Topological structure of solution sets for control problems governed by semilinear fractional impulsive evolution equations with nonlocal conditions

Abstract

Abstract This article investigates the topological structural of the mild solution set for a control problem monitored by semilinear fractional impulsive evolution equations with nonlocal conditions. The $R_{\delta }$-property of the mild solution set is obtained by applying the measure of noncompactness and a fixed point theorem of condensing maps and a fixed point theorem of nonconvex valued maps. Then this result is applied to prove that the presented control problem has a reachable invariant set under nonlinear perturbations. The obtained results are also applied to characterize the approximate controllability of the presented control problem.