Research Article | Open Access
Toufik Guendouzi, Ouahiba Benzatout, "Existence of Mild Solutions for Impulsive Fractional Stochastic Differential Inclusions with State-Dependent Delay", Chinese Journal of Mathematics, vol. 2014, Article ID 981714, 13 pages, 2014. https://doi.org/10.1155/2014/981714
Existence of Mild Solutions for Impulsive Fractional Stochastic Differential Inclusions with State-Dependent Delay
We study the existence of mild solutions for a class of impulsive fractional stochastic differential inclusions with state-dependent delay. Sufficient conditions for the existence of solutions are derived by using the nonlinear alternative of Leray-Schauder type for multivalued maps due to O’Regan. An example is given to illustrate the theory.
During the past two decades, fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, mechanics, and engineering [1–3]. There has been a great deal of interest in the solutions of fractional differential equations in analytical and numerical senses. One can see the monographs of Kilbas et al. , Miller and Ross , Podlubny , and Lakshmikantham et al.  and the survey of Agarwal et al. [7, 8].
To study the theory of abstract differential equations with fractional derivatives in infinite dimensional spaces, the first step is how to introduce new concepts of mild solutions. A pioneering work has been reported by El-Borai [9, 10]. Very recently, Hernández et al.  showed that some recent papers of fractional differential equations in Banach spaces were incorrect and used another approach to treat abstract equations with fractional derivatives based on the well-developed theory of resolvent operators for integral equations. Moreover, Wang and Zhou , Zhou and Jiao  also introduced a suitable definition of mild solutions based on Laplace transform and probability density functions.
On the other hand, the theory of impulsive differential equations or inclusions has become an active area of investigation due to its applications in fields such as mechanics, electrical engineering, medicine, biology, and ecology. One can refer to [14, 15] and the references therein. Recently, the problems of existence of solutions and controllability of impulsive differential equations and differential inclusions have been extensively studied [16, 17]. Benedetti in  proved an existence result for impulsive functional differential inclusions in Banach spaces. Obukhovskii and Yao  considered local and global existence results for semilinear functional differential inclusions with infinite delay and impulse characteristics in a Banach space. Some existence results were obtained for certain classes of functional differential equations and inclusions in Banach spaces under assumption that the linear part generates an compact semigroup (see, e.g., [20–22]). The existence results of impulsive differential equations and inclusions have been generalized to stochastic differential equations with impulsive conditions [23, 24] and for stochastic impulsive differential inclusions [25–27].
We would like to mention that the impulsive effects also widely exist in fractional stochastic differential systems [28–30], and it is important and necessary to discuss the qualitative properties for stochastic fractional equations with impulsive perturbations with state-dependent delay. However, to the authors’ knowledge, no result has been reported on the existence problem of impulsive fractional stochastic differential inclusions with state-dependent delay and the aim of this paper is to fill this gap.
Motivated by this consideration, in this paper we will discuss the existence of mild solutions for a class of impulsive fractional stochastic differential inclusions with state-dependent delay in Hilbert spaces. Specifically, sufficient conditions for the existence are given by means of the nonlinear alternative of Leray-Schauder type for multivalued maps due to O'Regan.
2. Preliminaries and Basic Properties
In this section, we provide definitions, lemmas, and notations necessary to establish our main results. Throughout this paper, we use the following notations. Let be a complete probability space equipped with a normal filtration , satisfying the usual conditions (i.e., right continuous and containing all -null sets). We consider two real separable Hilbert spaces , with inner product , and norm , . Let be a -Wiener process defined on with the linear bounded covariance operator such that . Assume that there exists a complete orthonormal system in , a bounded sequence of nonnegative real numbers such that , , and a sequence of independent Brownian motions such that and , where is the sigma algebra generated by . Let denote the space of all bounded linear operators from to equipped with the usual operator norm . For we define If , then is called a -Hilbert-Schmidt operator. Let denote the space of all -Hilbert-Schmidt operators . The completion of with respect to the topology is induced by the norm where is a Hilbert space with the above norm topology. Let be a Banach space of all strongly measurable, square integrable, -valued random variables equipped with the norm , where denote the expectation with respect to the measure . Let be the Banach space of all continuous maps from to satisfying the condition . Let denote the family of all -measurable, -valued random variables .
The purpose of this paper is to investigate the existence of mild solutions for a class of impulsive fractional stochastic differential inclusions with state-dependent delay of the form where is the Caputo fractional derivative of order , ; takes the value in the separable Hilbert space ; and is the generator of an -resolvent operator family , The history , , , belongs to an abstract phase space defined axiomatically; , , , and , , are given functions to be specified later. Here , , and represent the right and left limits of at , respectively. The initial data is an -measurable, -valued random variable independent of with finite second moments.
Recall the following known definitions. For more details see .
Definition 1. The fractional integral of order with the lower limit for a function is defined as provided the right-hand side is pointwise defined on , where is the gamma function.
Definition 2. Riemann-Liouville derivative of order with lower limit zero for a function can be written as
Definition 3. The Caputo derivative of order for a function can be written as If , then Obviously, the Caputo derivative of a constant is equal to zero. The Laplace transform of the caputo derivative of order is given as
Definition 4 (see ). A closed and linear operator is said to be sectorial if there are constants , , , such that the following two conditions are satisfied:(i),(ii), .
Definition 5 (see ). Let be a closed and linear operator with the domain defined in a Banach space . Let be the resolvent set of . We say that is the generator of an -resolvent family if there exist and a strongly continuous function , where is a Banach space of all bounded linear operators from to and the corresponding norm is denoted by , such that and where is called the -resolvent family generated by .
Definition 6. Let be an -resolvent operator family on Banach space with generator . Then, the following assertions hold:(i) and for all and ,(ii)for all , and , ,(iii) and if and only if , ,(iv) is closed, densely defined.
Definition 7 (see ). Let be a closed and linear operator with the domain defined in a Banach space and . We say that is the generator of a solution operator if there exist and a strongly continuous function such that and where is called the solution operator generated by .
The concept of the solution operator is closely related to the concept of a resolvent family. For more details on -resolvent family and solution operators, we refer the reader to .
Definition 8. We say that a function is a normalized piecewise continuous function on if is piecewise continuous and left continuous on .
We denote by the space formed by normalized piecewise continuous, -adapted measurable processes from into . In particular, we introduce the space formed by -adapted measurable, -valued stochastic processes such that is continuous at , and exists for .
In this paper, we assume that is endowed with the norm . Then is a Banach space .
We denote by , , the function given by Moreover, for , we denote by , , the set . It is proved in  that is relatively compact in if, and only if, the set is relatively compact in , for every . The notation stands for the closed ball with center at and radius in .
Throughout this paper, we assume that the phase space is a seminormed linear space of -measurable functions mapping into and satisfying the following fundamental axioms .(i)If , , , is continuous on and , then for every the following conditions hold:(a) is in ;(b);(c), where is a constant; , is continuous, is locally bounded, and , , are independent of .(ii)For the function in (i), the function is continuous from into .(iii)The space is complete.The next result is a consequence of the phase space axioms. The reader can refer to  for the proof.
Lemma 9. Let be an -adapted measurable process such that the -adapted process and . Then where and .
In what follows, we use the notations for the family of all nonempty subsets of and denote Now, we briefly introduce some facts on multivalued analysis. For details, one can see .
A multivalued map is convex (closed) valued, if is convex (closed) for all . is bounded on bounded sets if is bounded in , for any bounded set of , that is; .
For and , we denote by and , and the Hausdorff metric by .
A multivalued map is called upper semicontinuous (u.s.c. for short) on if, for each , the set is a nonempty, closed subset of and if, for each open set of containing , there exists an open neighborhood of such that .
is said to be completely continuous if is relatively compact, for every bounded subset .
If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph; that is, , , imply .
A multivalued map is said to be measurable if for each , the function is measurable function on .
Definition 10 (see ). Let be a multivalued map. Then is called a multivalued contraction if there exists a constant such that for each The constant is called a contraction constant of .
Next, we mention the statement of a nonlinear alternative of Leray-Schauder type for multivalued maps due to O’Regan.
Lemma 11 (see ). Let be a Hilbert space with an open convex subset of and . Suppose that(a) has closed graph;(b) is a condensing map with a subset of a bounded set in hold. Then either(i) has a fixed point in or(ii)there exist and with .
3. The Mild Solution and Existence
Let be the set of selections of for each , and , .
Definition 12. An -adapted stochastic process is called a mild solution of the system (3) if , for every , , , the restriction of to the interval is continuous, and there exists such that satisfies the following integral equation: where , , and denotes the Bromwich path. is called the -resolvent family and is the solution operator generated by .
The following result on the operator appeared and is proved in .
Theorem 13. If and is a sectorial operator, then for any and , one has where is a constant depending only on and .
In order to establish the results, we first assume that the function is continuous from into and we impose the following additional hypotheses.
If and is a sectorial operator, then for and , If , and , we have
The function is continuous from to and there exists a continuous and bounded function such that for each .
The multivalued map is Carathédory; that is,(i) is measurable for each ;(ii) is upper semicontinuous (u.s.c.) for almost all ,and for each fixed , the set of selections of is nonempty.
There exists a positive integrable function such that uniformly in for a nonnegative constant , where
The function is continuous and there exists such that
The functions are completely continuous and there exist constants such that for every , .
Remark 14. The condition is frequently verified by continuous and bounded functions. For more details, see, for instance,  (Proposition ).
Lemma 15. Let such that and . If (H2) holds, then where .
Lemma 16 (see ). Let be a compact interval and a Hilbert space. Let be a multivalued map satisfying (H3) and a linear continuous operator from to . Then the operator is a closed graph in .
Theorem 17. Assume that (H1)–(H6) hold and , with for every . Then the problem (3) has a mild solution on provided that
Proof. Consider the space endowed with the uniform convergence topology and define the multivalued map by such that
where and such that and on .
We shall show that has a fixed point, which is then a mild solution for the problem (3). To this end we show that satisfies all the conditions of Lemma 11. For the sake of convenience, we divide the proof into several steps.
Step 1. We show that there exists an open set with for and . Let and , then there exists an such that From assumption (H4), it follows that there exist two nonnegative real numbers and such that for any and , From assumption (H6), we conclude that there exist positive constants (), such that, for , Let We have By assumption (H5), (27) and (30), we have for Similarly, for any , , we have Then, for all , we have where By Lemmas 9 and 15, it follows that , , , and For each , we have where Since , we have Applying Gronwall’s inequality, we get Therefore, Then, there exists such that . Set . Thus, from the choice of , there is no such that for .
Step 2. has a closed graph.
Let , , , and . It is easy to see that uniformly for as . We need to show that . Now means that there exists such that, for each , We must show that there exists such that, for each , Set , and .
We have, for every , Consider the linear continuous operator defined by From Lemma 16 and the definition of , it follows that is a closed graph operator, and, for every , .
Since and is a closed graph operator, then there exists such that, for every , Similarly, for any , , we have We must show that there exists such that, for every , For every , , we have where Now, for every , , we consider the linear continuous operator , From Lemma 16, it follows that is a closed graph operator, and for every , .
Since and is a closed graph operator, then there exists such that, for every