Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces

Martina Pavlačková,Valentina Taddei

doi:10.3390/math10040672

Abstract

In this paper, the existence of a mild solution to the Cauchy problem for impulsive semilinear second-order differential inclusion in a Banach space is investigated in the case when the nonlinear term also depends on the first derivative. This purpose is achieved by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables obtaining the result under easily verifiable and not restrictive conditions on the impulsive terms, the cosine family generated by the linear operator and the right-hand side while avoiding any requirement for compactness. Firstly, the problems without impulses are investigated, and then their solutions are glued together to construct the solution to the impulsive problem step by step. The paper concludes with an application of the obtained results to the generalized telegraph equation with a Balakrishnan–Taylor-type damping term.

Highlights

The theory of impulsive differential equations has undergone considerable development in recent years since it serves as realistic mathematical descriptions of real situations containing abrupt changes, as well as other phenomena, such as harvesting or treatment of diseases
The investigation of impulsive differential equations and inclusions in infinite-dimensional Banach spaces has been undertaken by a lot of authors starting from the end of the last century—see, e.g., [9,10,11,12] and the references therein for the problems governed by first-order impulsive equations
The aim of this paper is to study the existence of mild solutions to second-order multivalued impulsive problems in infinite-dimensional Banach spaces without reducing the second-order problem to a first-order one

Summary

Introduction

The theory of impulsive differential equations has undergone considerable development in recent years since it serves as realistic mathematical descriptions of real situations containing abrupt changes, as well as other phenomena, such as harvesting or treatment of diseases. Several authors have been studying the semilinear second-order problems in Banach spaces—see, e.g., [17,18,19,20] or [21] In these papers, the existence of mild solutions to the second-order (impulsive) initial value problems in the case when the right-hand side (r.h.s.) does not depend on the first derivative of a solution was investigated. A lot of papers appeared, where the same technique was applied to study first- and second-order equations and inclusions (of functional and neutral types), fractional equations, controllability problems, and so on It was used in [36] to obtain the existence of a solution for a semilinear second-order equation in a Banach space with the r.h.s. not depending on the first derivative.

Preliminaries

Existence of a Mild Solution for the Cauchy Problem without Impulses

Existence of a Mild Solution for the Impulsive Problem

An Application

Conclusions and Future Studies