Abstract
In this paper we consider a class of fractional nonlinear neutral stochastic evolution inclusions with nonlocal initial conditions in Hilbert space. Using fractional calculus, stochastic analysis theory, operator semigroups and Bohnenblust–Karlin’s fixed point theorem, a new set of sufficient conditions are formulated and proved for the existence of solutions and the approximate controllability of fractional nonlinear stochastic differential inclusions under the assumption that the associated linear part of the system is approximately controllable. An example is provided to illustrate the theory.
Highlights
The purpose of this paper is to show the existence of solutions and the approximate controllability of fractional nonlinear stochastic differential inclusion of the form (2.1) in a Hilbert space under simple and fundamental assumptions on the system operators, in particular that the corresponding linear system is approximate controllable
We have investigated the approximate controllability of class of fractional neutral stochastic evolution inclusion with nonlocal initial conditions in Hilbert space
Impulsive fractional differential equations and inclusions have become important in recent years as mathematical models of many phenomena in both physical and social sciences [38]
Summary
The fractional differential equations and inclusions have attracted many physicists, mathematicians and engineers and there was an intensive development of both theory and applications of fractional differential equations (see [19,28,30,32,38]). Balasubramaniam et al [3] investigated the approximate controllability of fractional impulsive integro-differential systems with nonlocal conditions in a Hilbert space. Yan and Lu [45] considered the approximate controllability of a class of fractional stochastic neutral integro-differential inclusions with infinite delay in Hilbert spaces. To the best of our knowledge, so far no work has been reported in the literature about the existence of solutions and the approximate controllability of fractional nonlinear stochastic differential inclusions with nonlocal conditions and infinite delay of the form (2.1). The purpose of this paper is to show the existence of solutions and the approximate controllability of fractional nonlinear stochastic differential inclusion of the form (2.1) in a Hilbert space under simple and fundamental assumptions on the system operators, in particular that the corresponding linear system is approximate controllable.
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