Nonautonomous differential equations and Lipschitz evolution operators in Banach spaces

Abstract

A new class of Lipschitz evolution operators is introduced and a characterization of continuous infinitesimal generators of such evolution operators is given. It is shown that a continuous mapping $A$ from a subset $omega$ of $[a,b) x X into X$, where $[a,b)$ is a real half-open interval and $X$ is a real Banach space, is the infinitesimal generator of a Lipschitz evolution operator if and only if it satisfies a sub-tangential condition, a general type of quasi-dissipative condition with respect to a metric-like functional and a connectedness condition. An application of the results to the initial value problem for the quasilinear wave equation with dissipation is also given.