Second-order differential inclusions with two small parameters

Abstract

Consider in a real Hilbert space H the following problem, denoted (Pɛμ), −ɛu′′(t)+μu′(t)+Au(t)+Bu(t)∋f(t),0<t<T,u(0)=u0,u′(T)=0,where T>0 is a given time instant, ɛ>0,μ≥0 are small parameters, A:D(A)⊂H→H is a maximal monotone operator (possibly multivalued), and B:H→H is a Lipschitz operator (or monotone and Lipschitz on bounded sets). Consider also the following reduced problem, denoted (Pμ), μu′(t)+Au(t)+Bu(t)∋f(t),0<t<T,u(0)=u0,where μ>0, as well as the algebraic equation (inclusion), Au(t)+Bu(t)∋f(t),0≤t≤T.(E00)In this paper we are concerned with the following topics: (a) existence and uniqueness of solutions to the above problems and to equation (E00); (b) continuity of the solution to problem (Pɛμ) with respect to ɛ>0 and μ≥0; (c) convergence of the solution of problem (Pɛμ) to the solution of problem Pμ0, as ɛ→0+ and μ→μ0, where μ0 is a fixed positive number; (d) convergence of the solution of problem (Pɛμ) to the solution of the equation Au+Bu∋f(t) as ɛ→0+ and μ→0+; (e) applications.