Approximate Controllability of Finite Delay Fractional Functional Integro-Differential Equations with Nonlocal Condition

In this paper, we study the sufficient conditions for the approximate controllability of finite delay fractional functional integro-differential equations with nonlocal condition in a Hilbert space. We use the theory of fractional calculus, semigroup theory, 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}-norm, fractional power theory and Krasnoselskii’s fixed point theorem to obtain the results under the assumption that the corresponding linear system is approximate controllable. An example is presented to illustrate the main result.