Comparison theorems for evolution inclusions with maximal monotone operators. $L^2$-theory

Abstract

An evolution inclusion with time-dependent family of maximal monotone operators in considered in a separable Hilbert space. If the elements with minimum norm of the family of maximal monotone operators satisfy certain growth conditions, then the domains of definition of this family are closed convex sets. Hence the sweeping process is well defined, whose values are the normal cones of the domains of definition of maximal monotone operators. It is shown that if the sweeping process has a solution for each single-valued perturbation from the space of integrable functions, then the evolution inclusion with the maximal monotone operators and single-valued perturbations from the space of integrable functions is also solvable. Quite general conditions in terms of the properties of the family of maximal monotone operators that ensure the existence of solutions for the sweeping process are presented. All results obtained and the approach presented are new. They are used to prove an existence theorem for evolution inclusions with multivalued perturbations, whose values are closed nonconvex sets. Bibliography: 19 titles.