Existence Results for Neutral Functional Integrodifferential Equations With Infinite Delay in Banach Spaces

Abstract

This paper is concerned with the existence of mild solutions for partial neutral functional integrodifferential equations with infinite delay in Banach spaces. The results are obtained by using resolvent operators and Krasnoselski-Schaefer type fixed point theorem. An example is provided to illustrate the results.

Minimum Energy Problem in the Sense of Caputo for Fractional Neutral Evolution Systems in Banach Spaces

“…with A 2 and A 3 n × n matrices whose elements belong to L 2 (−1, 0); B is a constant n × r matrix; and the control u is an L 2 -function [1]. Nowadays, many researchers have investigated neutral differential equations in Banach spaces [2][3][4]. This interest is explained by the fact that neutral-argument differential equations have interesting applications in real-life problems: they appear, e.g., while modeling networks containing lossless transmission lines or in super-computers.…”

“…Using the same techniques as the previous authors, Qin et al have studied the controllability and optimal control of fractional dynamical systems of order 1 < q < 2 in Banach spaces [24]. Yan and Jia used stochastic analysis theory and fixed-point theorems with the strongly continuous α-order cosine family to study an optimal control problem for a class of stochastic fractional equations of order α ∈ (1,2] in Hilbert spaces [25]. In 2021, Zhou and He obtained, via the contraction principle and Shauder's fixed-point theorem, a set of sufficient conditions for the exact controllability of a class of fractional systems [26].…”

“…Using the probability density function and its Laplace transform [34] (see also [35,36]), we recall the definition of a mild solution for system (2).…”