Journal of Applied Mathematics and Stochastic Analysis
Volume 6 (1993), Issue 4, Pages 303-323

On second order discontinuous differential equations in Banach spaces

S. Heikkilä1 and S. Leela2

1University of Oulu, Department of Mathematics, Oulu 90570 , Finland
2SUNY College at Geneseo, Department of Mathematics, Geneseo 14454, NY, USA

Received 1 September 1993; Revised 1 December 1993

Copyright © 1993 S. Heikkilä and S. Leela. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.