Pergamon

Nonlinear Analysis, Theory, Mefhods & Appkolions, Vol. 22, No. 12, pp. 1511-1527, 1994 Copyright 0 1994 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0362-546X/94 $7.00+ .OO

BOUNDEDNESS IN FUNCTIONAL DIFFERENTIAL EQUATIONS

Bo ZHANC

Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville, NC 28301, U.S.A.

(Received 22 December 1992; received in revised form 9 March 1993; received for publication 16 June 1993)

Key words and phrases: Boundedness, Liapunov functionals, functional differential equations.

1. INTRODUCTION

WE CONSIDER a system of functional differential equations with finite or infinite delay

x’(t) = F(f, x,), x E R” (1)

and obtain some Liapunov-type theorems on the uniform boundedness (UB) and uniform ultimate boundedness (UUB) of solutions of (1). This work generalizes some very important results in the literature. For reference and history, see Burton [ 1, p. 25 11, Burton and Zhang [2], Hale [3, p. 1391, Hering [4], and Yoshizawa [5, p. 2021. During the last 40 years investigators have shown that if solutions of (1) are UB and UUB, and if F(t, 4) is o-periodic in t and solutions of (1) are continuous on initial conditions, then (1) has an w-periodic solution (see [6,7]). Here are some effective results concerning the boundedness of solutions of (1).

Let f: [0, +oo) x R” -+ R” be continuous and consider the system of ordinary differential equations

(0) x’(t) = f(t, x).

THEOREM A. Suppose that there exists a continuous function V: R+ x R” + R+, positive constants U and M such that

(i) W,(M) 5 W, x) 5 K(M), (ii) k&,0, X) 5 -K(lxl) + M, W,(u) > M,

(iii) W,(r)-++coasr-++c0. Then solutions of (0) are UB and UUB.

This theorem was discovered early, is easy to prove, and there are many important examples. Investigators have constructed many functionals V for (1) having analogs to (i)-(iii) although serious problems were encountered. An early result in extending theorem A to equation (1) with finite delay may be found in the book of Yoshizawa [5, p. 2021.

THEOREM B [5]. Suppose that there exists a continuous functional I/: R+ x C --t R+, and a con- stant U > 0 such that

w,(MO)I) 5 V(f, 4) 5 w,dwal) + w,(ll4ll), V(i,(C @J) 5 0 for l4(O)l 2 u,

W,(r) - W,(r) -+ +a, as r -+ +oo.

Then solutions of (1) are UB.

1511

1512 B. ZHANG

Burton and Zhang [2] considered the boundedness of solutions of (1) both for finite and infinite delay. Let R- = (-co, 01, R+ = [0, +oo) and R = (-0, +oo). Define

G = (g: R- + [l, +a): g is continuous, nonincreasing and g(0) = 1, g(u) -+ +03 as u --t -00)

and

C, = (4: R- + R”: q5 is continuous with 141, < +oo]

where 141, = sup]$(s)l/g(s). Then (C,,l* le) is a Banach space. SSO

THEOREM C [2]. Let V: R+ x C, -+ R+ be continuous and locally Lipschitz in 4 E C, . Suppose that there exists a: R+ + R+ with @ E L’(R+ ) and Q’(t) 5 0 such that

5

0

(Cl) @(t - s)W4(vg(.s)) ds =: H(t, v) --oo

is continuous and Q(u)W,(vg(-u)) E L’(R+) for each v > 0,

0

(Cd ~dlWl) 5 V(t, 4,) 5 w(l44o)l) + w, [I

w-wxl~(~)I) ds 3 -m 1 G) V,;,o, 4) 5 -Kddm) + M, (G) ~IW, w,w + +m as r + +co.

Then solutions of (1) are g-UB and g-UUB. A recent work by Hering [4] generalizes theorem B to (1) with infinite delay. Define an order in

G by g < go if and only if g, go E G, g(s) I g’(s) for s I 0 and lim [sup g(s)/g’(s - N)] = 0. N++m ss0

The following definition is also needed

63 = t (Y: [-h, +oo) + R+: 01 is continuous and there exist positive constants j3 and L

3’

t+L

such that p 5 (Y(S) d.s I 1 for all t E R+ . t 1

THEOREM D [4]. Suppose that there exists a continuous functional V: R+ x C, + R+, functions 17 E 03, go E G with g 5 go and positive constants K, M, U and y such that

~hfJ(m 5 vet, 4) 5 Kd#4O)l) + w3(141g4, V(i)@9 4) 5 -rlwK4(l~(O)l) + M,

W,(r) > y + ML + W,(U) for r >_ K, W,(r) + +a as r --, +w.

pW,(r) - FM L 0 with g % go, solutions of (1) are g-UB, and whenever

Wl(r) -

Then, whenever PW, - LM > 0 with g < g”, solutions of (1) are g-UUB.

Notice that the differential inequalities used in theorems C and D are quite different. In this paper, we generalize theorems B-D when a(t) = 1 using a unified approach and provide an important step in extending theorem A to (1). We follow Burton and Zhang [2] dividing the main results into two parts. The first part is for finite delay. Results in this section not only

Functional differential equations 1513

develop essential ideas for studying systems with infinite delays but also provide much weaker conditions for boundedness in finite delay systems. The second part deals with systems with infinite delays. The result in this section is a unification of theorems C and D.

2. EQUATIONS WITH BOUNDED DELAY

For x E R”, 1x1 denotes the Euclidean norm of x. For a given h > 0, C will be the space of continuous functions $: [-h, 0] + R” with the supremum norm ]]&]] = sup[&s)l: -h I s 5 0). If x is a continuous function of u defined on -h 5 u < A, A > 0, and if t is a fixed number satisfying 0 I t < A, then x, denotes the restriction of x to the interval [t - h, t] so that x, is an element of C defined by x,(19) = x(t + 0) for B E [-h, 01. We consider (1) with finite delay

x’(t) = F@,x(s), t - h 5 s 5 t), XER” (2)

where x’(t) is the right-hand derivative of x at t. It is assumed that F: R+ x C + R” is continuous so that a solution will exist for each (to, 4) E R+ x C. We denote by x(t,, 4) a solution of (1) with initial function 4 E C where x,,(t,, 4) = 4. The value of x(tO, 4) at t will be x(t) = x(t, t,, 4). We suppose that F takes a bounded set of R+ x C into a bounded set of R” so that the bounded solutions of (1) are continuable to t = + co.

Let V be a continuous functional defined on R+ x C. The upper right-hand derivative of I/ along solutions of (1) is defined by

v($, 4) = :$n+ SUP{ V(t + 6, X,+&r 4)) - vt, 4)1/d.

Definition 1. Solutions of (2) are uniformly bounded (UB) if for each B, > 0 there exists B, > 0 such that [t,, E R+, 11411 I B,, t L to] imply (x(i, to, 4)1 5 B,.

Definition 2. Solutions of (2) are uniformly ultimately bounded (UUB) for bound B if for each B, > 0 there exists T > 0 such that [t,, E R+, (1411 I B,, t 2 to + T] imply Ix(t, to, 4)1 I B.

Definition 3. W: Rf -+ R + is called a wedge if W is continuous and strictly increasing with W(0) = 0. Throughout this paper W, F (j = 1,2, . . .) will denote the wedges.

Letting W be a wedge and (Y E 63, we define

O IW4)Ia,t, = s

dt + ~)Wl4cel) ds for all 4 E C. -h

In particular,

i

0

I W4)l, = _h Wl4@)l) d.s forall+ECifa(t)= 1.

THEOREM 1. Suppose that there exists a continuous functional V: R+ x C + R+, a function CY E 03, and positive constants K, M, U with p W(U) - ML > 0, where /3 and L are given in the definition of &I such that

(I) wI(l4Kol) 5 WY 4) 5 w2(l4(o)l + lW4)l,,,,) + W3(ll4lh (11) Q;,@, 4) 5 -aWWldWl) + M,

(III) W,(r) - W,(r) > W,(U + ML + Mh) + ML for r 2 K, W,(r) -+ +m as r --f +a. Then solutions of (1) are UB and UUB.

1.514 B.ZHANG

Proof. Let B, > 0, r#~ E C with 11411 i B, and x(t) = x(t, to, 4). Define V(t) = V(t, xt) for t 2 I,. Then

V(GJ 5 W,(& + W(y&)) + W,(&) for some y > 0.

Without loss of generality, we assume that B, > K + U + M(h + L) and W(yB,) 2 U + ML + Mh, where L is defined in the definition of CR. We shall show that

V(f) 5 W,(B, + W(yB,)) + W,(B,) + M(L + h) =: V,

for all t L to. We prove (3) by induction. Define

(3)

z, = [to + n(L + h), t, + (n + l)(L + h)] =-: [t,, T,] forn =0,1,2 ,....

It is clear that V(t) I V, on I,. We assume that V(t) 5 V, on Z,, and show that (3) holds on Z n+l. In fact, on Z,+i we have either:

(A) V(t) I maxi V(s): s E ZJ for all t E I,+ 1 ; or (B) V(r,+,) > V(s) for some r,+i E Z,,, and all s E [f,, T,+J.

If (A) holds, then V(t) I V, on Z,+i by the assumption. Now suppose that (B) holds. We first claim that there exists s0 E [rn+i - L, T,+J

s E [G+~ - L G+J, then

n?I+,) 5 Q%+1 - L) -

5 V(T,+1 - L) -

with Ix(s,)] 5 U. In fact, if Ix(s)] > U for all

i

‘It+1

W(U) a(s) ds + ML 7,+1-L

VW(U) - ML1 < WG+I),

a contradiction. Thus, such s,, exists. Moreover,

.i

Tflt1 V(r,+,) 5 Qs, - h) - a(s) W(lx(s)l) ds + M(L + h).

so-h

Since V(r,+,) > V(s, - h), we have

5 So

T”n+l I WXSJ Ia = 4s)Wlx(4) ds 5

.I a(s) W((x(s)l) ds I M(L + h).

so-h so-h

Thus,

J’(r) s V(r,+r) 5 J’(s,) + ML 5 w,(~x(~,)~ + ~+‘(x,~)~,~,~,) + w,(k,~~) + ML

for all t E [t,, r,+r]. Let Ix(s*)I = (IxJ. Then

W,(lx(s*)l) I W,(U + ML + MA) + W,(/x(s*)j) + ML.

This yields that ]]xsOl] 5 K by condition (III). Hence,

V(t,+,) c: W,(U + ML + Mh) + W,(K) + ML 5 W,(B, + W(yB,)) + W,(B,) + ML.

Consequently, we have V(t) s V, on Z,,,, . By induction, we conclude that V(t) I V, for all t 1 t, and Ix(t)\ I W;‘(V,) for t 2 t,. Thus, solutions of (1) are UB.

We now show that solutions of (1) are UUB. Let B, > 0 and define Bz > 0 of uniform boundedness. Let 4 E C and ll~(] 5 B,, then (x(t, to, $)I I B, for t L t,,. We assume that B, > K, then there exists a constant YZ with 0 < q < p W(U) - ML such that

W,(r) - W,(r) > W,(U + ML + Mh + 17) + q + ML (4)

Functional differential equations 1515

for r E [K, II,]. We consider I, again. On each Z,, we have either (C) Ix(t)1 5 K for all t E Z, or

(D) k(y,)l = maxlk(s)I : s E Ll > K- If (C) holds, then by the uniform boundedness there exists B > 0 such that Ix(t)1 5 B for t 2 T, where B = B(K) is a function of K. Next we assume that (D) holds and define

V, = V/(6,) = maxi V(s): s E I,),

V It+1 = V(d,+,) = max(V(s): s E I,,,).

We claim that V a+1 5 V, - q with q given in (4). By the method of contradiction, suppose that

v,,, > K - rl* (5)

We first show that there exists a s0 E [a,,, - L, 6,+J with Ix(.q,)I 5 U. In fact, if Ix(s)I > U

on k%+r - L 4+J, then

c

6 “+I W,,,) 5 w,,, - L) - W(U) a(s) d3 + ML 5 V@,+, - L) - @W(U) - ML).

&+1-L

If 4+, - L E Z,,+l, then

V(&+,) 5 V(&+1 - L) - @W(U) + ML) < V(&+1)9

a contradiction. If a,+, - L E I,, then

V(&+,) 5 V(k+, - L) - (p W(U) - ML) = T/(6,+, - L) - ty - (/3 W(U) - ML - Yf)

I V(6,) - q - @W(U) - ML - q) < V(6,) - Y/

which contradicts (5). Thus, such s, exists. Moreover,

s

&I+ 1 V(&+,) 5 m, - h) - cY(.s)w(~x(s)~)ds + M(L + h).

so-h

This yields that

So I wso)I”(so) = s

49W(Ix(d) d.s 5 so-h 5

6”,1 4~)wlx(~)l) d.s

so-h

I V(s, - h) - V(S,+,) + M(L + h).

If s0 - h E I,, then

I Wx,,) I 01&J 5 K - rl - Kc+1 + M(L + h) + Y/ I M(L + h) + ?f.

If S, - h E la+l, then

I Wxs,) I cx(s,) I V(s, - h) - V(&+,) + M(L + h) < M(L + h) + ?f.

Thus, for t E I,,+1 we have

W,(l-e)l) 5 V(t) 5 W,,,) 5 V(%) + fkfL

5 W*[U + M(L + h) + q] + W~(llX,,ll) + ML. (6)

1516 B. ZHANG

Also,

Wl(lXWl) 5 uo 5 W,) 5 V&+1) + r

5 w,[U + M(L + h) + VI + w,dlxs,ll) + i%fL + rl

for all t E I,, . Therefore,

(7)

K(llx,,II) 5 w,[U + M(L + h) + VI + W(llx,,ll) + hfL + rl- This implies by (4) that ]]xsO(] I K. From (7) we also have

w,(k~,)]) 5 w,[u + ML + h) + sl + W,(K) + ML + 17

5 w,[U + M(L + h) + VI + w,e4Y,)l) + IbfL + rl which is impossible by (4) and (D). Thus, V n+l I V, - q if (D) holds. Let N be the first integer such that (N - 1)~ I V, < NV. Suppose that (D) holds on Ij for j = 1,2, . . . , N. Then for tEINwehave V(t)I~~_,-rlII/,_,-2~7... I V, - Nq < 0. This implies that there exists an integer k, 0 I k < N, such that (C) holds on Zk and Ix(t)/ I B for all t 2 Tk 2 TN = t,, + (N + l)(L + h) =: t, + T. Thus, solutions of (1) are UUB and the proof of theorem 1 is complete.

COROLLARY 1. Suppose that there exists a continuous functional I/: R+ x C + R+, positive constants M and U such that

(CA w(Im) 5 V(t, 4) 5 w,hm)I + I W4)Il) + w,(ll4ll), G) V(;,(t, 4) 5 -WkiW)l) + M W(u) - M > 0, (C,) W,(r) - W,(r) + +oo as r + +oo.

Then solutions of (1) are UB and UUB.

Example 1. The following equation has been studied extensively in the literature [l, p. 254; 4, p. 5331

i

* x’(t) = -a(t)x(t) + b(t) cc% x(s)) ds + At, xt)

t-h

where a, b: R’ + R, C: R’ x R + R are continuous scalar functions, and f: Rf x continuous. Suppose that the following conditions hold:

(Hi) ]C(t,.# 5 Q4)lx12 f or a function a E 63 and a positive constant Q, (H2) 2a(t) 2 /b(t)1 + NQa(t) + 1 for some constant N > 1, (H,) 1 - Ib(t)l(l + h)h2 L 0 for all t E R+,

0-h) If(t, 4)t 5 pbll + d401 1’2 for all (t, 4) E R+ x C and some p, q E 2p(l + h) I 1. Then solutions of (8) are UB and UUB.

Proof. For any continuous function r+~ R + R, we adopt the following notation

IIW(‘)IIIS’fl = sup(]y/(r)]: s I 5 I t]

and define a functional V: R+ x C --+ R+ by

O + r)]$(r)]2drds + llf#?(*)]]t”~Ol d&Y. -h

(8)

C-+Ris

R+ with

(9)

Functional differential equations 1517

Let x(t) = x(t, t,,, u/) be a solution of (8). Then for t L to we have

v(t, x,) = Ix(t)12 + ; 5 s

y *

* Qa(z)Ix(s)12 dt ds + 11x2( .)(I[‘,*] ds. h t+s t-h

Differentiate V(t, x,) for t L t, to obtain

V,$,xt) = 2-w -40x(t) + b(t) 5 * ‘3~s ~(9) ds + At, xt) + QaWlx(t)12 t-h >

; -- i’ 1, Q4.d4912 d.v + lx(O12 - llxt II2 + l;_h $ llx2(~)ll[s~*1 CLS

5 -24tMt)12 + IwNl.w12 + IW)lh 3

Iah .w)12 ds t-h

+ 2~WkfWt)l + QWkW12 - ; 1

; Q4dx(d12 h t h

+ lxw12 - Ilxtl12 + !; I, $ llX2(ws~*1 ds

5 (-Wt) + lW)l + Q4t) + l)h(t)i2 + 2bWl b-0, xt)l - ht iI2

- ;(I - lb(t)lh2) s * Q4@1~(41~ d.s + t-h s ;_, g llX2(~)p* ds.

For each fixed S, if 11x2( .)I([‘,*] = Ix(@I’ with s 5 8 < t and Ix(r)1 < Ix(e)1 for all 8 < T I t, then (d/dt)/ix2( .)I) k*] = 0 (see Hale [3, p. 1271). Now suppose that ((x2(.)((Is,*l = lx(t)12. Since

$x2(t) -i -2a(t)lx(t)12 + Jb(t)jJX(t))2 + Jb@)lh i

* law@~)12~ + 4xmf(t,xt)l, t-h

we have

Thus,

$ ~lx2(~)Ip’*l I Ib(t)lh s * Iw, wN2 d.s + 2lx@>llf<t, xt)l. t-h

Q(t, xt) 5 (-2&) + /WI + Q4t) + l)ix(t)12 - IIx,~~~ + 2(1 + h)Ix(t)IIf(t,x,)I

- ; [I - Ib(Ql(l + h)P] j

* Q4s)lx(d2 d.s t-h

5 (-2dt) + lb(t)l + Qa(t) + 1)lx(t)12 - 1jx,jj2

+ 2(1 + ~)lwl(PllxtII + ~[4011’2)

5 -Q(N - l)a(t)lx(t)12 + 2q(l + h)[a(t)]“21x(t)l

I -+Q(IV - l)a(t)jx(t)j2 + A4 for some A4 > 0.

1518

It follows from (9) that

B. ZHANC

’ (1 + Md(OV 5 V(t, 4) 5 MO)12 + Q4t + &dd2d~ + htbbi2. (11) J -h

Let W(r) = iQ(N - l)r2. Then for 14(0)12 I I w(+)I,(,, , we have

O lMw + i _h Q@ + ~)~~W12 dr = 14(0)i2 + & 1 W4d,,,, 5 1 +

hw9 + I WdA,(,,)

and

lvw)12 + & I W44,(,,

Define W,(r) + ((N + l)/(N - 1))r if 0 I r I 1 and W,(r) = ((IV + l)/(N - l))r2 if r 2 1. Then conditions (I)-(III) are satisfied with W,(r) = (1 + h)r2 and W,(r) = hr’. Thus, solutions of (8) are UB and UUB.

Remark 1. Neither theorem C nor theorem D is applicable to differential inequalities (10) and (11).

3. EQUATIONS WITH UNBOUNDED DELAY

In this section consider a system of functional differential equations with infinite delay

x’(t) = F(l, x,), x E R”, (12)

wherex,(s)=x(t+s)for-w<ssOandF:RxC, + R” is continuous for some g E G. We have the following assumptions on solutions of (12). The reader is referred to [6, 8-111 for extensive discussions of the fundamental theorems.

For each (to, 4) E R x C,, there exists an o > 0 and a continuous function x: (-03, to + a) -, R” such that x(t) satisfies equation (12) on [to, to + a) with xto = 4. x is called a solution of (12) and it is denoted by x(t,, 6). The value of x(t,, 4) at t is denoted by x(t) = x(t, to, 4).

For each (to, q5) E R x C,, x(t,, C#J) is defined on [to, +m) unless there exists to < p < +co such that lim s;pIx, (t, to, $)I = +a.

(13)

(14)

Definition 4. A seminorm II - lie on C, is said to have a fading memory with respect to I - lg if ~~~~~B I 141, for all 4 E C, and if for each E > 0 and D > 0 there exists an h > 0 such that

II& 5 max(ll~(~)ll’-“*O1~ &I whenever CJ L h and J$_,& I D,

where 14-,lg = supIr#$s - @I/g(s) = sup I4(u)l/g(u + a). SSO UC-0

Functional differential equations 1519

Example 2. Let 01: R- + R+ be continuous and g E G such that

s

0 cY(.s)g(.s) ds 5 1 and

-m s

0 a(s) dS I 3.

-co (15)

Define )I * lIB on C, by I1411e = j”_- c~(s)&s)I d.s for any 9 E C,. Then II- llB has a fading memory with respect to 1. lg.

Proof. For any $ E C,, we have

O 0

ll@4la = s

44M‘Ql dS.7 5 191, s

49g@) d.s 5 I&. -m -cc

Let E > 0 and D > 0 be given. Then there exists h > 0 such that 20 jIk o(s)g(s) ds < E. If r~ -1 h and I&_& I D, then

ll~LJ= O s

~(~MwI d&-T = -cc 5

0 4GI~(~)I d&s + -0 i

-0 49 Id@ I ds -cc

5

0 -0 I ~~f#J~~l-“~Ol 4s) ds + SUP b(u) I/& + 4 4&?(~ + 4 dJ- -r us-0 -cc

I

0 -0 5 ~~f#$“‘Ol a(s) dS + D

s dWg(s) ~

--m -cc

5 i IIf#II[-O,ol + i = max(ll+ll[-“*O1, E].

Thus, (( * (lB has a fading memory with respect to ( - lg.

Example 3. Let go E G and g < go. That is, g(s) I go(s) for all s E R- and

lim supg(s)/g”(s - N) = 0. N+ +m

Then 1. Jgo has a fading memory with respect to I - lg.

Proof. For any E > 0 and D > 0, there exists h > 0 such that supg(s)/g”(s - h) I E/D. SSO

If o 2 h and I@-,& I D, then

I+ Ip = maxt_oy~s o I +(G I /g°C% SUP l9(s) I /go@)) ssi--(r 5 max(l14(*)ll[-“‘01, SUPMU - f41/g0(u - Q.))

US0

5 max(llf$(~)ll[-“~O1, kkTlg SUP &wgO(u - @I US0

5 max(ll$(~)ll[-“~O1, E).

Also, I&o 5 k#& f or all q5 E C,. Therefore, I * lgo has a fading memory with respect to I * lg.

1520 B. ZHANG

Example 4. Let CY: R- -+ R+ be continuous such that

s 0 0

cY(s)g(s) ds I 1 and c a(s) d.s = +. --m --m

Define 1) * IIB on C, by

Il4Jlle= O .I

4~)II~(‘wo1 ds for any 4 E C, -m

(see [12]). Then 11. IIB h as a fading memory with respect to I * lg.

Proof. For E > 0 and D > 0, there exists h > 0 such that 20 jr: &)g(s) ds < E. If o 2 h and I+_,& I D, then

O 1143 = .i

a(s)ll~lpol ds + -CT a(s)l~~ll[s’o’ d.s -v 1’

5 I(#-“*ol ~yoCY@) b +-i -O cY(s)[ll$ll[-“90’ + llc##“‘-“l] d.s

5 ; llc#p’“’ + D i -u a(s)g(s) ds I f II#-090’ + ; -co

5 max(llq511[-“‘01, E].

For any 4 E C,, it also follows that

Il~lls= O \’ i

0

44114c)Il[“~01 d.Y 5 Id, 4M4 ds 5 144,. --m -cc Thus, I] - lie has a fading memory with respect to I * lg.

Next define

Q. = (01: 01 is a continuous real-valued function on --oo < s I t < +oo, a(t, s)/& 1 0, ]o(t, t)l I J for some constant J > 0 and all t E R, there exists a continuous function (Y*: R- + R+ and a wedge W such that a(t, t + s) I a*(s) for all s E R- and Jym a*(s) W(yg(s)) ds < +oo for any y 1 0).

We define H(y) = jfm a*(~) W(yg(s)) ds for any y E Rf.

THEOREM 2. Suppose that there exists a continuous functional V: R x C, + R+, a seminorm

II * IIB on C, having a fading memory with respect to I - lg, a function CY E Q. and positive

1521 Functional differential equations

constants K, M, and U such that

(I*) W,(l9(O)I) 5 V(t, 4) 5 w, I44O)l + ( s

O a(t, t + ~)W(I4(GI)~ + w3m4B)~ --oo > (II*) V,;& $1 5 -ww)I) + M, W(U) > M,

(III*) W,(r) - w,(r) > w,(u + MQ) for all r L K,

where Q = j!!- (Y*(S) ds and (Y* is defined in a. Then solutions of (12) are g-UB and g-UUB.

Proof. Let B, > 0, r#~ E C, with 141, I B, and x(t) = x(t, to, 4). Define V(t) = V(t, xi) for t L to. Since 11. IJB h as a fading memory with respect to 1. Ig, it follows that ll+lla 5 141, I B, . Thus,

V(t,) I W,(B, + H(B,)) + W,(B,) =: v,.

There exists U*, 0 < U* < U, such that W(U*) > M. Let /I = U - U*. Without loss of generality, we may assume that U < K. For E = U* and B, > 0, there exists h > 0 such that

llwlle 5 m~~llvII[-“~ol, U*I (16) if II,v_,~~ I B, with (T 1 h and

1’

-h -h

a@, t + s) W(B,g(s)) d.9 I

-03 5

Q*(U) W(B,g(u)) du I p. --m

On [to + h, +co) we have either (A*) V(u) I sup V(s) for all 24 2 to + h, or

t,ssato+h

(B*) there exists t > to + h such that

V(t) > V(s) for all to 5 s < t.

If (A*) holds, then V(u) I V. + Mh for all u L to + h. Now suppose (B*) holds. Multiply (II*) by a(t, s) and add a,(t, s) V(s) on both sides of (II*) to obtain

V’(s)a(t, s) + cY#, s)V(.s) I a,@, s)V(s) - a!Q, s)W(Ix(.s)I) + cYl(t, s)M.

This implies that

V(t)46 t) - V(to)4t, to)

i

t

.i

t I

s

t

4, s) w d.9 - 4t, ~)WM~)I) d.9 + kf 4t, d d&s to f 0 fo

5 V(t)(@, t) - 4, to)) - 4t, ~)wIx(4I) d.7 +

By the definition of a, we have

A4 s f a(t, s) ds. (17) to

= Q.

1522 B. ZHANG

It then follows from (17) that

I’

* 0 0

4t, ~)WI-wl) ds 5 -4t, to)(w) - V(fo)) + M a*(s) ds 5 A4 a*(s) ds = MQ.

to to-t -*

Next consider

i

to 4t, GWIx(4) ds =

-02 .r

0

4t, to + s)Wlx(~o + 41) ds

--oo

ET

.i

0

a(t, to + ~)Wkml) ds --m

i

to-t

= cr(t, t + u)W(B,g(t - to + u)) du -cc

r

-h

I a*(u) W(B,g(u)) du I /3. J --m

Hence,

i

t act, s)Wk)I) CI.Y s MQ + P.

-a3

Since V(s) < V(t) for all to I s < t, it follows that Ix(t)1 I U*. Thus,

W,(]x(O]) 5 I’(t, xt) 5 w,(U* + MQ + 8) + W(lixt tld. (18)

We claim that sup Ix(s)1 I K. Suppose there exists s* E [to, t] such that t05S5T

Ix@*)l = t w,Ix’“‘I > K. 0

Then by (16) with o = t - to, we have

llxtIIB 5 maMU*, Ix(s*)ll = Ix(s*)l. Using (18) we obtain

Wr(lx(s*)]) i V(s*) I I’(t) I W,(U + MQ) + W3(lx(s*)l)

which contradicts (III*). Thus, ]x(s*)l 5 K and

W,( Ix(s)I) 5 V(s) I V(t) 5 W,(U + MQ) + W,(K)

for all s E [to, t]. Define

B, = max(W-‘(I’, + hM), W-‘[W,(U + MQ) + W,(K)]).

Then, Ix(t)1 5 B, for all t 2 to. This proves the g-uniform boundedness of solutions of (12). Next we show that solutions of (12) are g-uniformly ultimately bounded. Let B, > 0 and

define B, > 0 of g-uniform boundedness. Let 4 E C, and 141g 5 B, ; then ]x(t, to, $J)] I B,

Functional differential equations 1523

for t L to. We assume that B2 > K, then there exists a constant q with q(1 + J) I U - U* such that

W,(r) - W,(r) > W,(U* + MQ + Y/J + q) + q

for r E [K, B,], where J is given in the definition of a. For any t L 1, , we have

(19)

Ixtlg 5 sup Ix(t + s)l + sup Ix(t + s)I/g(s)

-(t-t,)ssrO s 5 -(f--tO)

= sup Ix(s)1 + supIx(t, + u)l/g(u + t, - t)

tgssst US0

I sup Ix(.s)I + ldlg I B2 + B, =: B,. fgssst

For E = U* and D = B,, there exists h > 0 such that

Ilx,llE! I max{llxllt’-h”l, U*j I B, (20)

for any t L to + h and

s

-h

~*CGW(BsgW ds < II. -03

We consider Z, = [to + (n - l)h, to + nh], n = 1,2, . . . . On each Z,, we have either (C*) Ix(t)1 I K for all t E Z,, or (D*) Ix(v,)l = max(Ix(s)I : s E I,,) > K.

Let t, = to + nh. Suppose that (C*) holds. Then

and i

f ” 4t,, s)W(Ix(s)l) b 5 W(K)

t,-h t,-h

t,-h

s

-h

at, ~)wImI) ds = 4, t, + ~)Wlx(t, + d> du -co --m -- n

5 ~*WWB&)l cb 5 17. --m

Also, Jlx& I max(U*, t _.2yst Ix(d) 5 K. Thus, n n

V(t,) I W,(K + QW(K) + rj) + W,(K) =: V&.

Repeating the argument for g-uniform boundedness with to replaced by t, and V. replaced by V&, we conclude that there exists a B = B(K) > 0 such that Ix(t)1 I B for all t 2 t,. Now we assume (D*) holds and define

V, = V/(6,) = max (V(s): s E I,),

V n+l = V(S,+,) = max(V(s): s E Z,+r).

We claim that V n+l I V, - q with q given in (19). By the method of contradiction, suppose that

K,, > K - rl* (21)

1524 B. ZHANG

By (II*) we also have

5 W&-I) - rl - W,+,M&+, ,6-J + rl4z+, , &+A + MQ

5 [WJ - rl - W,+,k(~ n+~, G-d + rlJ + MQ 5 I;IJ + MQ.

By (20), for CJ = a,+, - t,_, we have

i

*n-1 &-%+1, ~)~(ImI) ds = --m 5

-0 4%+~,4t+, + u)Wlx@,+~ + 41) du --m

Therefore,

I

5

-h

~*(4 W&g(u)) du < rl. --m

i

6”,1 @,+us)W(Ix(GI)~ 5 ~(1 + J) + MQ. (23)

--m

By the definition of V(S,+,) and (II*) we have 1x(6,+,)1 I U*. Now we define Ix(s,)I =

SUP{ Ixw I : A+ 1 - h I s I d,,,). Then

I%%+,) 5 W,[U* + ~(1 + J) + MQI + W(maxtu*, Ix(~I). (24)

If s,, E 1,+1, then

WI(l~(~dl) 5 W,+J 5 W,[U* + ~(1 + J) + MQI + K(max(U*, Ix(~I).

If s0 E I,, , then

w,(lx(%)l) 5 W,) 5 w%+,) + rl

I W,[U* + ~(1 + J) + MQ] + q + W,(max(U*, Ix(s,)l)).

This implies by (19) that Ix(s,)I I K. From (25) we also have

W,(k’,)l) 5 V(Y,) 5 U&+,) 5 w,[u* + ~(1 + J) + MQI + rl + w,(K)

(25)

Functional differential equations 1525

which is impossible by (D*) and (19). Thus, V,,, _ .C V, - 7 if (D*) holds. Let N be the first integer such that (N - 1)~ I V, < NV. Suppose that (D*) holds on Ii forj = 1,2, . . . , N. Then for any t e I,we have V(t) 5 V,_, - q I V,_, - 2~ 5 ... I V, - NV < 0. This implies that there exists an integer k, 0 5 k I N, such that (C*) holds on Ik and Ix(t)\ s B for all t 2 t, 1 tN = t,, + Nh. Thus, solutions of (12) are g-UUB and the proof of theorem 2 is complete.

Example 5. Consider the following scalar integral-differential equation

s

t x’(t) = -a(t)x(t) + C(t, s, x(s)) d.s + At, xt)

--m

where a: R + R’, and C: R3 + R are continuous such that

I% s, 4 5 QW - hi

(26)

for a positive constant Q and a continuous function E: R+ + Rf with E E L’(0, +a). Define @: R+ -+ R+ by a’(s) = {imE dr. We assume that @ E L’(0, +a). Thus, by [13] there exists a function g E G such that

5

0

E(-@g(s) ds < +oo and -co s

0

a’( -s)g(s) ds < + 00. -cc

For the fixed g, we will consider the initial functions space C, and assume f: R x C, + R is well-defined. For any ly E C,, we also define

1 O ME = E<-~)ll~/(~)Il~“~~‘~ and IIv/IlB = --oJ i

O aq-s)llv/(‘)Il[s,o’ ds. --co

Without loss of generality, we assume that

i

0

5

0

a,(-s)g(.s)ds I 1 and @(-s) d.Y = J, O<Js+. --m -02

Thus, I( * lie has a fading memory with respect to I - Ig by example 4. We also define D: R x C, + R by D(t, C#I) = S”__ C(t, t + s, 4(s)) ds. Let C, denote the space C, with norm I * JE. Suppose the following conditions hold.

(Jr) a(t) - (Q + QJ + I)@(O) - S 2 0 for some 6 > 0, (J2) I f(t, +)I I pl$lE + q for any (t, 4) E R x C,, where p, q are positive constants with

(1 + J)p I 1, (J3) functions D and f are continuous on R x C,.

Then solutions of (26) are g-UB and g-UUB.

Proof. Let F(T, V) = -a(t)w(O) + D(t, I,@ + f(t, w). Then F is continuous on R x C,. It also follows that

O IVIE = s

~‘(-41wII[s’01 d.9 5 Yl wig --m

1526 B. ZHANG

where y = jfm @(-@g(s) &. It was noted in [12] that C, satisfies all axioms (Bi)-(B4) outlined in [ 111. Thus, for each (to, 4) E R x C, , there exists a continuous function x: (-43, to + a) + R”, (T > 0, such that x satisfies equation (26) on [f,, , t, + a) with xtO = 4. Since F: R x C, + R takes bounded set into bounded set, it follows that a solution is continuable in the future if bounded. For any (t, 4) E R x C,, define

V(t, 4) = k49l + (1 + J)Q i

0

O @(-~)ldG)l ds + -03 i

a+s)J~~(~)~p” ds. -cc

Let x(t) = x(t, to, w) be a solution of (26). Then for t z to we have

t f U&x,) = h(O + (1 + J)Q

i’ qt - s)Ix(.s)( ds + qt - s)~~x(*)p” d&v.

--m I’ -co

Let V(t) = V(t, xi). Differentiate V(t) for t 1 to to obtain

v’(t) 5 -a(t>Ix(t)I + 1 t Iw , s, W)t cb + If(t, xt)l + (1 + J)QW)LWI -cc

t f - (1 + J)Q E(t - MG ds + w)lx(~)I - qt - s)llx(~)Ip’l ds

-co -co

+ .i

f --m

qt - s) $ Ilx(#“Jl ds

s t 5 --IQ(~) - (Q + QJ + lkWO)l Ix(t>l - QJ at - d44 d.s + I.Iv, &)I -m t

t - s

E(t - s)ljx(*)Il[“Jl d&s + (27) --m .I --m

qt - s)$ IIX(*)pfl ds.

For each fixed s, if [1x( *)I( [‘,‘I = Ix(e)] withs I 0 < t and Ix(r)/ < Ix(e)/ for all 0 < r I t, then (d/dt)]lx(*)IltS3” = 0. If ]Ix(*)IIts~” = Ix(t)l, then

$ I(x(.)II[~,” I Q .i ’ at - M.9 ds + I_m, xt)l. (28) --m

Substituting (28) into (27) we obtain

V’(t) I -[a(t) - (Q + QJ + l)@(O)]Ix(t)l - QJ E(t - s)k)l d.s + I.f(tv xt)i --m

.r

t

s

f - E(t - s)IIx(.)II’~~~’ d.s + QJ E(t - s)Ix(d * + Jlf(t, xt)i

--co -m

5 -dx(Oi + (1 + J)b-@,x,)1 - PIE

5 -dWl + (1 + J)(~lxti, + 4) - Ix,IE 5 -+(Ol + (1 + J)q.

Functional differential equations

Since I+(O)lJ 5 S’!m Q(--~)ll+(~))l~“~~~ ds = I14jle, it follows that

(1 + J)b#4O)l 5 V(t, 4) 5 b(O)\ + (1 + J)Q i

O ~~-hiw d.s + IMB -02 >

152-l

= Ic#(O)l + (l +dJ)Q ( s O ~(-Nb#J(~)I d.s + IIdle. -co ) Let W,(r) = (1 + J)r, W,(r) = r, W,(r) = r, W(r) = 6r, and M = (1 + J)q. All conditions of theorem 2 are satisfied. Hence, solutions of (26) are g-UB and g-UUB.

1.

2.

3. 4.

5. 6.

I.

8.

9.

10.

II.

12.

13.

REFERENCES

BURTON T. A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, Orlando (1985). BURTON T. A. & ZHANG S., Unified boundedness, periodicity, and stability in ordinary and functional differential equations, Annali Mat. pura appl. 145, 129-158 (1986). HALE J. K., Theory of Functional Differential Equations. Springer, New York (1977). HERING R. H., Boundedness and periodic solutions in infinite delay systems, J. math. Analysis Applic. 163, 521-535 (1992). YOSHIZAWA T., Stability Theory by Liapunov’s Second Method, Math. Sot. Japan, Tokyo (1966). ARINO 0. A., BURTON T. A. & HADDOCK J. R., Periodic solutions to functional differential equations, Proc. R. Sot. Edinb. lOlA, 253-271 (1985). BURTON T. A. & ZHANG B., Uniform ultimate boundedness and periodicity in functional differential equations, Tohoku Math. J. 42, 93-100 (1990). HADDOCK J. R., Friendly spaces for functional differential equations with infinite delay, in Trends in the Theory and Practice of Non-linear Analysis (Edited by V. LAKSHMIKANTHAM), pp. 173-182. Elsevier, North-Holland (1985). HERING R. H., Boundedness and stability in functional differential equations, Dissertation, Southern Illinois University, Carbondale, IL (1988). HINO Y., MURAKAMI S. & NA~TO T., Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Vol. 1473. Springer, New York (1991). SAWANO K., Exponential asymptotic stability for functional differential equations with infinite delay, Tohoku Math. J. 31, 471-486 (1978). HUANG Q. & WANG K., Space C, , boundedness and periodic solutions of FDE with infinite delay, Scientia Sin. Series A XXX(S), 807-818 (1987). BURTON T. A. & GRIMMER R. C., Oscillation, continuation and uniqueness of solutions of retarded differential equations, Trans. Am. math. Sot. 179, 193-209 (1973).