DELAY DIFFERENTIAL SYSTEMS WITH

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1 Jurnal fa lied" 7, NurnberPPsprirlgl, 1994, Mathematics and Stchastic Analysis DELAY DIFFERENTIAL SYSTEMS WITH DISCONTINUOUS INITIAL DATA AND EXISTENCE AND UNIQUENESS THEOREMS FOR SYSTEMS WITH IMPULSE AND DELAY S.V. KRISHNA Andhra University Department f Mathematics Visakhapatnam, INDIA A.V. ANOKHIN Chelyabinsk State University Department f Mathematics Chelyabinsk , RUSSIA ABSTRACT The main purpse f this paper is t discuss sme qualitative aspects f differential equatins with delays and impulses. Such systems are encuntered in mdeling the dynamics f prices and cultured ppulatins. Hwever, any such discussin has t be based n sme existence and uniqueness results fr delay equatins with discntinuus initial data. This is the cntent f the first part f the paper. Fr an impulsive system, we bserve a phenmenn f existence f infinite number f slutins subject t impulses arbitrarily clse t a fixed time. Cnditins, when such slutins exist and when they d nt, are discussed. Key wrds: Impulsive delay equatins, initial data, cntinuatin. AMS (MOS) subject classificatins: 34A37, 34K INTRODUCTION Recently, the thery f impulsive differential equatins has gained much attentin and ppularity, mainly due t the large ptential such equatins have in prviding mre realistic mdels and als due t the mathematical challenges such equatins pse. Impulses can change the qualitative (nt t mentin the quantitative) prperties f the slutin rather drastically at times (cf. [5]). This can be taken as an advantage, fr this suggests the use f impulses as pssible 1Received: August, Revised: February, Printed in the U.S.A. (C) 1994 by Nrth Atlantic Science Publishing Cmpany 49

2 50 $.V. KRISHNA and A.V. ANOKHIN cntrls in the dynamics f the state f the system under study. It is well knwn that the respnse f a system f the inputs in real life prblems is nt instantaneus (delay) and depends n the histry f the system. This intrduces a delay. Many mdels in ecnmics, bilgy, and chemical kinetics fll int this categry. An example is a mdel fr the prices f several cmmdities in a speculative and unscrupulus envirnment where the custmer stcks fr speculative reasns and the trader hards the gds as his utility has reached a threshld value. This mdel cntains bth impulses and delays (cf. [4]). Thus the interest in impulsive equatins with delays is nt just theretical but practical t. In the practical situatin, the impulses are given at times determined by the slutin, and nt predetermined. This adds t the difficulties. In this situatin, slutins urging fr impulses t frequently, even in a small interval f time, is rather minus. One is als likely t lse the imprtant tl f equating the initial value prblem with an integral equatin, as will be explained elsewhere [4]. The study f delay differential equatins has been largely cnfined t situatins with cntinuus initial functin and when the delay is quite amenable (fr example, Vlterra delay). But, a mre general type f histry ught t be cnsidered while dealing with the mdels and in the presence f impulses, fr impulses create discntinuities in the initial functin right frm the first impulse time. In this paper we study the existence, uniqueness and pulse cntrl f impulsive differential equatins with delay. The existence and uniqueness therems are in a very general set-up whereas the pulse cntrl is btained in a much simpler case f Vlterra delay. These results can be extended t mre general delays with sme effrt, but we d nt attempt this. While prving the existence f a slutin f the nn-impulsive delay equatin, we use an idea f Azbelev [1] which we have used in earlier wrks [2, 3]. 2. EXISTENCE WITH DISCONTINUOUS INITIAL DATA Let f and h satisfy (C1) f: + x "" is a Carathedry functin,

3 Delay Differential Systems with Discntinuus Data 51 (C2) h: +---,i is Lebesgue measurable, h(t) <_ t fr all t I+. Let R+. We cnsider the fllwing initial value prblem: () = f(, (h(o)), > () = (), <, (2.2). where a is a functin, nt necessarily a cntinuus functin frm(-, t] t Our aim is t btain the existence f a lcal and glbal slutin f (2.1, 2.2), with the initial data being a discntinuus functin and t any time in +. We intrduce the fllwing peratrs, which have been first al. [1], and later by Ankhin and Krishna [2, 3]. define: used by Azbelev et Fr any functin z: [t, c)---", t R + fixed and T: ( c, t0l---,r" we (S(h;t)z)(t) = { x(h(t))o ifif h(t)h(t) >< tt (2.3) (h; t)(t)= { (h(t)) if h(t) < t 0 therwise. (2.4) We prve the fllwing lcal existence therem fr the initial value prblem (2.1, 2.2). Therem 2.1" Let (C1) and (C2) hld. Assume further (C3) there exists a Carathedry functin w: gt + x + --.gt + such that (a) w(t, y) is nndecreasina in y fr almst all t +, (b) fr any t, T +, t < T, and #r any y +, w(.,y) L[t, T], fr sme p > 1, (c) II f(t,x) II (t, II II) f t +, z ", Then, fr any (t, x) N+ x N", and fr any bunded Brel meurable functin -(-,t]n, there ezis e>0, such thet (2.1,2.2) h bslutely cntis slti [t, t + e] [0,T] stisfie the cnditin (t) = 0. (2.) Prf: Using (2.3) and (2.4), we can rewrite the initial value prblem (2.1, 2.2 and 2.5) as"

4 52 S.V. KRISHNA and A.V. ANOKHIN which is in turn equivalent t: x (t) = f(t, (h; t)(t ) + S(h; t)x(t)), t >_ t, Z(t) = Z (2.6) (t) = + f + s(h, (2.7) We intrduce the peratr T by (Tx)(t) = x + f f(s, (h, t)(s ) + S(h, t)x(s))ds (2.8) n the space f all abslutely cntinuus funcgins z n [t, T], < T < c. the initial value prblem is equivalent t he peratr equatin n the space f abslutely cntinuus functins n [t, T]. The x = Tx, (2.9) Let 5> 0 be fixed, and define B = {x C[t, T]: Vt [t, T], ][ z(t)- x ][ < 5}. Treating T as an peratr frm C[t, T t C[t, T], we shall prve that TB is precmpact in C[t, T ]. T his end, we bserve that fr each x C[t, T], S(h;t)x is a measurable and bunded functin n [t, T ]. Hence S(h;t)x L[t, T]. Als, frm the definitin f (h;t) and the hypthesis that is Brel measurable, axed h is Lebesgue measurable, it fllws that (h, t) is a measurable and bunded functin n [t, T ]. Hence (h;t) L[t, T] and v(h; t) + S(h; t)x e L[t, T]. Als, - [t, T] e [t0, T] e [t0, T] < ess sup [[ p(h; t)(t [[ + ess sup [I S(h, t)x(t)i[ Frm (C1) and then (C3)(c), it is measurable and that (2.10) fllws that t-f(t, (h;t)(t)+ S(h;t)x(t)) [[ f(t, y(h; t) + S(h; t)x(t))[[ <_ w(t, II (h; t)(t) + S(h; t)x(t ) I1 ) <- w(t, x(, Z, 5)), (2.11)

5 Delay Differential Systems with Discntinuus Data 53 where we have used (2.10) nd hypthesis (C3)(a). Further, (C3)(b) assures that w(., ) e L[t, T] and hence tf(t, (h; t)(t ) + S(h; t)x(t)) is in L"[t, TI. Als frm (2.8) we have Cnsequently, if x 6 B e [t O, T] which prves that TB is unifrmly bunded. Next, let t, t e [t, T]. Then, II (Tx)(t) (Tx)(t ) [I <- / II f(s, (h; t)(s ) + S(h; t)x(s)) II ds. p--1 1 <- f w(s, )ds < It - t l-w-. / w(s, )I Pds Since by w e L[t, T] by (C3)(b), the abve inequality establishes the equicntinuity f TB. Thus TB is precmpact in C[t, T]. Hence crrespnding t the > 0 we have already fixed, there exists e > 0 such that fr any x B (hence fr any Tx e TB), (Tz(t) Z fr all z e C[t, T]) II Tx(t)- x II < (2.12) fr all t 6 [t, t + el.. We nw cnsider the set B, = {x e C[t, t + el: II II t [t, t + e]}. If z B,, we can easily find segment [t, t + e], alng with its image. That is, and (s) = x(s) fr all s e [t, t + el, T. (s)= Tz(s) fr all s e [t, t + ]. B, which cincides with z n the Since TB is precmpact in C[t, T ], frm the identificatin suggested abve, it fllws that TB, is precmpact in C[t, t + e] and TB, c_c_ B,.

6 54 S.V. KRISHNA and A.V. ANOKHIN We nw apply Schauder s fixed pint herem he map T n B e and cnclude he existence f a fixed pin fr T which is a slutin n [t, t Obviusly he slutin is abslutely cntinuus n [t, t + el. This prves he herem. In he nex herem, we btain estimates fr he slutin f (2.1), (2.2), (2.5) and is grwth n he interval f existence which will be used prve glbal existence herem. Therem 2.2: Let t R+ and let (C1 -C3) hld. Suppse that (C4) there is a T > t such that fr any Y +, the maximal slutin f the IVP" =,(t,), (t)= U (e.a) exists n [t, T ]. Then fr any abslutely cntinuus slutin x f (2.1), (2.2), (2.5) existing n any interval [t, t+e]c_[t, T], the fllwing estimates hld: II (t)ii y(t, t, II II + II II)- II II II (t)ii (t, y(t, t, II II + II II ), t [t, t + ], -<t<t 0 e: r y i (.), (.9.), (.), (2.15) II s(h; t)x(t)1[ < max I[ x()ii < II (t II + f II ()II d. t <_S< We have frm (2.1)and he nndecreasing nature f w, using (2.16) II (t)ii = II f(t, (h; t)(t ) + S(h; t)x(t)) II (t, II!1 + II S(h; t)(t)ii ), (2.16) (t, II I! + II II + f II ()I! d). Let z(t)= II II + II 0 II + f II ()II d. Then 2(t)= II (t)ii and 2.(t) <_ w(t,z(t)), z(t)= II!1 + II X II. (2.17)

7 Delay Differential Systems with Discntinuus Data 55 ttence, frm (C41 and well knwn cmparisn therem [6], Therefre, II (t)ii = (t) _< (t, (t)) Cnsequently, which prves the therem. II (t)ii -< II II f / II ()II d = II II f / = z(t)- Z(t)+ II II = z(t)- II II t,!! I! + II II)- II II Therem 2.3, (Glbal existence therem)" Assume (C11-(C41. Then fr any t +, x R" and q any bunded Brel measurable functin n (- c,3, t) there exists a slutins x f(2.1), (2.2/, (2.5) n [t, T]. Prf: Lcal existence is already established. Suppse that the mximal interval f existence is [t, t*)_c [t, T]. Then, the estimates f Therem 2.2 shw that Iimt_t. x(t) exists and hence the slutin can be cntinued further by Therem 2.1. This prves glbal existence. 3. A UNIQUENESS THEOIM The last sectin establishes existence f an abslutely cntinuus slutin f a system f delay differential equatin under very general cnditins, particularly with a discntinuus initial data. This is extremely relevant t us as we are bund t encunter such situatin when we deal with impulsive differential equatins with delay. Hwever, t make any prgress in these difficult situatins, we must make sme assumptins regarding uniqueness f the nn-impulsive slutin. In this sectin we prve such a result, again under a very general hypthesis.

8 Therem 3.1" Suppse f satisfies the cnditin (C5) there exists a functin g:[t, T] x R +--, + such that g is Cthd, g(t, ) O, u (t) = (t, (h(t)),() = O, <_ t h zer slutin n [t, t + e] and [[ f(t, x)- f(t, y)[[ _< g(t, [[ x- y [I ), t [, T], x, y R". (3.1) Then, fr any bunded Brel measurable and any X R"; (2.1), (2.2), (2.5) has almst ne abslutely cntinuus slutin existing n [t, t + e] [t, T ]. 56 S.V. KRISHNA and A.V. ANOKHIN Prf: Fllws frm Therem EXISTENCE TttEOtMS FOR IMPULSWE EQUATIONS WITH DELAY In this sectin, we study the existence f slutins fr. equatins with bth impulses and delays. Let :+--R" and T:R"-+R+, A R+ x", f:+ x N"--,N" be chsen s as t assure existence and uniqueness f an abslutely cntinuus slutin f the initial value prblem (2.1), (2.2), (2.5), fr any bunded Brel measurable initial functin 9. We cnsider the fllwing prblem: where I: N+ x = f(t,x(h(t))), t >_ t (4.1) ((t) x(t_ ), (r r(x(t)) 7/= t)r (t,x(t_)) e A. cr < t, (4.:) (t + ) (t- ) = zx(t) = (t, (t,- )) (4.3) x N"N". This is an impulsive system with delay. Deletin 4.1" A slutin f (4.1) t (4.3) n [t0,t is a functin x: [t, T]R" such that (1) x is right cntinuus, and piecewise abslutely cntinuus n [t0,t], (2) (3) (4) (i.e., x e PAC[t, T]). Satisfies (4.1)fr almst all te [t0,t]. The nly pints f discntinuity f x are slutins f x(t x satisfies (4.3) at these pints f discntinuity. Definitin 4.2: We use the fllwing ntatin: _) (t).

9 Delay Differential Systems with Discntinuus Data 57 PAC[t, T]= {x:[t, T]--." such that there exists an abslutely cntinuus functin y: [t, T] x R", fl e N", r [t, T] such tha.t where is the unit step functin. (t) = u(t) + Hi(t) =.Z,.,(t)}, (.) 0 ift < r 1 if t > ri Pmark 1: By demanding the slutin t be PAC[t, T], and frm the definitin f the space PAC[t, T], it is clear that a slutin is necessarily right cntinuus. This is a deviatin, althugh n a serius ne, frm the usual definitin f slutin f an impulsive equatin which is taken t be left cntinuus (cf. [5]). Remark 2: In the literature s far, impulses were given when the slutins meets a surface f the frm t= r(x(t))r (t)= x. We included here the situatin where the impulses may be received when the slutin reches a certain value at a particular.time. Frm an applied pint f view this seems t be mre reasnable. Als, the earlier results can be recvered by setting = ((t, ). (t) = }, {(, ): t = ()}. Remark 3: An impulse is felt by the slutin x(t) nly when (t, x(t )) e A. If (t, x(t )) e A and after impulse, (t, x(t )) + I(t, x(t ))) e A, the slutin will nt receive any impulse, but mves n until it meets A gain. When we discuss the existence f a slutin f an I.D.E., we can frce the slutin f an N.I.D.E. nt t meet any impulsive surface (T, r A), s that an N.I.D.E. slutin is the slutin f an I.D.E. als. A number f cnditins assuring this can be envisaged frm trivial t mre sphisticated, but this wuld be aviding impulse and s des nt becme a part f the study f impulsive differential equatins. Hwever, we list such cnditins t satisfy curisity. 1) There exists c > 0 and e > 0 such that II -- (t)ii > e t < t < t +. 2) Fr C and g(t)= f f(s, (s- r))ds, t _> 0, there exists e > 0 such that

10 58 S.V. KRISHNA and A.V. ANOKHIN 3) 4) () = a(, +,). (4.6) Fr r C, there exists > 0 such that fr any x with Fr Cx, there exists e > 0 such that, (=) a- (=- =). (4.v) f(t, (h(t))) > ( (t) fr almst edl E [0, + e] (4.8) f(t, v(h(t))) < (t) fr almst all t E [t, t + el. What is very interesting in this cntext is, even after assuming the uniqueness f the slutin f the N.I.D.E., cnditins (4.6) r (4.8) cannt assure the uniqueness f impulsive slutins f I.D.E. as can be seen frm the fllwing. Example: () =, has tw slutins, namely x(t)= t, v(t) = t e [0,1] and t + e- " t E (3- "- 3- ") O, t=o. The secnd slutin y is extremely interesting. Such slutins will be called singular slutins. Nthing qualitative r quantitative can be said abut such slutins, nr d they have gd physical interpretatins. Thus, islating the circumstances in which such slutins can ccur and d nt ccur is an imprtant and useful task. Therem 4.1" Cnsider the impulsive system (4.1) t (4.3), with impulses being given t the slutin x when x(t_)= ((t). Suppse I and are cntinuus with I(0, (0)) = O, x = (0). If there exists a 6 > 0 such that x + f(s, (h(s)))ds- (t) I(t, ((t)) > 0 0 (4.9) fr all te (0,5), then I.D.D.E. (4.1)-(4.3) has a unique slutin n [0,5), which des nt suffer frm impulses.

11 Delay Differential Systems with Discntinuus Data 59 Therem 4.2: Cnsider the LD.D.E. as in Therem 4.1, except that (4.9) /s replaced by: There exists a > 0 such that and the functin + /(a, qa(h(s)))ds- (t) I(t, (t)) < 0 (4.10) 0 m(t)= z0+ f(s, q(h(s)))ds--(t) is either strictly increasing r decreasing in [0, 6). Then the LD.D.E. has exactly ne slutin n [0, 6) 0 (4.11) which has n impulses and has infinitely many singular (impulsive) slutins. Further, if is assumed t be differentiable, then [f(t, q(h(t))) (t)] I(t, (t)) < 0 implies (4.10) and (4.11) and hence the cnclusin f the therem. Prf f Therem 4.1" Let y be the unique slutin f the (nnimpulsive) D.D.E. Let us dente by B the set {t, (t) + I(t, (t)): t _> 0}. Frm the cntinuity f I and, the hypthesis (4.9)implies that the curves {(t,y(t))} and B are n the "same side" f {(t,(t))}. Hence y(t)des nt meet (t)fr t (0, ]. This prves the existence f a slutin fr I.D.D.E. which receives n impulses in [0, ). We shall next prve the uniqueness. Suppse that x is anther slutin f I.D.D.E. (4.1) t (4.3). Then, it must be an impulsive slutin; that is, a slutin which receives impulses as it meets (t). It is als clear that 0 must be a limit pint f the impulse times f this slutin x. Let t > 0 be a discntinuity f x which means t I is an impulse time f x. Fr definiteness, suppse that I(t,(t)) > 0. Then by (4.9)it fllws that X+ ff(s,(h(s)))ds-(t)> 0 tv-,i(t,(t)) is cntinuus at t 1. Hence there is e such that 0 < e < tx, and there exists a cnstant k, such that I(t,(t))> k, fr all t (e,t). If in any finite interval, x has an infinite number f discntinuities, then there exists sequence, {s,} C_ {e,t) f discntinuities f z. The jump t each si is I(s,,x(,-))= I(s,(s))> k,. Since the slutin zpac, there exists an abslutely cntinuus functin z such that

12 60 S.V. KRISHNA and A.V. ANOKHIN. where fli axe the jumps f x at the discntinuity times sl. Hence, : = =(, + -(,-)= z(,,) < This shws that fli diverges. This is a cntradictin. Hence n any finite interval, x can have at mst a finite number f discntinuities. Nw, let > 0 be as in the hypthesis, and let the discntinuities f x be > t > t >... > t > t, + >...---,0 as n---. Then we have: (t-) = (t -) + z(t, (t)) + f f(, (()))d ((tl) + I(tl, (tl) ) + f f(8, 9(h(s)))ds I 1 fr any k_l. limit f (4.13) as = (t) + k.i(t,, (t,)) + i f(s, =1 tk Since x(t- )-((t) and I and are cntinuus and taking the (t) = + (t,, (t,)) + i=1 0 f(s, 9(h(s)))ds > x + / f(s, (h(s)))ds x(t ) = (t). This is a cntradictin. Hence the uniqueness fllws. 0 Remark 4: If the slutin receives impulses whenever it meets the curves i(t), i = 1,2,..., instead f just ne curve, the cnclusin f Therem 4.1 can be and in additin we have: btained if (4.9) hlds fr all If j is the integer such that then there exists c > 0 and 5 > 0 such that (0, (0))= 0, /0)=, (4.14)

13 Delay Differential Systems with Discntinuus Data 61 I0- (0) > c er.a i # j. Prf f Therem 4.2: Frm (4.10) and (4.11), we may assume, withut lss f any generality, that re(t)>0 fr all t(0,). Obviusly, then I(t,(t)) < 0 fr all t fi (0,8). Existence and uniqueness f a nn-impulsive We shall shw that the I.D.D.E. has an infinite number slutin fllw trivially. f impulsive slutins existing lcally. Le [0, T) be he interval f existence f the nn-impulsive slutin y. 5 = rain(5, T) > 0. Fix t I (0, 5 ) and define Let x(t) = (t)- f f(s, qa(h(s)))ds, t e [0, tl). Then xx is differentiable n [0,t) and 2(t)= f(t,(h(t))). Thus x is a slutin f the delay differential equatin n [0,t). fr x(t)- (t) = (t)- f f(s, (h(a)))da- {(t) tl = [(t) f f(, (h())) ]- [(t) f 0 0 = F/g(t)"--m(l) < 0 Cnsider the difference x(t)- (t), y(, (h()))d 0] since m is strictly increasing and t < t. Thus x(t)# (t) n (0, tl). Let g(t) = x(t)- (t)- I(t, (t)), t E [0, t]. and g(t) re(t)- m(tx) I(t, (t)). We bserve that" Assume then g is a() = r()-.(t)- z(. ()) = r()-.(t) <. g(t) = I(t,(t)) > O, cntinuus since we assume I(t,(tl) ) < 0. Hence, there exists t2, t < t2 < t such that g(t2) 0; ha is, z(t2) (t2) + I(t2, (t2) ). Obviusly, z cann mee A between t2 and t. We nw define

14 62 S.V. KRISHNA and V. ANOKHIN 2 x(t) = ((t) / f(s, v(h(s)))ds t e [0, t). We prceed as earlier. Repeating this prcess, we btain a sequence {x,} f functins and a sequence {t,} C [0, t) such that" x,(t) = (t,) J f(s, t e [,.) Defie Xn(7n + 1) = (n + 1) -" I(tn + 1, (:n + 1))" If t g = 0 fr sme psitive integer N, then x is a slutin f I.D.D.E. with impulse times ty_,...,t, and x(o)=x(ty)=(ty)+i(ty,(ty))=xo (by hypthesis). If {t,,} is infinite, bviusly t decreases t 0 (if t, cnverges t sme nn-zer value, x will cincide with the nn-impulsive slutin which is impssible by the cnstructin f By the cntinuity f and I, it then fllws that (0)-..( =..((t.) + z(t, (t.))) = 0. Hence x is a slutin f the I.D.D.E. (4.1) t (.43). We next claim that there are an infinite number f such singular slutins. Fr, if t and t are in (0, 6 ) with t{ <tl, say and if z and z{ are ghe tw slutins btained as in the last paragraph, then n impulse pint f ne can be an impulse pint f the her, r else t = t. Hence if t E (0, t), there exist m and n such that If x(t)--x*(t), then we shuld have (after sme cmputatins) m(tm)--m(t,) which is nt pssible since t, 7 t, and m is sgricgly increasing. Hence z and z* are differen slutins. Lasgly, if " is differentiable, rn is increasing if and nly if m (t) > 0, if and nly if f(t,(h(t))). I(t,(t))< 0. This cmpletes he prf f the therem. We cnclude this sectin with the fllwing bservatin.

15 Delay Differential Systems with Discntinuus Data 63 A negative result: Le ( be cntinuus. Suppse ghere sequence {ti}, ti 0 as ic and exists i(t) = z + f f(s, (h(s)))ds, i = 1, 2, 0 U I(t,(,)) c > 0 fr 11 t (0,e), fr sme e > 0 and c > 0, then (4.1) t (4.3) cannt hve slutin n any interval (0, Let O(t) = {i(t):i = 1, 2,..., } fr each t = {(t, =) e [0,,] +-= e = {(t, ) e [0, ] x /. e ()}. Then, A, C_ M, C_C_ A,, where dentes clsure. Prpsitin: Let there exist > 0 such that +. Fr each e > 0, let t, x + f(s, (h(s)))ds Then the LD.D.E. has a lcal slutin. M,, fr all t e [0, e). 5. CONTINUATION OF THE SOLUTION In the last sectin we discussed the lcal existence f slutin f the I.D.D.E. If the slutin f the nn-impulsive equatin des nt meet any impulsive surface, the situatin is simple and needs n special ttentin. If the slutin des meet an impulse surface (curve), then Therem 4.2 gives a cnditin under which a chs may develp at the time f impulse. Therem 4.1 is t strng negatin f Therem 4.2, since in this therem, the nn-impulsive slutin des nt meet the impulsive surface fr ny t > 0. We wuld be interested in result in which the nn-impulsive slutin des meet an impulsive surface ( but can cntinue further. We shall prve such therems in this sectin. Therem 5.1: Let there exist p L( + ) and S: bunded subsets f + such that II f(t, x)ii p(t). (!1!1) bunded n

16 64 S.V. KRISHNA and AV. ANOKHIN fr all t >_ 0 and x ". Let h(t) g t- r, r > 0 fr all >_ O. Let be cntinuus with (0) Z. Let I be such that fr any T > O, there exists c(t) > 0 with I(t,(t)) > c(t) fr all t e [0,T]. Then, the slutin f LD.D.E. (4.1) t (4.3) can be extended t Prf: Frm the hypthesis, lcal existence is assured. Let [0,T] be the interval f existence. Let x be the slutin. Then fr 0 _< t _< r, Xl(t ) X 0 "- J f(s, q(h(a)))da. + I(t, 0 where t. are such that x(ti-)= (ti). Using (5.1) () x + s( II II)"/P()d + (, ()) 0 We first bserve that the summatin f the R.H.S. f (5.2) has at mst finite number f terms. T see this, suppse x meets at ti, in [O,r]. We can, withut lss f generality, assume that ti--+t* <_ r as i--,cxz. Since is cntinuus, we can find a 5 > 0 such that fr t, s e (t* 5, t*), [(t)- (s)[ < () Hence by hypthesis regarding I, (,) (t) z(, ()) ()1 > 2 fr all t,s (t"-5, t*). Frm he cnvergence f Ti g t, here exists a psitive integer N s.t. ti (t*-5, t*) fr all i >_ N and hence, Let i > N. fllws that" I(t + ) I(t, (t,)) (t) > c..r), i>n. Then xa(t +1 -) -- (ti + 1), Xl(ti) -= (ti) + [(ti, (ti)) and it On the ther hand, Xl(ti + 1 xx(t, + -)- xx(t,) > (") (5.) 2 ti+l )- x(ti) f = f(s, q(h(s)))ds fr all i > N. Hence, s(!i II)" up p(t). (t, + t,) t[ti, ti+ 1] x(ti+x)-xx(ti)lo as i--.c, which cntradicts (5.3).

17 Delay Differential Systems with Discntinuus Data 65 This prves ur claim. Thus, t x() I + s( II II) J() +. >_t<r We nte that B is a cnstant depending nly n x0,, h, I and [r, 2r], we have xz(t) = X + f f(s, 9(h(s)))ds + t O<_tj<r I(tj, (tj)) Fr + f f(s, x(h(s)))ds + I(t, (tl)). r r>_tj<_2r We have, using (5.4). The cnstant B 2 depends n B1, r, I, h and Since, 2r x:(t) <_ B / S(Bx)fp(s)d / r r>_tj<_2r I(t, (t.)) = B1 itself depends n %v and X, B2 depends n, x0, r, h, and I. By inductin, we cnclude, that fr any t e [kr,(k + 1)r] and fr any psitive integer k, there exists a cnstant B(, X, h,, I) such that Hence fr any T > O, xz<(t) xx(t) < B(T,,x, h,,i) fr all t [O,T]. _< Further, if 0 < t < t z(t)- z(t ) < T, then = S(B(T)). fp(s)ds + t<ti<t II(t,,5(t,))l. Hence as t t O, zx(t) z(t ) 0. As we can find a left neighbrhd f T in which z has n discnthauity, frm (5.6) we cnclude that z(t_) exists. If z(t_)# (T), ghe slutin cntinuus beynd T. If z(t_) (T), we csider he prblem: x (t) = f(t,x(h(t))), t > T X(O ) = Xl(O" ) (7 < T

18 66 S.V. KRISHNA and A.V. ANOKHIN _ x(t) = (T) + I(T, (T)) ( # (T)) Under the hypthesis, this prblem has a slutin existing t the right f T. This prcedure can be cntinued. Therem 5.2: Assume the hyptheses f Therem 5.1 fr f and h. Suppse there are a finite number f s at which impulses are given t the slutins. Let be cntinuus, i = 1, 2,..., m. Fr any i, j, 1 i, j <_ m, suppse that i 7 j implies that (t) #,(t) + (t,,(t)) f, t +. If X,(0) fr any i, then the LD.D.E. (4.1) t (4.3) has a unique slutin z PAC(R + ). Prf: Let t [0,r], y(t,o, x) be an abslutely cntinuus slutin f the crrespnding delay equatin. Since x 0 # (0), and as is cntinuus, here exists an interval [0,5] such that y(t,o, x)i(t) fr all t[0,5] fr all i = 1, 2,..., m. If Y(t) (t) fr any j fr all t e [0, r], take x(t) = y(t, 0, z0) fr t E [0,r]. If there exists j such that fr sme t i E (0,r], y(t)=.l(tl) and y(t) i(t) fr any t e [0,t), i= 1,2,...,m, we cnsider y(t)= y(t,t,(t)+ If yl(t)i(t)fr any t e[tl, r) fr any /--1,2,...,m, then we take (t) = u(t), u(t), If nt, prceed as earlier t btain t2, ta, t e [0,t) t e [t,,-]. (5.s) This prcess must terminate after a finite number f steps. Otherwise, we btain a sequence {t,} (0, r] such that,(t- ) = (t,) and,(t,) = (t,) + (t,, ((t,)) fr each i and Yi is abslutely cntinuus n [ti, ti + x). We can assume, lsing n generality, that ti--t* <_ r as i--+c. Frm the abslute cntinuity, we have

19 Delay Differential Systems with Discntinuus Data 67 ti+l Yi(ti + ) = Yi(ti) + [ f(s, q(h(s)))ds. Frm (5.7) and the cntinuity f I and, fr any c > 0, there exists such that,(,)- es(t) z(,, ts(t)) > > 0 fr all s, t (t*, t*]. Hence chsing i large enugh fr ti, t + (t", t*) As ti+ 1 --t -- Yi(t,) Y,i( ti + + ti+l ) / f(s, q(h(s)))ds ti+l <_ ( f p()d). S( II II ). can be made arbitrarily small, this leads t a cntradictin. Since the prcess terminates after a finite number f steps, the functin x defined as in (5.8) is a piecewise abslutely cntinuus slutin f the I.D.D.E. n [0, r]. Ig is nw easy frm he estimates fr z g bserve gha he slutin z(t) can be extended t any interval [0, kr], k being any psitive integer. REFERENCES [] [2] [3] Azbelev, N.V., Maksimv, V.P. and Rakhmatulina, L.F. (lussian), Inirduclin ihe Thery f Funciinal Differential Equatins, Nauka, Mscw, lussia Krishna, S.V. and Ankhin, A.V., Impulsive cntrllability f differential equatins with deviating argument, Dynamical Systems and Applicatins, (t appear). Krishna, S.V., Ankhin, A.V., Impulsive cntrllability f equatins with bth retarded and advanced argument, (in preparatin). [] Krishna, S.V., Cntrl by impulse, delay and histry, Prceedings f Dynamical Systems and Applicatins I, Dynamic Publishers, Atlanta, USA 1993 (t appear). Lakshmikantham, V. Bainv, D.D. and Simenv, P.S., Thery f Impulsive Differential Equatins, Wrld Scientific Publishing C., Singapre [6] Lakshmikantham, V. and Leela, S., Differential and Integral Inequalities. Vl. I, Academic Press, New Yrk 1969.