Approximate controllability for finite delay nonlocal neutral integro-differential equations using resolvent operator theory

In this paper, our purpose is to study the approximate controllability of abstract nonlocal neutral integro-differential equations with finite delay in a Hilbert space using the resolvent operator theory. We derive a variation of parameters formula for representing a solution of the given neutral integro-differential system in the form of resolvent operators and then define a mild solution of the system. We also study the existence of a mild solution of the system with the help of resolvent operator theory. The fractional power 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, resolvent operator theory, semigroup theory and Krasnoselskii’s fixed point theorem are used to prove the approximate controllability of the system. Finally, we illustrate our results with the help of an example.