Investigating the Existence Results for Hilfer Fractional Stochastic Evolution Inclusions of Order $$1<{\mu }<2$$

Abstract

The objective of this article is to investigate the issue of existence results for Hilfer fractional stochastic differential inclusions of order \(1<\mu <2\) in Hilbert spaces. Our discussion is based on fractional calculus, multivalued analysis, sine and cosine operators, and Bohnenblust–Karlin’s fixed point theorem. At first, we investigate the existence of a mild solution for the Hilfer fractional stochastic differential system of order \(1<\mu <2\). After that, we developed our system with Sobolev-type, and we provided the existence results of a mild solution for the considered system. Then, the ideas of nonlocal conditions are applied in the Sobolev-type Hilfer fractional stochastic system. Finally, an example is offered in order to illustrate the effectiveness of the main theory.