Impulsive fractional differential equations with variable times

K e y w o r d s I m p u l s i v e functional differential equations, Variable times, Fixed point. 1. I N T R O D U C T I O N This note is concerned with the existence of solutions, for the initial value problems (IVP for short), for first-order functional differential equations with impulsive effects y' ( t )=f( t , yt), a.e. t e J = [ O , T ] , t¢Tk(y(t)), k = l , . . . , m , (1) y(t +) = Ik(y(t)) , t = ~k(y(t)), k = 1 , . . . , -~, (2) y(t) = ¢(t) , t e I--r, 0], (3) where f : J × D --* R ~ is a given function, D = {~b : [-r , 0] --+ R'~; ¢ is continuous everywhere except for a finite number of points ~ at which ¢(t-) and ¢(t+) exist and ¢ ( t ) = ¢(~)}, ¢ e D, 0898-1221/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. Typeset by ~4A/~S-~X doi:10.1016/j.camwa.2004.06.013 1660 M. BENCHOHRA et al. 0 < r < c~, 7k : N n --* R, Ik : R n --* ]R n, k = 1 , 2 , . . . , m are given functions satisfying some assumptions that will be specified later. For any function y defined on I-r , T] and any t E J, we denote by Yt the element of D defined by yt(0) = y(t + 0), 0 e I-r,0]. Here Yt(') represents the history of the state from time t r, up to the present time t. Impulsive differential equations have become more important in recent years in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. There has been a significant development in impulse theory, in recent years, especially in the area of impulsive differential equations with fixed moments; see the monographs of Bainov and Simeonov [1], Lakshmikantham et al. [2], and Samoilenko and Perestyuk [3], and the references therein. The theory of impulsive differential equations with variable time is relatively less developed due to the difficulties created by the state-dependent impulses. Recently, some interesting extensions to impulsive differential equations with variable times have been done by Bajo and Liz [4], Frigon and O'Regan [5-7], Kaul et al. [8], Kaul and Liu [9,10], Lakshmikantham et aI. [11,12], and Liu and Sallinger [13]. The main theorem of this note extends problem (1)-(3) considered by Benchohra et el. [14] when the impulse times are constant. Our approach is based on the Schaefer's fixed-point theorem (see [15, p. 29]). 2. P R E L I M I N A R I E S In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. By C(J, ]Rn), we denote the Banach space of all continuous functions from J into ]R n with the norm Ilylloo := sup{[y(t)] : t E J ) . Also, D is endowed with norm II" II defined by I1¢11 := sup{l¢(o)I : r < 0 < 0}. In order to define the solutions of (1)-(3), we shall consider the space f~ ={y : [-r , T] ~R" : there e x i s t 0 = t 0 < t l < . . < t m < t , ~ + l = T , suchthat , tk = ~'k (y( tk) ) , y (t-~) and y (t +) exist, with y (t~-) = y (tk), k = 1 , . . . , m , and y E C( [ tk , tk+ l ] ,R") , k = 0 , . . . , m } . DEFINITION 2.1. A map f : J x D ~ N '~ is said to be L1-Carathgodory if (i) t ~ ~ f ( t , u) is measurable for each u E D; (ii) u ~ f ( t , u) is continuous for almost a11 t E J; (iii) for each q > 0, there exists h a 6 L I ( J ,R+) , such that If ( t ,u ) l < hq( t ), l'or a11 Ilu]l <_ q and t'or almost a11 t e J. In what follows, we will assume that f is an L1-Carath~odory function. The consideration of this paper is based on the following Schaefer's fixed-point theorem (cf. [15]). THEOREM 2.2. Let X be a Banach space and N : X --~ X be a completely continuous map. If the set £ ( N ) = {y e X : y = AN(y) , for some 0 <)~ < 1} is bounded, then N has a fixed point. Impulsive Functional Differential Equations 1661 3. M A I N R E S U L T Let us start by defining what we mean by a solution of problem (1)-(3). DEFINITION 3.1. A function y E Q, is said to be a solution of (1)-(3) if y satisfies the equation y'(t) = S(t,y~), a.e. on J, t # ~k(y(t)), k = 1 , . . . , , ~ , and the conditions y(t+) = Ik(y(t)), t = 7k(y(t)) k = 1 , . . . , m and y(t) = ¢(t) on I-r , 0]. We are now in a position to state and prove our existence result for problem (1)-(3). For the study of this problem, we first list the following hypotheses. (H1) The functions ~'k E CI(N'~,R), for k = 1 , . . . , m. Moreover, 0 < rl(X) < . . . < TIn(X) < T, for all x E R'L (H2) There exist constants ck, such that IIk(x)l _< ck, k = 1 , . . . , m for each x E 1R ~. (H3) There exists a continuous nondeereasing function ¢ : [0, oo) ---. (0, oo) and p E LI(J, R+), such that lf(t,u)] < p(t)¢ (llull), for a.e. t E J and each u E D with ~ ds ¢ (---~ = oo. (H4) For all (t, x) E [0, T] x R '~ and for all Yt E D, we have (r~(x) , f ( t , yt)) ¢ l, for k = l , . . . , m , where (., .) denotes the scalar product in R n. (H5) For all x E IR '~, ~k (Ik(z)) _< rk(z) < ~k+~ (irk(~)), for k = 1 , . . . , , ~ . THEOREM 3.2. Assume that Hypotheses (H1)-(HS) hold. Then, the IVP (1)-(3) has at least one solution on [-r, T]. PROOF. The proof will be given in several steps. STEP 1. Consider the following problem y'(t) = f (t, y~), a.e. t e [0, T], (4) y(t) = ¢(t), t e I-r, 0]. (2) Transform problem (4),(5) into a fixed-point problem. Consider the operator N : C( [ r , T], R ~) C([-r, T], N n) defined by: ¢(t), f0 if t E [-r , 0]; t N ( y ) ( t ) = ¢ ( 0 ) + l ( s , ys)ds, i f t E [0,T]. We shall show that the operator N is completely continuous. CLAIM i . N N continuous. Let {Yn} be a sequence, such that yn --* y in C( [ r , T], ]Rn). Then, 2 I N ( y , ) ( t ) N ( y ) ( t ) l < I f ( s , y , 8 ) f ( s , ys)l ds <_ lY (s,y~8) f (s,y~)l ds. Since f is an L1-Carath6odory function, we have by the Lebesgue dominated convergence theorem IIN (Y~) N(Y)Iloo << Ilf (',Yn(.~) -f (',Y<'))IIL1--~ 0, as n --* oo. 1662 M. BENCHOHRA et al. CLAIM 2. N maps bounded sets into bounded sets in C ( [ r , T], Rn). Indeed, it is enough to show tha t for any q > 0, there exists a positive constant g, such that for each y e Bq = {y • C([-r,T],~[n) : ]fyl[oo <_ q}, we have [[g(y)l[o~ < ~. By Definition 3.1 (iii), we have for each t • [0, T], ~0 t IN(y)( t) l ~ 14,(o)1 ÷ If(s, ys)l ds ]I¢IL ÷ IlhqllL~. Thus, [[N(y)II~ ~ I1¢11 ÷ ][hql{L~ := £. CLAIM 3. N maps bounded sets into equicontinuous sets of C([-r , T], Rn). Let ll, 12 C [0, T], 11 < I2, Bq be a bounded set of C([-r, T], ~n) as in Claim 2, and let y E Bq. Then, f/ iN(y) (12) N(y) (/1)[ _< hq(s) ds. As 12 ) ll, the right-hand side of the above inequality tends to zero. The equicontinuity for the cases ll < 12 _< 0 and 11 _< 0 _< 12 is obvious. As a consequence of Claims 1-3, together with the Arzela-Ascoli theorem, we can conclude tha t N : C([-r,T],]~ n) ~ C( [ r ,T] ,~ ~) is completely continuous. CLAIM 4. Now it remains to show tha t the set $ (N) := {y e C ( [ r , T ] , R n ) : y = AN(y), for some 0 < A < 1} is bounded. Let y C g (N) . Then, y = AN(y) for some 0 < A < 1. Thus, for each t e [0, T], ( /0 ) u ( t ) = A ¢ ( 0 ) + f ( s , ys) ds . This implies by (H2),(H3) that for each t e J , we have lY(t)] <-I1¢11 + p(s)C (llysll) as. We consider the function # defined by i~(t) =sup{[y(s)[ : r < s < t} , O < t < T . Let t* e [ -r , t I be such tha t #(t) = [y(t*)[. If t* e [0, T], by the previous inequality, we have for t e [o, T] . ( t ) <__ I!¢11 + p(s)~(~(s)) ds. If t* e [ r , 0], then #(t) = I1¢1] and the previous inequality holds. Let us take the right-hand side of the above inequality as v(t). Then, we have c = v ( O ) = N ¢ [ [ , # ( t ) < v ( t ) , t E [ O , T ] , and ~'(t) =p(t)~(~(t)) , a.e. t • [O,T]. Impulsive Functional Differential Equations 1663 Using the nondecreasing character of 4, we get v'(t) < p( t )¢(v( t ) ) , a.e. t e [0, T]. This implies that for each t E [0, T] v(t) ds < p(s) ds < Jr(0) ¢ ( s ) (0) ¢ ( s ) Thus, there exists a constant K, such that v(t) <_ K , t E [0, T], and hence, #(t) < K, t E [0, T]. Since for every t e [0,T], Ilyt]l -< #(t), we have ]lylloo -< g ' = ma~ {]]¢1] , K } , where K ' depends on T and on the functions p and 4This shows that £(N) is bounded. Set Z := C([-r , T], R~). As a consequence of Schaefer's theorem (see [15, p. 29]), we deduce that N has a fixed-point y which is a solution to problem (4),(5). Denote this solution by YtDefine the function rk,l(t) = ~-k (yl(t)) t, for t >_ 0. (H1) implies that