Perturbed evolution problems with absolutely continuous variation in time and applications

This paper is devoted to the existence and uniqueness of absolutely continuous solutions in evolution problems of the form \(-\frac{\mathrm{{d}}u}{\mathrm{{d}}t}(t) \in A(t)u(t) + f(t, u(t))\) in a new setting. For each t, \(A(t) : D(A(t)) \rightarrow 2 ^H\) is a maximal monotone operator in a Hilbert space H and the perturbation f is separately integrable on [0, T] and separately Lipschitz on H. It is assumed that \(t \mapsto A(t)\) has absolutely continuous variation, in the sense of Vladimirov’s pseudo-distance. Some extensions are also provided allowing new applications of our results to a larger number of problems modeled by maximal monotone operators. In particular, we solve evolution problems with multivalued upper semicontinuous perturbations, by using a fixed point theorem.