

in view of 1 ,C 0,nz 0,nz for all 1.n n

50

Dividing equation (3.1.1) by n lf y and summing from 1n to

1,n we obtain

1

1

11

2 22 11 1 2 1 1

2 1

,nn s s s s sn

n n

s nn l s l s ln l

z a z g y y yza a

f y f y f yf y

1

1n

s

s n

q

This implies that

1

1

1

22nn

n n

n l n l

zza a

f y f y

1

1n

s

s n

q

3.2.1

In view of the conditions 2 3,C C and from the inequality (3.2.1),

we obtain 2

n na z as .n Therefore, there exists an integer

2 1n n and 1 0k such that

2

1,n na z k for all 2n n 3.2.2

Summing the inequality (3.2.2) from 2n to 1,n we have

2n nz z 2

1

1

1,

n

s n s

ka

for all 2.n n 3.2.3

In view of the condition 3 ,C and from the inequality (3.2.3), we

obtain nz as ,n which is a contraction to the fact that

0,nz for all large .n z Infact 0,ny 0,n my for all large ,n z the

proof is similar, and hence omitted. Hence this shows that .R

51

Corollary 3.2.1

In addition to the conditions of theorem 3.2.1,

If 4C 1k and 1 0na hold then .R

Where is the forward difference operator defined by

1 .n n ny y y

Where ,k l are fixed non-negative integers ,n np q and nh are

real sequences with respect to the difference equation (3.1.1) throughout

then it is assumed that the following condition hold.

1 :H ,n np q and 0nh for infinitely many value of .n

2 :H :f R R is continuous and 0yf y for all 0y

3 :H There exists a real valued function g such that

f u f v ,g u v u v for all 0u , 0v and

, 1g u v 0 R

4 :H max ,M k l and 0n be a fixed nonnegative integers

5 :H :nR y S there exists a integer n Z such that

0n ny y for all n N and S is the set of all nontrivial solution of

(3.1.1), By a solution of equation (3.1.1), we mean a real sequence ny

satisfying (3.1.1) for 0.n n A solution of ny is said to be oscillatory if

it is neither eventually positive nor negative otherwise it is called

asymptotic behaviour of non-oscillatory.

52

3.3 EXAMPLES

Example 3.3.1

Consider the difference equation

52 3

1 2

1 12 3 2 0,

1 2

nn n

n n

n yy y n

n n y y

3n 3.3.1

Here 1

,nan

1

,n

np

n

2 3 2nq n

and

5

2,

1 2

nn

n n

yf y

y y

3, 1, 0k l

All the conditions of the theorem 3.2.1 are satisfied, and hence all

solutions of equation (3.1.1) are not in R. One such solution of equation

(3.1.1) is 1

.ny Rn

53

53

CHAPTER 4

ASYMPTOTIC PROPERTIES THIRD ORDER NONLINEAR

NEUTRAL DELAY DIFFERENCE EQUATION

4.1 INTRODUCTION

To concerned with the oscillatory properties of all the solutions of

a third order nonlinear neutral delay difference equation of the form

2

( ) ( 1) ( 1 ) 0n n n n k n n lp y h y q f y

4.1.1

4.2 MAIN RESULTS

Theorem 4.2.1

With respect to the difference equation (4.1.1) assumes that the

following hold:

1 : nC h is non-negative and non-decreasing for all n Z

0

2 1lim p: su sn

s n

qC

0: n Z

0 0

3

1 1:

s n s ns s

Cq h

Then R

Proof

Suppose that the equation (3.1.1) has a solution Rny since

0n ny y for all n N implies that ny is a non-oscillatory without loss

of generality we can assume that exists an integer 1 0n n such that

0,ny 0,ny 0n my and 0n my for all 1.n n

54

In fact, for all 0, 0n n my y for all large ,n z the proof is similar

set n n n n kz y h y then in view of 1 0,nC z 0nz and

2 0nz for

all 1.n n Dividing the equation (4.1.1) by 1 1nf x and summing from

1n to 1,n we have

1 1

1

2 22 11

1 1 1

,nn n s s s l s l s ln n

s nn l n l s l s l

p z p z g y y yp z

f x f x f y f y

1

1

1

n

s

s n

q

and hence

1 1

22

1 1

n nn n

n l n l

p zp z

f x f x

1

1

1

n

s

s n

q

Thus from 2C we find 2

n np z as .n This implies that

2

1,n np z k 1 0k . summing this last inequality from 2n to 1n , we

obtain 2

1,n np z k 1 0k summing the last inequality from 2n to 1n .

We obtain

2

nz2 2

2

12

1

1,

n

n n

s n s

k p zp

2n n

Thus from 3C 2

nz as n

Which is contradict to the assumption that 2 0nz for all large n.

Example 4.2.1

Consider the difference equation

2

1

11 2 3 2 0n n n

nn y y n f y

n

4.2.1

55

1 1

1 ,n

n n

p n

1

1 1

2 3 2n

n n

q n

Hence all the assumption of theorem holds. Hence the equation

(4.2.1) has a solution 1 Rnny such 0.n ny y

Theorem 4.2.2

With respect to the difference equation (4.1.1) assume that in

addition to the condition 3C the following holds

4C 1k and 1 0nh

5C 0nq for all 0n n

0

1

6 1limn

sn

s n

C q

Then R

Proof

As in theorem as have a solution ny R such that 0,ny

0,ny 0n my and 0n my for all 1 0.n n n Again set

n n n n kz y h y then in view of 4C and the fact ny R we have

n n k n n kz y h y then in view of 4C and the fact that ny R we have

0n n k n n kz y h y for all 1.n n

Since equation (4.1.1) is the same as 2

n np z 1 1 1 ,n nq f x

1.n n From condition 3C it follows that 2

n np z is non-increasing for

56

all 1.n n now suppose that 2 0n np z for all 1,n n 2

2 ,n np z k

2 0,k summing the last inequality from 3n to 1.n We have

2

nz2

3

12

2 3

1,

n

n

s n s

k z n np

Let n and because of 3C we see that 2

3,nz k 3 0k

summing the last inequality from 4n to 1.n We obtain

nz4

4 4

1 1

3

1n m

n

s n k n ks

k zp

Let nand because of 3C we see that nz contradicts.

Thus 2 0n np z now following as in the theorem and using the

condition 5 ,C we obtain

2

1

lim n n

nn l

p z

f x

.

Which is the required contradiction.

57

57

CHAPTER - 5

ASYMPOTIC OF OSCILLATION SOLUTIONS OF THIRD

ORDER NONLINEAR DIFFERENCE

EQUATIONS WITH DELAY

5.1 INTRODUCTION

Sufficient conditions for the asymptotic oscillation of some third

order nonlinear difference equations of the form

2 0,nn n n nr x q f x 0,1,2,n 5.1.1

Where, ∆ denotes the forward difference operator 1 ,n n ny y y

nq is a sequence of real numbers, n is a sequence of integers such

that

1 : lim ,nn

C n

where nr is a sequence of positive numbers

1

2

0

1:

n

n

k

C Rrk

asn

3 : :C f R R is a continuous with 0 0uf u u

By a solution of equation (5.1.1) we mean a sequence ,nx which

is defined for 0

min ii

N i r

and satisfies equation (5.1.1).

A nontrivial solution nx of (5.1.1) is said to be oscillatory if for

every 0 0n there, exists 0n n such that

1 0.n nx x Otherwise it is called

non-oscillatory.

58

5.2 MAIN RESULTS

Theorem 5.2.1

Assume that

4 :C 0nq and 1

n

n

q

5 :C inli fm 0u

f u

Then every solution of equation (5.1.1) is oscillatory.

Proof

Assume that equation (5.1.1) has non-oscillatory solution nx and

we assume that nx is eventually positive. Then there is a positive integer

0n such that

0nnx for 0n n 5.2.1

From the equation (5.1.1) we have,

2

n nr x 0,nn nq f x 0n n

and so 2

n nr x is an eventually non-increasing sequence.

We first show that 2

00, .n nr x n n In fact, if there is an 1 0n n

such that

1

2 0n nr x c

and 2

n nr x c for 1n n

59

that is 2

n

n

cx

r

and hence nx1

1

1 1n

n

k n k

x cr

nx2

1 1 1

1 1 1 1m m n

s n

s m s m k n ks

x c xr

as ,n m

which contradicts the fact that 0nx for 1.n n

hence 00, .n nr x n n therefore we obtain,

0,nnx 2 0,nx 2 0n nr x for 0.n n

Let lim .nn

L x

Then 0L is finite or infinite.

Case (i)

If 0L is finite

From the continuity of function f u we have

limnn

nf x

0.f L

Thus we may choose a positive integer 3 0n n such that

nnf x

1,

2f L 3n n 5.2.2

By substituting (5.2.2) into equation (5.1.1) we obtain

2 10,

2n n nr x f L q 3n n 5.2.3

60

Summing up both sides of (5.2.3) from 3n to 3 .n n We obtain

3 3

3

1 1

10

2

n

n n n n i

i n

r x r x f L q

and so 3

1

2

n

i

i n

f L q

3 3,n nr x 3n n contradicts.

Case (ii)

If L

For this case, from the condition 1C

We have liminf 0nn

nx

and so we may choose a positive

constant ‘c’ and a positive integer 4n sufficiently large such that

nnf x c for 4n n 5.2.4

Substituting (5.2.4) into equation (5.1.1), we have

2 0,n n nr x cq 4.n n

Using the similar argument as that of case 1 we may obtain a

contradiction to the condition 1 .C This completes the proof.

5.3 BOUNDEDNESS AND OSCILLATION

Theorem 5.3.1

Assume that 6 : 0nC q and 0

n n

n

R q

, then every bounded

solution of (5.1.1) is oscillatory.

61

Proof

Proceeding as in the proof of theorem 5.2.1 with assumption that

nx is a bounded non-oscillatory solution of (5.1.1) we get the inequality

(5.2.3) and so we obtain

2 10,

2n n n n nR r x f L R q 3n n 5.3.1

It is easy to see that

2

n n nR r x 2 2

n n n n n nR r x r x R 5.3.2

From inequalities (5.3.1) and (5.3.2) we deduce

3 3

2 2 10,

2

n n n

k k k k k k

k n k nk n

R r x x f L R q

3.n n

This implies

3

1

2

n

k k

k n

f L R q

3 3 3 3

2

1 3,n n n n nx R r x x n n

Hence there exists a constant ‘c’ such that 3

,n

k k

k n

R q c

3.n n

Contrary to the assumption of the theorem.

Theorem 5.3.2

Assume that

7( ) : nC n is non-decreasing, where 0,1,2,n there is a

subsequence of ,n say kn such that 1,

knr 0,1,2,k

62

8

0

( ) : ,n

n

C q

9( ) :C f is non-decreasing and there is a nonnegative constant M

such that

0

limsupu

uM

f u 5.3.3

Then the difference 2

nx of every solution nx of equation

(5.1.1) oscillates.

Proof

If not, then equation (5.1.1) has a solution nx such that its

difference 2

nx is non-oscillatory. Assume that the sequence 2

nx is

eventually negative.

Then there is positive integer 0n such that 2 0,nx

0n n and so

nx decreasing for 0n n which implies that nx is also non-oscillatory.

Set nw

2

,

n

n n

n

r x

f x

1 0n n n 5.3.4

Then nw

1

2 2

1 1

1 n n

n n n n

n n

r x r x

f x f x

1

1

212

1 1

1

n n

n n n

n nn nn n

n n n

f x f xr xr x

f x f x f x

5.3.5

63

2

,

n

n nn

n

r xq

f x

1.n n

Summing up both sides of (5.3.5) from 1n to ,n we have

11n nw w

1

n

i

i n

q

and by (5.3.1), we get

lim nn

w

5.3.6

This implies that eventually

0nnf x 5.3.7

and therefore 0.nnx

By (5.3.6), we can choose 2 1 ,n n such that

1 ,nw M 2.n n

2 1 0,nn nr M f x

2n n 5.3.8

Set lim nn

x L

then 0.L

Now we prove that 0.L If 0L then we have

limnn

nf x

0f L

By the continuity of ,f u choosing an 3n sufficiently large, such that

nnf x

1,

2f L 3n n 5.3.9

64

And substituting (5.3.9) into (5.3.8) we have

2 11 0,

2nx M f L

rn 3n n 5.3.10

Summing up both sides of (5.3.10) from 3n to n , we get

3

3

1

1 11 0

2

n

n n

i n i

x x M f Lr

This implies that lim .nn

x

This contradicts (5.3.7).

Hence lim 0.nn

x

By the assumptions, we have

limsup .n

n

n

nn

xM

f x

From this we can choose 4n such that

n

n

n

n

x

f x

1,M

that is 41 ,n nn nx M f x n n

and so from (5.3.8) we get

2 0,

nn n nr x x 4n n 5.3.11

In particular, from (5.3.11) for a subsequence knr satisfying the

condition 5C of theorem 5.2.1, We have

65

1k k k kn n n nx x x r 1 0,

k k k k kn n n n nr x x x r

For k sufficiently large, implies that

10 0k k k kn n n nx x r x for all large .k

This is a contradiction.

The case that 2

nx is eventually positive can be treated in a

similar fashion and so the proof of theorem 5.3.2 is completed.

66

66

CHAPTER 6

UNCOUNTABLY MANY POSITIVE SOLUTIONS OF FIRST

ORDER NONLINEAR NEUTRAL DELAY

DIFFERENCE EQUATIONS

6.1 INTRODUCTION

We consider the difference equation of the form

0n n n nnx p x q f x 6.1.1

where, 0,n n 0 are integers.

also 0 , , 0, ,a C t , 0,p R

and ,f C R R

where f is non decreasing function for 0,f x 0.x

we are concern with the first order nonlinear neutral delay

difference equation (6.1.1).

1 0( ) , , 0, ,nH r C n 1

s n rs

2 0( ) , , 0, 0nH p C n p

3( ) , , 0,H C 0 / 0)x

4( ) , , , ,H f x C 0 / 0)f x x

5( )H G x

0 0 :f x

x

G x is non decreasing in ,0

6 0( ) , 0, ,H G n C n g n n

67

A non trivial solution nx is said to be oscillatory if it has

arbitrarily large zeros otherwise nx is said to be non-oscillatory. The

proof is an adaptation of that given (6.1.1) where the special case

g n n was considered.

6.2 KRASNOSELSKII’S FIXED POINT THEOREM

Let X be a Banach space. Let Ώ be a bounded close convex subset

of X and let 1 2,s s be maps of Ώ into X such that 1 2s x s y for every

, .x y

If 1s is contractive and 2s is completely continuous. Then the

equation 1 2 .s x s x x

Theorem 6.2.1

Suppose that there exist bounded from below and from above by

the function 0, , , 0,n nu v c n constant 2 10, 0c k k and

1 0n n m such that

0,n nu v n n 6.2.1

1 1 0,n n n nv v u u 0 1n n n 6.2.2

1

1( n s s n

s n

u k p f v au n

2

1( ) 1,n s sv k p f u c

v n

1n n 6.2.3

68

Then equation (6.1.1) has uncountable many positive solution

which are bounded by the function , .u v

Proof

Let 0 , , )c k R be the set of all continuous bounded functions with

the norm 0

|| || sup .n n nx x Then is Banach space.

We define a close bounded and convex subset of 0 , ,c n R as

0 0Ώ , , ) : ,n n n nx x c n R u x v n n

For 1 2,k k k we define two maps 1 2 0& : Ώ ,s s c n as follows

1 ns x 1

1

1 0 1

n n

n

k a x n n

s x n t n

6.2.4

2 ns x

1 1

1

2 0 1

s s

s n

n n n

p f x n n

s x v v n t n

We will show that for any , Ώx y we have 1 2 Ώs x s y for

every , Ώx y and 1t t with regard to (6.2.3) we obtain

1 2n ns x s y n n s s

s n

k a x p f y

n n s s

s n

k a v p f y

2n nk v k v

69

For 0 1,n n n we have

1 2n ns x s y 1 1 11 2n n n ns x s y v v

1 1n n nv v v nv

Further more for 1n n we get

1 2n ns x s y n n s s

s n

k a u p f v

nk u t k u

Let 0 1,n n n with regards to (6.2.2) we get

1 1,n n n nv v u u 0 1n t n

Then 0 1,n n n and any , Ώx y we obtain

1 2n ns x s y 1 1 11 2n n t ts x s y u u

1 1n n n nu u u u

Then we have prove that 1 2 Ώs x s y for any , Ώx y .

We will show that 1s is a contraction mapping on Ώ for , Ώx y

and 1.n n We have

1 2n ns x s y | || | || ||n na x c x y

This implies 1 2s x s y || ||c x y

Also for 0 1,n n n the above inequalities are valid.

We conclude that 1s is a contraction mapping on Ώ.

70

We now show that 2s is completely continuous.

First we show that 2s is continuous. Let Ώii

nx x be such that

in nx x as .n Because x is close Ώnx x for 1n n we have

2 2

i

n ns x s x i

s s s

s n

p fx f x

1

i

s s

s n

fx f x

Since 0i

s sf x f x as i by applying the lebeague

dominant convergence theorem we obtain 2 2lim 0.i

is x s x

This means

2s is continuous.

We now show that 2s is relatively compact in Ώ, it is sufficient to

show by Arzela ascolic theorem that the family of functions 2 : Ώs x x

is uniformly continuous. Then

2

s s

s n

p f x

Then Ώ,x 2 1N N n

where 2 2 2 1s x N s x N 2 1

s s s s

s N s N

p f x p f x

2 2

2 2 2 1s x N s x N 1

s s

s N

p f x

2 1max ,s sp f x N N 1 2N N n

71

then there exist 1s

M

where max s sM p f x such that

1 2N N n

2 2 2 1s x N s x n if 2 1 10 N N s .

Next we show that equation (6.1.1) has uncountable many positive

solutions Ώ. Let 1 2,k k k

be such that .k k

we assume that , Ώx y

1 2 1 2,s x s x x s y s y y

,n n s sn

s n

x k a x p f x

1n n

,n n s sn

s n

y k a y p f y

1n n .

It follow that there exist a 2 1n n satisfying

2

s s s

s n

p f x f y

|| k k

In order to prove that the set of bounded positive solution of

equation (6.1.1) is constant. It is sufficient to verify that x y for 2n n ,

we get

(1 ) || ||x x y | | [ ]s s s

s n

k k p f x f y

Corollary 6.2.1

Suppose that there exist bounded from below and from above by

function 0, , 0,u v C n that 0,C 2 1 0,k k 1 0n n m such

that (6.2.1), (6.2.3) holds.

72

Proof

0,n nv u 1 2n n n

H t 1 1n n n nv v u u

H t 0n nv u

nH t 0 .

6.3 EXAMPLE WITH DIAGRAMMATIC REPRESENTATION

1 1

10n n nx x x

n

Input

1

1 2 1 1 0x n x n x n x nn

Graph of the difference equation is:

73

Values:

n 0 1 2 3 4

x n 0 1 2 2.5 2.33333

74

74

CHAPTER - 7

ASYMPTOTIC OF NON OSCILLATORY BEHAVIOUR

OF THIRD ORDER NEUTRAL DELAY

DIFFERENCE EQUATIONS

7.1 INTRODUCTION

We are concerned with the oscillatory properties of all the

solutions of a third order nonlinear neutral delay difference equation of

the form

2

1 0n n n n k n np y h y q f y 7.1.1

Where ∆ is the forward difference operator defined by

1 ,n n ny y y where ,k l are fixed nonnegative integers and np , nq

and nh are real sequences with respect to the difference equation

(7.1.1).

Throughout we shall assume that the following conditions hold:

1 : n nH p q and 0nh and 0nq for infinitely many value of n

2 :H f R R is continuous and 0yf y for all 0y

3H There exists a real valued function g such that f u f v

,g u v u v for all 0u and 0v and ,g u v 0L R

4H M max ,k l and 0n be a fixed non-negative integer

5H R :ny S there exists an integer N S such that 0n ny y

for all n N and ‘S’ is the set of all nontrivial solution of (7.1.1).

75

By a solution of equation (7.1.1), real sequence ny satisfying

(7.1.1) for 0.n n

A solution nh is said to be oscillatory if it is neither eventually

positive nor negative, otherwise it is called non-oscillatory.

7.2 ASYMPTOTIC AND NON-OSCILLATION THEOREMS

Theorem 7.2.1

With respect to the difference equation (7.1.1) assumes that the

following hold:

1 nC h is non-negative and non-decreasing for all n Z

0

1

2 1limsup ;n

sn

s n

C q

0n Z

0 0

3

1 1( )

s n s ns s

Cq h

then R

and equation (7.1.1) is asymptotic behaviour of non-oscillatory.

Proof

Suppose that the equation (7.1.1) has a solution ny R . since

0n ny y for all n N implies that ny is a non-oscillatory. Without

loss of generality we can assume that there exists an integer 1 0n n such

that

0,ny 0.ny

0n my and 0n my for all 1n n

76

In fact for all 0,ny 0n my for all large ,n z the proof is

similar.

Set n n n n kz y h y then in view of 1 , 0nC z , 0nz and

2 0nz for all 1.n n

Dividing the equation (7.1.1) by n lf x and summing from 1n to

1n , we have

1 1

1

22

1

n nn n

n n

p zp z

f x l f x l

1

211 1,

1

ns s s s l s l

s n s l

p z g y y y

f y f s l

1

1

1

n

s

s n

q

and hence

1 1

1

22n nn n

n l n l

p zp z

f x f x

1

1

1

n

s

s n

q

Thus from 2C we find 2

n np z as n . This implies that

2

1 1, 0n np z k k .

Summing this last inequality from 2n to 1n , we obtain

2

1 1, 0n np z k k , summing the last inequality from 2N to 1n , we

obtain

2

nz2 2

2

12

1 1

1n

n n

s n s

k p z np

Thus from 2

3 nC z as .n Which is a contradict to the

assumption that 2 0nz for all large n.

77

Theorem 7.2.2

With respect to the difference equation (7.1.1) assumes that in

addition to the condition 3C the following holds:

4( ) 1C k and 1 0nh

5( ) 0nC q for all 0n n

0

1

6 1( ) limn

sn

s n

C q

Then R

Then equation (7.1.1) is asymptotic behaviour of non-oscillatory.

Proof

As in theorem (7.2.1)we have a solution ny R such that

0,ny 0ny and 0n my and 0n my for all 1 0n n n .

Again set n n n n kz y h y then in view of 4C and the fact

that ny R .

We have n n k n n kz y h y then in view of 4C and the fact

that ny R .

We have 0n n k n n kz y h y for all 1.n n Since equation (7.1.1)

is the same as

2

1,n n n n lp z q f x n n

78

From condition 5C it follows that 2

n np z is non increasing for

all 1n n .

Now suppose that 2 0n np z for all 1n n ,

2

2 2, 0n np z k k

Summing the last inequality from 3n to 1n , We have

2

nz2

3

12

2 3 1

1,

n

n

s n s

k z n np

Let n and because of 3C we see that

2

nz that 2

3 3, 0nz k k summing the last inequality

from 4n to 1n , we obtain

nZ4

4 4

1 1

3 4

1,

n m

n

s n k n s

k z n npk

We see that nz , which a contradiction. Let n and

because of 3C we have 2 0n np z . Now following as in theorem 7.2.1

and using the condition 5C we obtain

2

lim n n

nn l

p z

f x

Which is the required contradiction.

79

79

CHAPTER - 8

OSCILLATORY AND NON OSCILLATORY PROPERTIES

OF FOURTH ORDER DIFFERENCE EQUATION

8.1 INTRODUCTION

Consider the fourth order difference equation of the form

2 2 2

1 1 2 2 0n n n n n np y q y r y 8.1.1

where ,n np q and nr are real sequences satisfying 0,np 0nq

and 0nr for each 0n and the forward difference operator ∆ is defined

by 1n n ny y y also .ny y n

Definition 8.1.1

Let ny be a function defined on ,N we say k N is a generalized

zero for ny if one of following holds:

i 0ny

ii 1k N and 1 0,n ny y 1 ,k N and there exists an integer

,m such that 1 .m k

iii 1 0m

k m ny y and 0jy for all 1, 1 .j N k m k

A generalized zero for ny is said to be of order 0, 1, or 1m

according to whether condition (i), (ii) or (iii), respectively holds. In

particular, a generalized zero of order 0 will simply be called a zero, and

a generalized zero of order one will again be called a node.

80

Obviously, if y a 1y a 2y a 3y a 0 , for some

a N then 0ny is the only solution of (8.1.1). Thus, a nontrivial

solution of (8.1.1) can have zeros at not more than three consecutive

values of k. In definition 8.1.1, we shall show that a nontrivial solution of

(8.1.1) cannot have a generalized zero of order 3.m However, a

solution of (8.1.1) can have arbitrarily many consecutive nodes, as it is

clear from 1n

ny which is a solution of (8.1.1).

The following properties of the solutions of (8.1.1) are fundamental

and will be used subsequently.

1S If ny is a nontrivial solution of (8.1.1) and if

a 0ny b 0ny c 2

1 0ny d 3

2 0ny

for some 2k a N then 1 1 1, ,a b c and 1d holds for all

,k N a with strict inequality in a for all 2 ,k N a strict

inequality in 1b for all 1k N a and strict inequality in 1c and

1d for all 3k N a . Furthermore,

2 2 2

1 1 2 2 0n n n n n np y q y r y for all k N a 8.1.2

With strict inequality for all 2 ,k N a and ,ny ny , 2

ny all

tends to as .k

In 2S , If ny is a nontrivial solution of (8.1.1) and if

81

1a 0ny 1b 0ny 1c2 0ny 1d

3 0ny

for some k a N , then 1 1 1, ,a b c and 1d holds for all

k N a with strict inequality in 1 1 1, ,a b d for all 3k N a and

in 1c for all 4 .k N a

Furthermore, 4 0ny for all k N a 8.1.3

With strict inequality for all 2k aa N and ,ny ny and

2

ny all tends to as .k

3S If ny is a nontrivial solution of (8.1.1) and if

2 0na y 2 1 0nb y 2

2 1 0nc y 3

2 1 0nd y

for some 3k a N then (8.1.2) holds for all 2,k N a and

2 2 2

1 1 2 2 0n n n n n np y q y r y for all 2,k N a 8.1.4

Furthermore, 0 1 0y y and 0 0y . Strict inequality

holds in 2a and (8.1.3) for all 2, 2k N a , if 4a N in 2b for

all 2, 1k N a and in 2C for all 2, 3k N a , if 5 .a N

4S Let 2a N

If ny is a solution of (8.1.1) with 0,y a 1 0,y a

1 0,y a 1y a and 1y a not both zero, then at least one of the

following conditions must be true.

82

(i) Either 0ny for all 2k N a or

(ii) 0ny for all 0, 1 .k N a In particular, ny cannot have

generalized zeros of any order at both and where

0, 1N a and 2 .N a An analogous statement

holds for the hypothesis 1 0y a and 1 0.y a

8.2 Main Theorem on Oscillatory and Non-Oscillatory Properties of

Fourth Order Difference Equation

Theorem 8.2.1

If ny is a nontrivial solution of (8.1.1) with zeros at three

consecutive values of k, say ,a 1a and 2a then ny has no other

generalized zeros. If 3 0 0y a then 0 0ny for all k, and the

inequality is strict if 2k N a or 0, 1k N a In particular, if

0, 1N a and 3 ,N a then 0.y y

Proof

Clearly 2 0.y a y a Since the solution ny is nontrivial, we

may assume that 3 0.y a Thus, 3 0y a and by 2S ny is

positive and strictly increasing on 3 .N a Next, Let n nv y , then

1 0,v a 0,v a 2 0v a and 3 0.v a If 2a N , then

3S implies that nv is positive and strictly decreasing on ,0 .N a

83

Thus ny is negative and strictly increasing on ,0 .N a If 1,a

then we again assume that 3 4 0.y a y Then by (8.1.1)

4 0 2 ,y r 2 0.y

But 4 0 4 0 ,y y y

So 0 4 0y y and 0 1 0 0y y y as claimed.

If 0,a then the part of the conclusion concerning 1k a is

empty. This completes the proof.

Theorem 8.2.2

Let 1 ,a N suppose that ny is a solution of (8.1.1) with

0 0,y 1 0,y a 2 0,y a but 2a is a generalized zero for

.ny Then ny has no other generalized zeros.

If 2 0 0 ,y a then 0 0ny for all ,k N with strict

inequality for all 2k N a or 0, 1 .k N a In particular, if

0, 1N a and 2 ,N a then 0.y y

Proof

Since 2 0,y a we can assume that 2 0.y a

Since y a 1 0,y a 2a cannot be a generalized zero of

order 1 or 2, and theorem (8.2.1) implies that the order cannot be greater

than 3.

84

Thus, 2a is a generalized zero of order 3, which implies that

1 0.y a now since from (8.1.1), we have 2 0,n np y it follows

that 3 0,y a clearly 2 0,y a 0y a and 0,y a thus by

2 ,Sny is positive and strictly increasing on 3 .N a

For 0, ,k N a Let .n nv y

Then 0,v a 1 0v a , 2 1 0v a and

3 1 0v a . If 3 ,a N

then as in equation (8.1.1), 3S yields the results.

If 2,a then 2 3 0,y y 1 0y , 4 0y and 1 0y

By (8.1.1) we have 4 0 0.y

But, 4 0y 4 4 3 6 2 4 1 0y y y y y

4 4 1 0 ,y y y

and so 4 1 0y y 4 0.y

Hence, 0 4,y 1 0y and 0 1 3 1 0.y y y

Therefore, 0 0y and 0 0y as claimed. If 1,a then

1 2 0,y y 3 0y and 2 3a is a generalized zero. It follows

from the definition of a generalized zero that this must be a generalized

zero of order 3, so that if 3 0y then 0 0.y

Hence 0 0.y Which completes the proof.

85

Corollary 8.2.1

If ny is a nontrivial solution of (8.1.1) with generalized zeros at

and and zero at ,a where 1 1,a then 1 1 0.y a y a

In particular, ny does not have a generalized zero at 1.a

Proof

Since 1 1a from theorem (8.2.1) it follows that 1y a

and 1y a both cannot be zero. If 1 1 0,y a y a then 4S

implies that ny cannot have generalized zeros at both and which is a

contradiction. Thus, 1 1 0.y a y a

Corollary 8.2.2

If ny is a nontrivial solution of (8.1.1) with

0,y y a y where 1a then 1 0.y a

Corollary 8.2.3

If a nontrivial solution ny of theorem (8.2.1) has a zero at and a

generalized zero at where , then cannot have consecutive

zeros at , 1a a where 1.a

Theorem 8.2.3

If two nontrivial solutions ny and nv of (8.1.1) have three zeros in

common, then ny and nv are linearly dependent, i.e., specifying any three

zeros uniquely determines a nontrivial solution up to a multiplicative

constant.

86

Proof

If y y a 1y a v v a 1 0v a for some

and ,a where 0 ,a then by theorem 8.2.1, 2 0u a and

2 0.v a

Define w n 2v a y n 2 .y a v n Since w n is a linear

combination of y n and v n , it is a solution of (8.1.1). However,

w w a 1w a 2 0,w a and so w n must be the trivial

solution of (8.1.1) by theorem (8.2.1). Since 2u a and 2v a are

nonzero, u n and v n must be constant multiples of each other.

Next, if y y a y v v a 0,v where

1a then by corollary 8.2.3, 1 0y a and 1 0.v a

Define w n 1 1 .v a y n y a v n

Clearly, w w a 1w a 0,w which contradicts

corollary 8.2.2 unless 0w n . But this means y n and v n are

constant multiples of each other. This completes the proof.

Definition 8.2.1

A solution y n of (8.1.1) is called recessive if there exists an

a N such that for all .k N a

0,y n 0,y n 2 0y n and 3 0y n 8.2.1

87

Let my n be the solution of (8.1.1) satisfying my m

1my m 2 0my m and 0 1my .where 1 .m N

For each , mm y n exists and is unique. The existence is clear from

theorem 8.2.1 and normalization. While the uniqueness follows from

theorem 8.2.3.

Note that by construction

0 1my n for all 0, 2k N m 8.2.2

Also, Theorem (8.2.1) implies that

1m my n y n for all .k N 8.2.3

We now consider m sequence 1 .my

By (8.2.1), 0 1 1my for all 1m N

Thus lim sup 1m

my

exists, we call it 1 .y Then, there exists a

subsequence 1 1lm N such that

2 22 1m m

m k m k m k m k m k m ky k p y y k q y r y 8.2.4

Consider (8.2.4) with 2k and m replaced by 3 .lm we can

conclude that 3lim 5 5 .lm

ly y

Proceeding inductively, we conclude

that 3lim lm

ly k y k

exists for any .k N

88

Replacing m by 3lm in (8.2.4) and letting ,l we conclude that

y k is a solution of (8.1.1).

Also, 1 0y k y k 8.2.5

This follows from (8.2.3) by replacing m by 3 ,lm fixing ,k and

letting .l From (8.2.5) we conclude that

limk

y k

exists, and we shall call it L 8.2.6

We will now show that this y k is a recessive solution of (8.1.1).

8.3 OSCILLATION THEOREM BY USING MONOTONIC

PROPERTY

Theorem 8.3.1

The solution y k constructed above is a recessive solution of

(8.1.1). In addition 2,y k y k and 3 y k all monotonically approach

zero as .k

Proof

We will first show that (8.2.1) is satisfied. By (8.2.3) and theorem

8.2.1, 3

3 3 0.lm

ly m

Choosing 3 3lm and using 3S with 3 1,la m we can conclude

that for any k such that 32 1,lk m 3 1 0,lm

y k 32 1 0lmy k

and 33 1 0.lmy k

89

Letting l implies that y k satisfies (8.2.1) for 1a and is

recessive. We note that y k also satisfies (8.2.1) for 0.a Concerning

the monotonicity, we choose any 2k N and any 3 .lm k

Then, 32 1 0lmy k which means 3lm

y k 3 1 ,lmy k and

hence 3 30 1 .l lm my k y k Taking the limit as l implies

that y k is monotonically decreasing in absolute value. By (8.1.1),

Since y k monotonically approaches a finite limit, 0y k as

.k The argument that 2 y k and 3 y k monotonically approach

zero is similar.

By theorem 8.3.1 this recessive solution y k of (8.1.1) can be

written as

2 2 2

1 1n n n np y q y 1

  1 2 36 l k

l l k l k l k r l y l

8.3.1

Corollary 8.3.1

If 3

1

,l r l

then the recessive solution of (8.1.1)

constructed above approaches zero as .k

Corollary 8.3.2

Suppose that y k and v k are two recessive solutions of (8.1.1)

such that .y a v a If y k v k for all k N a then .y k v k

90

Proof

Let limk

l y k

and limk

h v k

.

By hypothesis, l h .

Thus, if w k y k v k , then

From (8.3.1) with 2 k a , we have

2

10 1 1 0

6 l a

l m l a l a l a r l w l

From this we conclude that .y k v k

Conclusion

The oscillatory property of fourth order difference equation

becomes Oscillate.

91

91

CHAPTER - 9 CONCLUSIONS

1. The asymptotic Behaviour of the solution of the third order nonlinear

Neutral Delay difference equation using Riccati transformations was

found to be asymptotic.

2. Asymptotic Behaviour of third order nonlinear Neutral Delay

difference equation using Riccati transformations were found to obtain

several oscillation criteria.

3. The Asymptotic property of third order Nonlinear Delay Difference

Equations were found to become Oscillate using Schwarz’s Inequality.

4. The Asymptotic property of Third order Nonlinear Neutral Delay

difference Equation was found to become Oscillate using Schwarz’s

Inequality.

5. Uncountably many positive solutions of first order Nonlinear Neutral

Delay Difference Equations were found using Krasnoselskii’s fixed

point theorem.

6. Asymptotic of Non- oscillatory Behaviour of fourth order Neutral

Delay Difference Equations were found using Double sequence

Techniques.

92

92

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LIST OF PUBLICATIONS

1) Dr. B. Selvaraj, Dr. P. Mohankumar and V. Ananthan, Asymptotic and

Oscillatory Behavior of third order nonlinear Neutral Delay Difference

Equations, International Journal of Nonlinear Science,

Volume.13(2012) No.4.pp.472-474.

2) Dr. B. Selvaraj, Dr. P. Mohankumar and V. Ananthan, Asymptotic

properties of third order nonlinear Neutral Delay difference equation,

Journal.Comp.& Math.Sci.Vol.4(5), 356-359(2013).

3) Dr. P. Mohankumar, V. Ananthan and Dr. A. Ramesh, Asymptotic

properties of third order nonlinear Neutral Delay difference Equation,

International Journal of Mathematical Archive-5(8), 2014, 188-190.

4) Dr. P. Mohan Kumar, V. Ananthan and Dr. A. Ramesh, Asymptotic

of Oscillation solutions of third order nonlinear difference equations

with delay, International Journal of Mathematics And Computer

Research, Volume 2 Issue 8 August 2014, Page No 581-586,

ISSN2320-7167.

5) V. Ananthan and Dr. S. Kandasamy, Uncountably Many Positive

solutions of First Order Non-Linear Neutral Delay difference

105

equations, International Journal of Mathematical Archive-7(8), 2016,

144-147.

6) V. Ananthan and Dr. S. Kandasamy, Asymptotic of Non-oscillatory

Behavior of Third Neutral Delay Difference Equations, Journal of

Chemical and Pharmaceutical Sciences, ISSN: 0974-2115(2016).

7) V. Ananthan and Dr. S. Kandasamy, Oscillatory and Non-oscillatory

properties of fourth order difference equation, Int.J.Chem.Sci:14(4),

2016, 3005-3012. ISSN.0972-768X.