Density and co-density of the solution set of an evolution inclusion with maximal monotone operators

An evolution inclusion defined on a separable Hilbert space and containing a time-dependent maximal monotone operator and a perturbation is considered in the paper. The perturbation is given by the sum of two terms. The first term is a demicontinuous single-valued operator with a time-dependent domain. It is measurable along a continuous function valued in the domain of the maximal monotone operator and satisfies nonlinear growth conditions. The sum of this operator with the identity operator multiplied by a square integrable nonnegative function is a monotone operator. The second term is a measurable multivalued mapping with closed, nonconvex values satisfying conventional Lipschitz conditions and linear growth conditions. Along with this (original) inclusion we introduce an alternative (relaxed) inclusion by convexifying the original multivalued perturbation. We prove the existence of solutions for the original inclusion and establish the density (relaxation theorem) and co-density of the solution set of the original inclusion in the solution set of the relaxed inclusion. Also, we give necessary and sufficient conditions for the closedness of the solution set of the original inclusion in the case when the values of the perturbation are closed nonconvex sets. For the class of perturbations we consider, all our results are completely new.