Results on Neutral Partial Integrodifferential Equations Using Monch-Krasnosel’Skii Fixed Point Theorem with Nonlocal Conditions

Chokkalingam Ravichandran,Natarajan Valliammal,Kasilingam Munusamy,Kottakkaran Sooppy Nisar

doi:10.3390/fractalfract6020075

Abstract

In this theory, the existence of a mild solution for a neutral partial integrodifferential nonlocal system with finite delay is presented and proved using the techniques of the Monch–Krasnosel’skii type of fixed point theorem, a measure of noncompactness and resolvent operator theory. For this work, we have introduced some sufficient conditions to confirm the existence of the neutral partial integrodifferential system. An illustration of the derived results is offered at the end with a filter system corresponding to our existence result.

Highlights

We establish the solution of the existence of Equations (1) and (2) with finite delay dD(v, zv ) = AD(v, zv ) + dv Z vPublished: 31 January 2022Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims inH (v − s)D(s, zs )ds+ φ v, zv, h(v, s, zs )ds, for v ∈ I = [0, b], z0 = φ + g(z) = C ([−r, 0]; X ). (1) (2)published maps and institutional affiliations
In [9], the authors proved the solutions of neutral functional integrodifferential equations with an initial condition in finite delay, and in [4], the authors proved the existence of the mild solution for a class of neutral partial integrodifferential equations using resolvent operator theory and measure of noncompactness and proved the existence using the Monch–Krasnosel’skii type of fixed point theorem with initial conditions
Let S be a bounded subset of X and let φ be a function defined on S called the measure of noncompactness (MNC), such that φ(S) = 0 if and only if S is relatively compact

Summary

Introduction

We establish the solution of the existence of Equations (1) and (2) with finite delay d. Among many other applications, semigroup theory has been widely used in the study of control and stability of systems governed by differential equations on Banach space It has been discussed by many authors for A is densely defined. In [9], the authors proved the solutions of neutral functional integrodifferential equations with an initial condition in finite delay, and in [4], the authors proved the existence of the mild solution for a class of neutral partial integrodifferential equations using resolvent operator theory and measure of noncompactness and proved the existence using the Monch–Krasnosel’skii type of fixed point theorem with initial conditions. The contribution of this article is extended from the neutral integrodifferential equation, including an integral term in functional and taking nonlocal conditions with finite delay.

Results on Measure of Noncompactness

Important Results on Fixed Point Theorem

Results on Existence

Application I

Application II—Filter System

Conclusions