On second order discontinuous differential equations in Banach spaces. (1993)

• Record Type: Journal Article
• Authors: Heikkilä Heikkilä, S. S.; Leela Leela, S. S.
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 6:Number 4(1993)
• Page Start: 303
• Page End: 323
• Publication Date: 1993
• Subjects: second order semilinear initial value problems -- discontinuous nonlinearities -- weak and mild solutions; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953393000279
• Languages: English
• ISSNs: 1048-9533

Abstract

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.