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 - d u d t (t) ∈ A (t) u (t) + f (t , u (t)) in a new setting. For each t, A (t) : D (A (t)) → 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 ↦ 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. [ABSTRACT FROM AUTHOR]