On second order discontinuous differential equations in Banach spaces (Q1317193)

This article deals with the Cauchy problem \(x''= Ax+ g(t,x,x')\), \(x(0)= x_ 0\), \(x'(0)= x_ 1\) in which the linear operator \(A\) is the infinitesimal generator of a strongly continuous cosine family in a Banach space \(\mathbb{E}\), the nonlinearity \(g(t,x,x')\) is a Carathéodory function. The existence, uniqueness and continuous dependence on initial data theorems for weak solutions are given; then in the particular case when \(g(t,x,x')\) does not depend on \(x'\) and \(\mathbb{E}\) is an order Banach space the existence and some properties of extremely mild solutions are described. At the end of the article some applications to quasilinear hyperbolic differential equations are considered.