- Open Access
Existence results for impulsive neutral stochastic functional integro-differential inclusions with infinite delays
© Park and Jeong; licensee Springer. 2014
Received: 11 September 2013
Accepted: 17 December 2013
Published: 15 January 2014
In this paper, we prove the existence of mild solutions for a class of impulsive neutral stochastic functional integro-differential inclusions with infinite delays in Hilbert spaces. The results are obtained by using the fixed-point theorem for multi-valued operators due to Dhage. An example is provided to illustrate the theory.
where the state takes values in a separable real Hilbert space H with inner product and norm , A is the infinitesimal generator of a compact analytic resolvent operator , , in the Hilbert space H. Suppose that is a given K-valued Brownian motion or Wiener process with a finite trace nuclear covariance operator and denotes the space of all bounded linear operators from K into H. Further , , and are given functions, where , is the family of all nonempty subsets of and denotes the space of all Q-Hilbert-Schmidt operators from K into H, which will be defined in Section 2. Here, () are bounded functions. Furthermore, the fixed times satisfies , and denote the right and left limits of at . represents the jump in the state x at time , where determines the size of jump. The histories , , which are defined by setting , belong to the abstract phase space , which will be defined in Section 2. The initial data is an -measurable, -valued random variables independent of with finite second moment.
The theory of impulsive integro-differential inclusions has become an active area of investigation due to their applications in the fields such as mechanics, electrical engineering, medicine biology, ecology, and so on (see [1, 2] and references therein).
The existence of impulsive neutral stochastic functional integro-differential equations or inclusions with infinite delays have attracted great interest of researchers. For example, Lin and Hu  consider the existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions. Hu and Ren  studied the existence results for impulsive neutral stochastic functional integro-differential equations with infinite delays.
Motivated by the previous mentioned papers, we prove the existence of solutions for impulsive neutral stochastic functional integro-differential inclusions with infinite delays.
If , then ψ is called a Q-Hilbert-Schmidt operator. Let denote the space of all Q-Hilbert-Schmidt operator . The completion of with respect to the topology induced by the norm , where is a Hilbert space with the above norm topology.
Let be the infinitesimal generator of a compact, analytic resolvent operator , . Let . Then it is possible to define the fractional power for as a closed linear operator with its domain being dense in H. We denote by the Banach space endowed with the norm , which is equivalent to the graph norm of .
Lemma 2.1 ()
If , the and the embedding is continuous and compact whenever the resolvent operator of A is compact.
- (ii)For every , there exists a positive constant such that
Lemma 2.2 ()
A multi-valued mapping is called upper semicontinuous (u.s.c) if for any , the set is a nonempty closed subset of H and if for each open set G of H containing , there exists an open neighborhood N of x such that . Γ is said to be completely continuous if is relatively compact for every bounded subset of . If the multi-valued mapping Γ is completely continuous with nonempty compact values, then Γ is u.s.c. if and only if Γ has a closed graph, i.e., , , imply .
is measurable for each ,
is u.s.c. for almost all and ,
- (iii)for each , there exists such that
for all and for a.e. .
The following lemma is crucial in the proof of our main result.
Lemma 2.3 ()
Theorem 2.1 ()
is a contraction,
is u.s.c. and completely continuous.
the operator inclusion has a solution for , or
the set is unbounded.
Lemma 2.4 ()
for every and every such that and is the Gamma function.
3 Main result
Let . First, we present the definition of the mild solution of problem (1.1)-(1.3).
is measurable and -adapted for each ,
- (iii)has càdlàg paths on a.e. and there exists a function such that
on satisfies .
Now, we assume the following hypotheses:
is measurable for each ;
is continuous for almost all ;
- (iii)There exists a constant such thatfor all , andfor almost all , where , is continuous and increasing withand(H5) The multi-valued mapping is an -Carathéodory function that satisfies the following conditions:
- (i)For each , the function is u.s.c. and for each fixed , the function is measurable. For each , the set
- (ii)There exists a positive function such that
where . In what follows, we show that the operators and satisfy all the conditions of Theorem 2.1.
Lemma 3.1 Assume that the assumptions (H1)-(H6) hold. Then is a contraction and is u.s.c. and completely continuous.
Proof We give the proof in several steps:
Step 1. is a contraction.
and so is a contraction.
Now, we show that the operator is completely continuous.
Step 2. is convex for each .
Step 3. maps bounded sets into bounded sets in .
Thus, for each , we get .
Step 4. maps bounded sets into equicontinuous sets of .
The right-hand side of the above inequality tends to zero as with ε sufficiently small, since is strongly continuous and the compactness of for implies the continuity in the uniform operator topology. Thus, the set is equicontinuous. Here we consider the case , since the case or is simple.
Step 5. maps into a precompact set in H.
Therefore letting , we can see that there are relative compact sets arbitrarily close to the set . Thus, the set is relatively compact in H. Hence, the Arzelá-Ascoli theorem shows that is a compact multi-valued mapping.
Step 6. has a closed graph.
Let , and . We prove that .
Therefore has a closed graph and is u.s.c. This completes the proof. □
Lemma 3.2 Assume that the assumptions (H1)-(H2) hold. Then there exists a constant such that for all , where K is depends only on b and the functions ψ and .
which indicates that . Thus, there exists a constant K such that , . Furthermore, we see that , . □
Theorem 3.1 Assume that the assumptions (H1)-(H6) hold. The problem (1.1)-(1.3) has at least one mild solution on J.
where is a constant in Lemma 3.2. This show that G is bounded on J. Hence from Theorem 2.1 there exists a fixed point for Φ on , which is a mild solution of (1.1)-(1.3) on J. □
4 An example
Then (4.1)-(4.4) can be rewritten as the abstract form as the system (1.1)-(1.3). If we assume that (H2)-(H6) are satisfied, then the system (4.1)-(4.4) has a mild solution on .
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