Impulsive Evolution Equations with Causal Operators

Tahira Jabeen,Donal O’Regan,Ravi P Agarwal,Vasile Lupulescu

doi:10.3390/sym12010048

Abstract

In this paper, we establish sufficient conditions for the existence of mild solutions for certain impulsive evolution differential equations with causal operators in separable Banach spaces. We rely on the existence of mild solutions for the strongly continuous semigroups theory, the measure of noncompactness and the Schauder fixed point theorem. We consider the impulsive integro-differential evolutions equation and impulsive reaction diffusion equations (which could include symmetric kernels) as applications to illustrate our main results.

Highlights

Let R be the set of real numbers and let R+ be the set of non-negative real numbers
In this paper we study the class of impulsive evolution equations involving causal operators
In the result we show that, under the conditions (H1) and (H2), we can construct a sequence of successive approximations which converges to a mild solution of (4)

Summary

Introduction

Let R be the set of real numbers and let R+ be the set of non-negative real numbers. Let E be a real. We denote by C ([0, T ], E) the Banach space of continuous functions from [0, T ] into E endowed with the norm ku(·)k = sup ku (t)k. Theoretical aspects regarding existence, stability or periodicity of solutions of differential equations with causal operators in finite or infinite dimensional spaces were presented in a series of works, such as: [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39].

Preliminaries

Existence Result

Applications

Consider the following impulsive reaction diffusion equation

Conclusions