Fractional Stochastic Evolution Equations In Infinite Dimensional Spaces

Abstract

The research reported in this thesis deals with the problem of fractional stochastic differential equations and inclusions in Hilbert spaces. We have discussed the existence and uniqueness result for an impulsive fractional stochastic evolution equation involving Caputo fractional derivative and fractional partial neutral stochastic functional integro-differential inclusions with state-dependent delay and analytic resolvent operators. Sufficient conditions for the existence are established by using the nonlinear alternative of Leray-Schauder type for multivalued maps due to O’Regan and the fractional power of operators. The main results are obtained by means of the theory of operators semi-group, fractional calculus, fixed point technique and stochastic analysis theory and methods adopted directly from deterministic fractional equations.

We also investigate the approximate controllability for a class of fractional neutral stochastic functional integro-differential inclusions involving the Caputo derivative in Hilbert spaces. A new set of sufficient conditions are formulated and proved for the approximate controllability of fractional stochastic integro-differential inclusions under the assumption that the associated linear part of system is approximately controllable. The main techniques rely on the fractional calculus, operator semigroups and Bohnenblust-Karlin’s fixed point theorem. An example is given to illustrate the obtained theory.