Local and Global Existence of Mild Solution for Impulsive Fractional Stochastic Differential Equations

In this paper, the local and global existence of mild solutions are studied for impulsive fractional semilinear stochastic differential equation with nonlocal condition in a Hilbert space. The results are obtained by employing fixed-point technique and solution operator. In many existence results for stochastic fractional differential systems, the value of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is restricted to 12<α≤1;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2} < \alpha \le 1;$$\end{document} the aim of this manuscript is to extend the results which are valid for all values of α∈(0,1).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,\,1).$$\end{document} An example is provided to illustrate the obtained theoretical results.