On second order discontinuous differential equations in Banach spaces

S Heikkilä,S Leela

doi:10.1155/s1048953393000279

Abstract

In this paper we study a second order semilinear initial value problem (IVP), where the linear operator in the differential equation is the infinitesimal generator of a strongly continuous cosine family in a Banach space E. We shall first prove existence, uniqueness and estimation results for weak solutions of the IVP with Carathéodory type of nonlinearity, by using a comparison method. The existence of the extremal mild solutions of the IVP is then studied when E is an ordered Banach space. We shall also discuss the dependence of these solutions on the data. A characteristic feature of the results concerning extremal solutions is that the nonlinearity is not assumed to be continuous in any of its arguments. Moreover, no compactness conditions are assumed. The obtained results are then applied to a second order partial differential equation of hyperbolic type.

Highlights

INTRODUCTIONIn this paper we consider the second order semilinear initial value problem x" = Ax + g(t,x,x’), (o)-
In this paper we consider the second order semilinear initial value problem x" = Ax + g(t,x,x’), (o)- (1.1)Printed in the U.S.A. (C) 1993 The Society of Applied Mathematics, Modeling and SimulationS
Remark 3.2" The hypotheses given for q in (91) ensure ha the minimal solution u of he initial value problem (IVP) (3.5) and its derivative u’ are nondecreasing wila respect go u0 and u, and hat boh of them tend to zero uniformly over t J as

Summary

INTRODUCTION

In this paper we consider the second order semilinear initial value problem x" = Ax + g(t,x,x’), (o)-. The existence of mild solutions of (1.1) is considered in [16] when g is continuous. Uniqueness and estimation results for weak solutions of the initial value problem (IVP) (1.1), by using a comparison method and assuming that g satisfies Carathodory conditions. The existence of the extremal mild solutions of (1.1) is studied when E is an ordered Banach space, and when g does not depend on x’. A characteristic feature of the results concerning extremal solutions is that g is no assumed to be continuous in any of its arguments. The obtained results are applied to a second order partial differential equation of hyperbolic type

PRELIMINARIES

ON EXTREMAL MILD SOLUTIONS

ON THE EXISTENCE OF MINIMAL OR MAXIMAL MILD SOLUTION

DEPENDENCE ON THE DATA