S. Heikkilä, S. Leela, "On second order discontinuous differential equations in Banach spaces", International Journal of Stochastic Analysis, vol. 6, Article ID 756762, 21 pages, 1993. https://doi.org/10.1155/S1048953393000279
On second order discontinuous differential equations in Banach spaces
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 . 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 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.
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