European Scienific Journal Sepember 214 /SPECIAL/ ediion Vol.3 ISSN: 1857 7881 (Prin) e - ISSN 1857-7431 CONTRIBUTION TO IMPULSIVE EQUATIONS Berrabah Faima Zohra, MA Universiy of sidi bel abbes/ Algeria Absrac In his paper, we show he validiy of he mehod of upper andlower soluions o obain an exisence resul for a firs order impulsive differenial equaions wih variable momens. Keywords: Impulsive differenial equaion, variable imes, upper and lower soluions. Inroducion In his paper we will consider he following sysem of differenial equaions wih impulses a variable imes: I. The invesigaion of heory of impulsive differenial equaions wih variable momens of ime is more difficul han he impulsive differenial equaions wih fixed momens. This paper concerns he exisence of soluions for he funcional differenial equaions wih impulsive effecs a variable imes. We consider he firs order iniial value problem (IVP for shor): y = f, y a. e, T, τ k y, k = 1, m y = I k y = τ k y, k = 1,, m (1,1) y = φ [ r, ] Here f: [, T) D R n is a given funcion. We le D = ψ: r, R n, ψ is coninuous everywhere excep for a finie number of poins a which ψ and ψ exis, and ψ = ψ( )} φ D ; < r <, τ k : R n R, I k : R n R n, k = 1,, m are given funcions saisfying some assumpions ha will be specified laer. Impulsive differenial equaions have been sudied exensively in recen years. Such equaions arise in many applicaions such as spacecraf conrol, impac mechanics, chemical engineering and inspecion process in operaions research. Especially in he area of impulsive differenial equaions and inclusions wih fixed momens; see he monographs of Bainov and Simeonov, Lakshmikanham e al, and Samoilenko and Peresyuk, he papers of Benchohra e al and he references herein. The heory of impulsive differenial equaions wih variable ime is relaively less developed due o he difficulies creaed by he sae-dependen impulses. Recenly, some ineresing exensions o impulsive differenial equaions wih variable imes have been done by Bajo and Liz, Frigon and O Regan, Kaul e al, Kaul and Liu, Lakshmikanham e al, Liu and Ballinger and he references cied herein. Preliminaries Consider Ω a = {y: a r, T R n, a r < T, y is coninuous everywhere excep for some k a which y k and y k, k = 1,, m exis and y k = y k } 393
European Scienific Journal Sepember 214 /SPECIAL/ ediion Vol.3 ISSN: 1857 7881 (Prin) e - ISSN 1857-7431 Ω 1 a = { y Ω a, y is differeniable almos everywhere on a r, T, and y 1 L loc a r, T } AC loc, T, R n is he se of funcions y C(, T, R n which are absoluely coninuous on every compac subse of [, T) Throughou his secion we will assume ha he following condiions hold: H1) f: [, T) D R n 1 is an L loc, T Carahéodory funcion, by his we mean a) The map f(, y) is measurable for all y D b) The map y f(, y) is coninuous almos all [, T) c) Foe each r > here exiss μ r L 1 loc [, T) such ha y < r implies f(, y) μ r () for almos all [, T) H2) he funcions τ k C 1 (R n, R) for k = 1,, m. Moreover < τ 1 x < τ 2 x < < τ m x < T for all x R n H3) here exis consans c k such ha I k (x) c k, k = 1,, m, for each x R n. H4) f(, y) q ψ( y ) For almos all [, T) wih ψ:, (, ) a Borel measurable funcion; 1 L1 ψ loc [, ) and q L 1 loc (,, R ) and du q s ds < for any < T and φ = φ φ() ψ(u) H5) for all, x [, T] R n and for all y D τ k x, f(, y ) 1 for k = 1,, m Where.,. denoes he scalar produc in R n. H6) for all x R n τ k I k x τ k x < τ k1 I k x for k = 1,, m Theorem 1 under he assumpions (H1)-(H6), he problem (1.1) has a leasone soluion on [,T] Proof. The proof will be given in several seps; Sep 1; Consider he problem y = f, y a. e, T y = φ r, (1.2) which will be needed when we examine he IDE (1.1) we use he Schauder-Tychonoff heorem o esablish exisence resuls of (1.2) for compleeness we sae he fixed poin resul. Theorem 2 Le K be a closed convex subse of a locally convex linear opological space E. Assume ha f: K K is coninuous and ha f(k) is relaively compac in E. Then f has a leas one fixed poin in K. Transform he problem (1.2) ino a fixed poin problem. Consider he operaor N: Ω Ω defined by φ [ r, ] N y = φ f s, y s ds [, T] Le K = {y Ω : y b,, T } 394
European Scienific Journal Sepember 214 /SPECIAL/ ediion Vol.3 ISSN: 1857 7881 (Prin) e - ISSN 1857-7431 Where b = J 1 z q x dx dx J z = φ ψ(x) Noice K is a closed, convex, bounded subse of Ω. We nex claim ha N maps K ino K. To see his le y K. Noice for < T ha Ny φ q s ψ y s ds φ = φ b s ds = b() q s ψ b s ds Thus, Ny K and so N: K K. I remains o show ha N: Ω Ω is coninuous and compleely coninuous. Claim I: is coninuous Claim II: maps bounded se ino bounded se in Ω Claim III: maps bounded ses ino equiconinuous ses of Ω As a consequence of Claims I o III ogeher wih he Arzela-Ascoli heorem we can conclude ha N: Ω Ω is compleely coninuous. The Schauder-Tychonoff heorem implies ha N has a fixed poin in, i.e. (1.2) has a soluion y Ω. denoe his soluion by y 1. Define he funcion r k,1 = τ k y 1, [, T] (H2) implies ha r k,1 for k = 1,, m If r k,1 on [, T], k = 1,, m; i,e. τ k (y 1 ) on [, T] and for k = 1,, m; hen y 1 is a soluion of problem (1.1). I remains o consider he case when r 1,1 = for some [, T] Since r 1,1 () and r 1,1 is coninuous, here exiss 1 > such ha r 1,1 1 = and r 1,1 () for all [, 1 ). Thus by (H2) we have r k,1 (), for all [, 1 ) and k = 1,, m. Impulsive Funcional Differenial Equaions In his secion various exisence resuls are esablished for he impulsive funcional differenial equaion y = f, y a. e 1, T y 1 = I 1 y 1 1 2.1 y = y 1 [ 1 r, 1 ] Transform problem (2.1) ino a fixed poin problem. Consider he operaor N 1 : Ω 1 Ω 1 defined by y 1 if [, 1 ] N 1 y = I 1 y 1 f s, y s ds if ( 1, T] 1 395
European Scienific Journal Sepember 214 /SPECIAL/ ediion Vol.3 ISSN: 1857 7881 (Prin) e - ISSN 1857-7431 As in secion 1, we can show ha N 1 is compleely coninuous. and he se K 1 = {y Ω 1, y b, 1 r, T } is closed, convex, bounded subse of Ω 1 where b = J 1 J z = 1 z φ q x dx dx ψ(x) Thus N 1 : K 1 K 1. As a consequence of he Schauder-Tychonoff heorem, we deduce ha N 1 has a fixed poin y which is a soluion o problem (2.1). Denoe his soluion by y 2. Define r k,2 = τ k y 2 for 1 If r k,2 on 1, T, for all k = 1,, m Then y = y 1 if, 1, y 2 if ( 1, T] is a soluion of problem (2.1). I remains o consider he case when here exiss > 1 wih r k,2 =, k = 1,, m by (H6) we have r k,2 1 = τ k y 2 1 1 = τ k I 1 y 1 1 1 > τ k 1 y 1 1 1 τ 1 y 1 1 1 = r 1,1 1 = Since r k,2 is coninuous, here exiss 2 > 1 such ha r k,2 2 =, r k,2 for all ( 1, 2 ) Suppose now ha here is ( 1, 2 ) such ha r 1,2 = from (H6), i follows ha r 1,2 1 = τ 1 y 2 1 1 = τ 1 I 1 y 1 1 1 τ 1 y 1 1 1 = r 1,1 1 = Thus he funcion r 1,2 aains a nonnegaive maximum a some poin 1 ( 1, T]. Since y 2 = f(, y 2 ) Then r 1,2 1 = τ 1 y 2 1 y 2 1 1 = τ 1 y 2 1 f 1, y 2 1 1 = Therefore τ 1 y 2 1, f( 1, y 2 1 ) = 1 which is a conradicion by (H5). Coninue his process and he resul of he heorem follows. Observe ha if T < he process will sop afer a finie number of seps aking ino accoun ha y m1 y [m,t] is a soluion o he problem y = f, y a. e m, T y m = I m y m 1 m (3.1) y = y m 1 [ m r, m ] 396
European Scienific Journal Sepember 214 /SPECIAL/ ediion Vol.3 ISSN: 1857 7881 (Prin) e - ISSN 1857-7431 The soluion y of he problem (1.1) is hen defined by y 1 if [ r, 1 ] y y = 2 if ( 1, 2 ] y m1 if ( m, T] References: Bainov D. D. & Dishliev A. B., Sufficien condiion for absence of beaing in sysems of differenial equaions wih impulse, Appl. Anal. 18 (1984), 67-73. Bainov D. D. & Dishliev A. B., Condiions for he absence of phenomenon beaing for sysems of impulse differenial equaions, bull. Ins. Mah. Acad. Sinica. 13(1985), 237-256. Bainov D. D. & Simeonov P. S., Sysems wih Impulse Effec, Sabiliy Theory and Applicaions. Ellis Horwood, Chicheser (1989). Benchohra M., Henderson J., Nouyas S. & Ouahab A., Impulsive Funcional Differenial Equaions wih Variable Times, Compuers & Mahemaics wih Applicaions. 47(24), no. 1-11, 1659-1665. Benchohra M., Henderson J., Nouyas S. & Ouahab A., Impulsive funcional dynamic equaions on ime scales wih infiniy delay, Dynam. Sysems and Appl. 14(25), no. 1-1. Franco D., Liz E. & Nieo J. J. A conribuion o he sudy of funcional differenial equaions wih impulses, Mah. Nachr. 28(2), 49-6. Frigon M. & O'Regan D., Impulsive Differenial Equaions wih Variable Times. Nonlinear Analysis: Theory, Mehods & Applicaions, 26(12):1913-1922, 1996. Granas and J. Dugundji, Fixed Poin Theory. SpringerMonographs inmahemaics, Springer, New York, 23. Lakshmikanham V., Bainov D. D. & Simeonov P. S., Theory of Impulsive Differenial Equaions. World Scienific Press, Singapore (1989). Samoilenko A. M. & Peresyuk, Impulsive Differenial Equaions. World Scienific, Singapore (1995). Vasala A. S. & Sun Y., Periodic boundary value problem of impulsive differenial equaions, Applacable Analysis. 44(1992), no.3-4, 145-158. 397