Investigating the Existence Results for Hilfer Fractional Stochastic Evolution Inclusions of Order 1<μ<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<{\mu }<2$$\end{document}

The objective of this article is to investigate the issue of existence results for Hilfer fractional stochastic differential inclusions of order 1<μ<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<\mu <2$$\end{document} 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<μ<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<\mu <2$$\end{document}. 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.