On second order discontinuous differential equations in Banach spaces

[摘要] 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 spaceE. 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 whenEis 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.

[学科分类] 应用数学