Journal of Applied Mathematics and Stochastic Analysis
Volume 6 (1993), Issue 4, Pages 303-323
doi:10.1155/S1048953393000279
On second order discontinuous differential equations in Banach spaces
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.
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.