Fractional Differential Inclusions with Impulses at Variable Times

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1 Advances in Dynamical Systems and Applications ISSN , Volume 7, Number 1, pp (212) Fractional Differential Inclusions with Impulses at Variable Times Mouffak Benchohra and Farida Berhoun Université de Sidi Bel Abbes Laboratoire de Mathématiques BP 89, 22 Sidi Bel Abbes, Algeria benchohra@univ-sba.dz, berhoun22@yahoo.fr Juan J. Nieto Universidad de Santiago de Compostela Departamento de Análisis Matematico Facultad de Matemáticas Santiago de Compostela, Spain juanjose.nieto.roig@usc.es Abstract In this paper, we establish sufficient conditions for the existence of solutions for a class of initial value problem for impulsive fractional differential inclusions with variable times involving the Caputo fractional derivative. AMS Subject Classifications: 26A33, 34A37, 34A6. Keywords: Initial value problem, impulses, variable times, Caputo fractional derivative, fractional integral, differential inclusion. This paper is dedicated to Johnny Henderson on the occasion of his 61st birthday. 1 Introduction This paper deals with the existence of solutions for the initial value problems (IVP for short), for fractional order 1 < α 2 differential inclusions with impulsive effects c D α y(t) F (t, y(t)), for a.e., t J = [, T ], t τ k (y(t)), k = 1,..., m, (1.1) Received July 13, 211; Accepted August 19, 211 Communicated by Martin Bohner

2 2 M. Benchohra, F. Berhoun and J. J. Nieto y(t + ) = I k (y(t)), t = τ k (y(t)), k = 1,..., m, (1.2) y (t + ) = I k (y(t)), t = τ k (y(t)), k = 1,..., m, (1.3) y() = y, y () = y, (1.4) where c D α is the Caputo fractional derivative, F : J R P(R) is a compact convex valued multivalued map, (P(R) is the family of all nonempty subsets of R, τ k : R (, T ), I k, I k : R R, k = 1,..., m are given continuous functions and y, y R. Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [21, 22, 26, 32, 38]). There has been a significant development in fractional differential and partial differential equations in recent years; see the monographs of Kilbas et al [28], Lakshmikantham et al [3], Miller and Ross [33], Samko et al [42] and the papers of Agarwal et al [1], Ahmad and Nieto [3], Belmekki et al [8,9], Benchohra et al [1,12,14], Chang and Nieto [16], Delbosco and Rodino [18], Diethelm et al [19], Podlubny et al [41], and the references therein. Applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contain y(), y (), etc. the same requirements of boundary conditions. Caputo s fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann Liouville and Caputo types see [25, 4]. Integer order Impulsive differential equations have become important in recent years as mathematical models of phenomena in both the physical and social sciences. There has a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments; see for instance the monographs by Benchohra et al [13], Lakshmikantham et al [29], and Samoilenko and Perestyuk [43] and the references therein. Particular attention has been given to differential equations at variable moments of impulse; see for instance the papers by Bajo and Liz [6], Belarbi and Benchohra [7], Benchohra et al [11], Frigon and O Regan [2], Graef and Ouahab [23], Nieto et al [34 37], Stamova [44], Zhano and Feng [46] and the references therein. Very recently Agarwal et al [2], and Benchohra and Slimani [15] have initiated the study of impulsive fractional differential equations at fixed moments. These results were extended to the multivalued case by Ait Dads et al [4]. The aim of this paper is to extend these studies when the impulses are variable. This paper is organized as follows. In Section 2 we present some backgrounds on fractional derivatives and integrals and multivalued analysis. Section 3 is devoted to the existence of solutions for problem (1.1) (1.4). In Section 4 we present an example showing the applicability of our assumptions.

3 Impulsive Fractional Differential Inclusions at Variable Times 3 2 Preliminaries In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis which are used throughout this paper. By C(J, R) we denote the Banach space of all continuous functions from J into R with the norm y := sup{ y(t) : t J}. In order to define the solution of (1.1) (1.4) we shall consider the space of functions Ω = {y : J R : there exist = t < t 1 <... < t m < t m+1 = T such that t k = τ k (y(t k )), y(t + k ) exists, k =,..., m and y k C((t k, t k+1 ], R), k =,..., m}, where y k is the restriction of y over (t k, t k+1 ], k =,..., m. L 1 (J, R) denotes the Banach space of measurable functions x : J R which are Lebesgue integrable and normed by x L 1 = T x(t) dt for all x L 1 (J, R). Let P cl (R) = {Y P(R) : Y closed}, P b (R) = {Y P(R) : Y bounded}, P cp (R) = {Y P(R) : Y compact} and P cp,c (R) = {Y P(R) : Y compact and convex}. A multivalued map G : R P (R) is convex (closed) valued if G(x) is convex (closed) for all x R, G is bounded on bounded sets if G(B) = x B G(x) is bounded in R for all B P b (R) (i.e. sup{sup{ y : y G(x)}} < ). G is called upper x B semicontinuous (u.s.c.) on R if for each x R, the set G(x ) is a nonempty closed subset of R, and if for each open set N of R containing G(x ), there exists an open neighborhood N of x such that G(N ) N. G is said to be completely continuous if G(B) is relatively compact for every B P b (R). If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e. x n x, y n y, y n G(x n ) imply y G(x )). G has a fixed point if there is x R such that x G(x). The fixed point set of the multivalued operator G will be denoted by F ixg. A multivalued map G : [, 1] P cl (R) is said to be measurable if for every y R, the function t d(y, G(t)) = inf{ y z : z G(t)} is measurable. For more details on multivalued maps see the books of Aubin and Frankowska [5], Deimling [17] and Hu and Papageorgiou [27]. Definition 2.1. A multivalued map F : J R P(R) is said to be L 1 -Carathéodory if (i) t F (t, u) is measurable for each u R;

4 4 M. Benchohra, F. Berhoun and J. J. Nieto (ii) u F (t, u) is upper semicontinuous for almost all t J; (iii) for each q >, there exists ϕ q L 1 (J, R + ) such that F (t, u) = sup{ v : v F (t, u)} ϕ q (t) for all u q and for a.e. t J. For each x Ω, define the set of selections of F by S F,x = {v L 1 (J, R) : v(t) F (t, x(t)) a.e. t J}. Lemma 2.2 (See [31]). Let F : J R P cp,c (R) be an L 1 -Carathéodory multivalued map and let L be a linear continuous mapping from L 1 (J, R) to C(J, R), then the operator L S F : C(J, R) P cp,c (C(J, R)), x (L S F )(x) := L(S F,x ) is a closed graph operator in C(J, R C(J, R). Definition 2.3 (See [28, 39]). The fractional (arbitrary) order integral of the function h L 1 ([a, b], R + ) of order α R + is defined by I α a h(t) = a (t s) α 1 h(s)ds, where Γ is the gamma function. When a =, we write I α h(t) = [h ϕ α ](t), where ϕ α (t) = tα 1 for t >, and ϕ α(t) = for t. ϕ α (t) δ(t) as α 1, where δ is the delta function. Definition 2.4 (See [28, 39]). For a function h given on the interval [a, b], the αth Riemann Liouville fractional-order derivative of h, is defined by (D α a+h)(t) = ( ) n 1 d t (t s) n α 1 h(s)ds. Γ(n α) dt a Here n = [α] + 1 and [α] denotes the integer part of α. Definition 2.5 (See [28,39]). For a function h given on the interval [a, b], the αth Caputo fractional-order derivative of h, is defined by ( c D α a+h)(t) = 1 Γ(n α) a (t s) n α 1 h (n) (s)ds. Here n = [α] + 1 and [α] denotes the integer part of α.

5 Impulsive Fractional Differential Inclusions at Variable Times 5 3 Existence of Solutions Let us define what we mean by a solution of problem (1.1) (1.4). Definition 3.1. A function y Ω is said to be a solution of (1.1) (1.4) if there exist a function v L 1 (J, R) such that v(t) F (t, y(t)) a.e. t J and satisfies the equation c D α t k y(t) = v(t), for a.e. t (t k, t k+1 ), t τ k (y(t)), k = 1,..., m, and conditions y(t + ) = I k (y(t)), y (t + ) = I k (y(t)), t = τ k (y(t)), k = 1,..., m, and y() = y, y () = y are satisfied. For the existence of solutions for the problem (1.1) (1.4), we need the following auxiliary lemmas. Lemma 3.2 (See [45]). Let α >, then the linear differential equation c D α h(t) = has solutions h(t) = c +c 1 t+c 2 t c n 1 t n 1, c i R, i =, 1, 2,..., n 1, n = [α] + 1. Lemma 3.3 (See [45]). Let α >, then I α c D α h(t) = h(t) + c + c 1 t + c 2 t c n 1 t n 1 for some c i R, i =, 1, 2,..., n 1, n = [α] + 1. Let us introduce the following hypotheses which are assumed hereafter: (H 1 ) The function F : J R P cp,c (R) is L 1 -Carathéodory; (H 2 ) There exist a continuous non-decreasing function ψ : [, ) (, ) and a function p C(J, R + ) such that F (t, u) p(t)ψ( u ) for each (t, u) J R; (H 3 ) The functions I k, I k : R R, k = 1,..., m are continuous and nondecreasing; (H 4 ) There exists a number M > such that { } M M min y + T y 1 + p T α ψ(m), I Γ(α+1) k (M) + T I k (M) + p T α ψ(m) Γ(α+1) > 1, where p = sup{p(t) : t J};

6 6 M. Benchohra, F. Berhoun and J. J. Nieto (H 5 ) The functions τ k C 1 (R, R) for k = 1,..., m. Moreover < τ 1 (y) <... < τ m (y) < T for all y R; (H 6 ) For all y R τ k (I k (y)) τ k (y) < τ k+1 (I k (y)) for k = 1,..., m; (H 7 ) For a J fixed, y Ω, v S F,y and for each t J we have τ k(y(t)) a (t s) α 2 v(s)ds Γ(α 1) for k = 1,..., m. Theorem 3.4. Assume that hypotheses (H 1 ) (H 7 ) hold. Then the IVP (1.1) (1.4) has at least one solution. Proof. The proof will be given in several steps. Step 1: Consider the problem c D α y(t) F (t, y(t)), for a.e., t J (3.1) y() = y, y () = y. (3.2) Transform the problem (3.1) (3.2) into a fixed point problem. Consider the operator N : C(J, R) P(C(J, R)) defined by N(y) = {h C(J, R) : h(t) = y + y t + 1 (t s) α 1 v(s)ds, v S F,y }. We shall show that N satisfies the assumptions of nonlinear alternative of Leray Schauder type. Claim 1: N(y) is convex for each y C(J, R). Indeed, if h 1, h 2 belong to N(y), then there exist v 1, v 2 S F,y such that for each t J, we have h i (t) = y + y t + 1 Let d 1. Then for each t J, we have (dh 1 + (1 d)h 2 )(t) = y + y t + 1 (t s) α 1 v i (s)ds (i = 1, 2). (t s) α 1 [dv 1 (s) + (1 d)v 2 (s)]ds.

7 Impulsive Fractional Differential Inclusions at Variable Times 7 Since S F,y is convex (because F has convex values), then dh 1 + (1 d)h 2 N(y). Claim 2: N maps bounded sets into bounded sets in C(J, R). Indeed, it is enough to show that for any q >, there exists a positive constant l such that for each y B q = {y C(J, R) : y q}, we have N(y) l. Then for each h N(y), there exists v S F,y such that By (H 2 ) we have for each t J, h(t) = y + y t + 1 h(t) y + T y + 1 y + T y + 1 y + T y + p ψ(q) t (t s) α 1 v(s)ds (t s) α 1 v(s) ds (t s) α 1 p(s)ψ( y(s) )ds y + T y + T α p ψ(q) Γ(α + 1) (t s) α 1 ds := l. Claim 3: N maps bounded sets into equicontinuous sets of C(J, R). Let t 1, t 2 J, t 1 < t 2, B q a bounded set of C(J, R) as in Claim 2 and let y B q and h N(y). Then t 2 h( t 2 ) h( t 1 ) y ( t 2 t 1 ) + 1 ( t 2 s) α 1 v(s) ds t t 1 [( t 2 s) α 1 ( t 1 s) α 1 ] v(s) ds y ( t 2 t 1 ) + ψ(q) t 2 ( t 2 s) α 1 p(s)ds t 1 + ψ(q) t 1 [( t 2 s) α 1 ( t 1 s) α 1 ]p(s)ds. As t 1 t 2, the right-hand side of the above inequality tends to zero. As a consequence of Claims 1 to 3 together with the Arzelá Ascoli theorem, we can conclude that N : C(J, R) P(C(J, R)) is completely continuous. Claim 4: N has a closed graph.

8 8 M. Benchohra, F. Berhoun and J. J. Nieto Let y n y, h n N(y n ) and h n h. We need to show that h N(y ). h n N(y n ) means that there exists v n S F,yn such that, for each t J h n (t) = y + y t + 1 (t s) α 1 v n (s). We must show that there exist v S F,y such that for each t J, h (t) = y + y t + 1 (t s) α 1 v (s). Clearly (h n φ) (h φ) as n, where φ(t) = y + ty. Consider the continuous linear operator defined by Γ : L 1 (J, R) C(J, R) v (Γv)(t) = (t s) α 1 v(s)ds. From Lemma (2.2), it follows that Γ S F is a closed graph operator. Moreover, we have ( ) h n (t) φ(t) Γ(S F,yn ). Since y n y, it follows from Lemma 2.2 that h (t) = y + y t + 1 Claim 5: A priori bounds on solutions. (t s) α 1 v (s). Let y C(J, R) be such that y λn(y) for some λ (, 1). Then, there exists v L 1 (J, R) with v S F,y such that, for each t J, From (H 2 ) y(t) = λy + λy t + λ (t s) α 1 v(s)ds. y(t) y + T y + 1 y + T y + 1 t (t s) α 1 v(s) ds (t s) α 1 p(s)ψ( y(s) )ds y + T y + p ψ( y ) (t s) α 1 ds y + T y + T α p ψ( y ). Γ(α + 1)

9 Impulsive Fractional Differential Inclusions at Variable Times 9 Thus y y + T y + p T α Γ(α+1) ψ( y ) 1 Then by (H 4 ), y M. Let U = {y C(J, R) : y < M}. The operator N : U P(C(J, R)) is upper semicontinuous and completely continuous. From the choice of U, there is no y U such that y λn(y) for some λ (, 1). As a consequence of the nonlinear alternative of Leray Schauder type [24], we deduce that N has a fixed point in U which is a solution of the problem (3.1) (3.2). Denote this solution y 1. Define the function (H 5 ) implies that If r k,1 (t) = τ k (y 1 (t)) t for t. r k,1 () for k = 1,..., m. r k,1 (t) on J for k = 1,..., m, then y 1 is solution of the problem (1.1) (1.4). Now we consider the case when r 1,1 (t) = for some t J. Since r 1,1 () and r 1,1 is continuous, there exists t 1 > such that Thus by (H 5 ) we have r 1,1 (t 1 ) = and r 1,1 (t) for all t [, t 1 ). r k,1 (t) for all t [, t 1 ) and k = 1,..., m. Step 2: Consider the problem c Dt α 1 y(t) F (t, y(t)), for a.e., t [t 1, T ] (3.3) y(t + 1 ) = I 1 (y 1 (t 1 )) (3.4) y (t + 1 ) = I 1 (y 1 (t 1 )). (3.5) Consider the operator N 1 : C([t 1, T ], R) P(C([t 1, T ], R)) defined by N 1 (y) = {h C([t 1, T ], R)} with h(t) = I 1 (y 1 (t 1 )) + (t t 1 )I 1 (y 1 (t 1 )) + 1 (t s) α 1 v(s), t 1

10 1 M. Benchohra, F. Berhoun and J. J. Nieto and v S F,y. As in Step 1 we can show that N 1 satisfies the assumptions of the nonlinear alternative of Leray Schauder type and deduce that the problem (3.3) (3.5) has a solution denoted by y 2. Define r k,2 (t) = τ k (y 2 (t)) t for t t 1. If then r k,2 (t) on (t 1, T ] for k = 1,..., m, y(t) = { y1 (t), if t [, t 1 ], y 2 (t), if t (t 1, T ] is solution of the problem (1.1) (1.4). It remains to consider the case when By (H 6 ) we have r 2,2 (t) =, for some t (t 1, T ]. r 2,2 (t + 1 ) = τ 2 (y 2 (t + 1 )) t 1 = τ 2 (I 1 (y 1 (t 1 ))) t 1 > τ 1 (y 1 (t 1 )) t 1 = r 1,1 (t 1 ) =. Since r 2,2 is continuous, there exists t 2 > t 1 such that r 2,2 (t 2 ) = and r 2,2 (t) for all t (t 1, t 2 ). By (H 5 ) we have r k,2 (t) for all t (t 1, t 2 ) and k = 2,..., m. Suppose now that there exists s (t 1, t 2 ) such that r 1,2 (s) =. From (H 6 ) it follows that r 1,2 (t 1 ) = τ 1 (y 2 (t 1 )) t 1 = τ 1 (I 1 (y 1 (t 1 ))) t 1 τ 1 (y 1 (t 1 )) t 1 = r 1,1 (t 1 ) =. Also, from (H 5 ) we have r 1,2 (t 2 ) = τ 1 (y 2 (t 2 )) t 2 < τ 2 (y 2 (t 2 )) t 2 = r 2,2 (t 2 ) =. Since r 1,2 (t 1 ), r 1,2 (t 2 ) < and r 1,2 (s) = for some s (t 1, t 2 ), the function r 1,2 attains a nonnegative maximum at some point s (t 1, t 2 ). Since y 2(t) = 1 Γ(α 1) t 1 (t s) α 2 v(s)ds,

11 Impulsive Fractional Differential Inclusions at Variable Times 11 we have Therefore which contradicts (H 7 ). r 1,2(s) = τ 1(y 2 (s))y 2(s) 1 =. s τ 1(y 2 (s)) (s s) α 2 v(s)ds = Γ(α 1), t 1 Step 3: We continue this process and taking into account that y m+1 := y [tm,t ] is a solution to the problem c D α t m y(t) F (t, y(t)), for a.e., t (t m, T ], (3.6) y(t + m) = I m (y m 1 (t m)), (3.7) y (t + m) = I m (y m 1 (t m)). (3.8) The solution y of the problem (1.1) (1.4) is then defined by y 1 (t), if t [, t 1 ], y y(t) = 2 (t), if t (t 1, t 2 ],... y m+1 (t), if t (t m, T ]. 4 An Example c D α y(t) F (t, y(t)), for a.e., t J = [, 1], t τ k (y(t)), k = 1,..., m, 1 < α 2, (4.1) y(t + ) = I k (y(t)), t = τ k (y(t)), k = 1,..., m, (4.2) y (t + ) = I k (y(t)), t = τ k (y(t)), k = 1,..., m, (4.3) where y() =, y () =, (4.4) 1 τ k (y) = 1 2 k (1 + y 2 ), ( ] 1 I k (y) = b k y, I k (y) = b k y, b k >, b k 2, 1, k = 1, 2,..., m. We have τ k+1 (y) τ k (y) = 1 2 k+1 (1 + y 2 ) > for each y R, k = 1, 2,..., m.

12 12 M. Benchohra, F. Berhoun and J. J. Nieto Hence Also, and Finally, < τ 1 (y) < τ 2 (y) <... < τ m (y) < 1, for each y R. τ k (I k (y)) τ k (y) = τ k+1 (I k (y)) τ k (y) = b 2 k 1 2 k (1 + y 2 )(1 + b 2 k y2 ) 1 + (2b 2 k 1)y2 2 k+1 (1 + y 2 )(1 + b 2 k y2 ), for each y R, F (t, y) = {v R : f 1 (t, y) v f 2 (t, y)}, >, for each y R. where f 1, f 2 : J R R. We assume that for each t J, f 1 (t, ) is lower semicontinuous (i.e, the set {y R : f 1 (t, y) > µ} is open for each µ R), and assume that for each t J, f 2 (t, ) is upper semicontinuous (i.e the set {y R : f 2 (t, y) < µ} is open for each µ R). Assume that there are p C(J, R + ) and ψ : [, ) (, ) continuous and nondecreasing such that max{ f 1 (t, y), f 2 (t, y) } p(t)ψ( y ), t J, and all y R. Assume there exists a constant M > such that M M(b k + b k ) + > 1, p ψ(m) Γ(α+1) where p = sup p(t). It is clear that F is compact and convex valued, and it is upper t J semicontinuous (see [17]). If (H 7 ) holds, then by Theorem (3.4) the problem (4.1) (4.4) has at least one solution. References [1] R.P Agarwal, M. Benchohra and S. Hamani, Boundary value problems for fractional differential equations, Adv. Stud. Contemp. Math. 16 (2) (28), [2] R. P. Agarwal, M. Benchohra and B.A. Slimani, Existence results for differential equations with fractional order and impulses, Mem. Differential Equations Math. Phys. 44 (28), [3] B. Ahmad, J.J. Nieto, Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems Volume 29 (29), Article ID 78576, 11 pages. [4] E. Ait Dads, M. Benchohra and S. Hamani, Impulsive fractional differential inclusions involving the Caputo fractional derivative, Fract. Calc. Appl. Anal. 12 (1) (29),

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