Existence and approximate boundary controllability of some partial functional integrodifferential equations with nonlocal initial condition in Banach spaces

Abstract

This work concerns the study of the existence of solutions and the approximate boundary controllability for some nonlinear partial functional integrodifferential equations with nonlocal initial condition arising in the modelling of materials with memory, in the framework of general Banach spaces. We give sufficient conditions that ensure the approximate boundary controllability of the system by supposing that its linear part is approximately controllable, admits a resolvent operator in the sense of Grimmer, and by making use of the Banach fixed-point Theorem and the continuity of the resolvent operator in the uniform norm-topology. As a result, we obtain a generalization of several important results in the literature, without assuming the compactness of the resolvent operator and the uniform boundedness of the nonlinear term. An example of applications is given for illustration. Keywords: approximate boundary controllability; semigroup; functional integrodifferential equation; nonlocal condition; resolvent operator; Banach fixed-point theorem.