An Existence Result for Impulsive Stochastic Functional Differential Equations with Multiple Delays

In this paper we consider Impulsive stochastic neutral functional differential equations with multiple delays. By using Schaefer’s fixed point theorem, we prove the existence of solutions for stochastic differential equations with impulses.


Introduction
The theory of impulsive differential equations is an important area of scientific activity. Many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly. These short term perturbations act instantaneously, that is in the form of impulses. For example, that many biological phenomena involving thresholds, optimal control models in economics and frequency modulated systems, do exhibit impulsive effects. So the impulsive differential equations appear as a natural description of observed evolution phenomena of several real world problems. Existence of solutions of impulsive differential equations has been studied by many authors. If the impulses are random the solution becomes a stochastic process. Existence of solutions of differential equations with random impulses have been studied by many authors [1][2][3].
Furthermore, besides impulsive effects, stochastic effects likewise exist in real systems. There is a wide range of interesting process in robotics, economics and biology that can be described as differential equations with non-deterministic dynamics such phenomena are described by stochastic differential equations. The solution of stochastic differential equation is a stochastic process. However the solution of differential equation with random impulses is different from the solution of stochastic differential equations. Existence, Uniqueness and qualitative analysis of solutions of stochastic differential equations have discussed by several authors [4,5].
Since both impulsive and stochastic effects exist it is very difficult to investigate the existence of solution of impulsive stochastic differential equations. In [6] Anguraj and Vinodkumar discussed the existence, uniqueness and stability of impulsive stochastic semi linear neutral functional differential with infinite delays. Lakrib [7] discussed about the existence results for impulsive neutral functional differential equations with multiple delays. Based on the existing literature, stochastic impulsive differential equations involved mainly on controlability and stability. To the best of our knowledge, there is no work reported on impulsive stochastic differential equations with multiple delays. The purpose of this paper is to discuss about the existence results of impulsive stochastic neutral functional differential equations with multiple delays. Our approach is based on Schaefer's fixed point theorem.
In this paper we study the existence results for stochastic impulsive differential equations with multiple delays where E denotes the expectation of stochastic process [8][9][10]. The initial Let PC(J, R n ) the space of piecewise continuous functions x: J → R n such that x is continuous everywhere except for t=t k at which ( )

Definition 2.2:
(Schaefer fixed point theorem). Let X be a normed linear space and let Γ: X → X be a completely continuous map, that is, it is a continuous mapping which is compact on each bounded subset of X. If the set ζ={x ∈∈ X: λx=Γx for some λ>1} is bounded, then Γ has a fixed point [11].
Hypotheses H 1 : The function f, g: J × R n → R n is such that H 2 : The function a: J × R n → R n is Caratheodory, that is, a t x is continuous for a.e t ∈ J.
H3: There exists a function q ∈ L 1 (J, R n ) with q(t)>0 for a.e t ∈ J and a continuous non decreasing function ψ : [0, ∞) → [0, ∞) such that for a.e t ∈ J and each x ∈ R n with   for all x ∈ B, that is Γ is bounded on bounded subsets of Ω.
Let B be as in Step 1 and x ∈ B. Let t and 0 h ≠ be such that t, t + h ∈ J\{t 1 , t 2 , .. The right hand side of the above inequality tends to zero as h → 0. The equicontinuity on J 0 follows from the uniform continuity of φ on this interval.
Step 3: Now we show that Γ is continuous Let {x n }⊂ Ω be a sequence such that x n → x. We will show that Γx n → Γx.
For t ∈ J Using H 3 it can be easily shown that the function is Lebesgue integrable. By the continuity of f and I k , k=1, 2, ...m and the dominated convergence theorem, the right hand side of inequality (3.2) tends to zero as n → ∞, which completes the proof that Γ is continuous [15][16][17][18].
As a sequence of steps 1 to 3, together with the Arzela-Ascoli theorem, we conclude that Γ is completely continuous.
To complete the proof of the theorem, it suffices to prove the following step.