BV continuous solutions of an evolution inclusion with maximal monotone operator and nonconvex-valued perturbation. Existence theorem
This well-written and illuminating paper is devoted to a thorough study of evolution inclusions governed by set-valued perturbations of time-dependent maximal monotone operators in a separable Hilbert space. These set-valued perturbations are assumed to have nonempty, closed and not necessarily convex point images, and the dependence of the maximal monotone operators on time is described with the help of the pseudo-distance introduced by \textit{A. A. Vladimirov} [Nonlinear Anal., Theory Methods Appl. 17, No. 6, 499--518 (1991; Zbl 0756.34064)]. This pseudo-distance is assumed to be of bounded variation as a function of time with an upper bound given by a nonatomic positive Radon measure \(\mu\). A solution of the evolution inclusion is a continuous function of bounded variation, the differential measure of which is absolutely continuous with respect to \(\mu\), so that the values of the density of this differential measure with respect to \(\mu\) belong to the right-hand side of the inclusion almost everywhere. Using a variety of auxiliary results, which are of independent interest, the author establishes the existence of such solutions and describes several properties of the solution sets.
scientific article; zbMATH DE number 7345309
scientific article