On mild solutions of first order discontinuous semiliear differential equations in banach spaces

Abstract

In this paper we shall consider a first order semilinear initial value problem (IVP) when the evolution term in the differential equation is the generator of a strongly contionuous family of bounded linear operators in Banach space E. We shall first study existence, Uniqueness and estimation of mild solutions of the IVP when the nonlinearity of the differential equation is for Carathéodory type. By assuming that E is ordrered, we shall then derive existence and comparison results for extremal solutions the IVP in the case when the nonlinearity can be discontinuous in all its arguments. The obtained results are then applied to initial value problems of first and second order partial differential equations. No kind of compactness conditions are assumed.

Keywords:

• semilinear initial value problems
• discontinuous nonlinearities
• mild solutions

AMS:

• 34G20
• 47H07
• 35D05