Abstract
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.
Highlights
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 [1,2,3]
We study the existence of mild solutions for a class of impulsive fractional stochastic differential inclusions with state-dependent delay
There has been a great deal of interest in the solutions of fractional differential equations in analytical and numerical senses
Summary
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 [1,2,3]. 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. 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 statedependent delay in Hilbert spaces. Chinese Journal of Mathematics conditions for the existence are given by means of the nonlinear alternative of Leray-Schauder type for multivalued maps due to O’Regan
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