Second-order differential inclusions with two small parameters

Consider in a real Hilbert space H the following problem, denoted (P-epsilon mu), {-epsilon u ''(t) + mu u'(t + Au(t) + Bu(t) (sic) f(t), 0 < t < T, u(0) = u(0), u'(T) = 0, where T > 0 is a given time instant, epsilon > 0, mu >= 0 are small parameters, A : D(A) subset of H -> H is a maximal monotone operator (possibly multivalued), and B : H -> is a Lipschitz operator (or monotone and Lipschitz on bounded sets). Consider also the following reduced problem, denoted (P-mu), {mu u'(t) + Au(t) + Bu(t) (sic) f(t), 0 < t < T, u(0) = u(0), where mu > 0, as well as the algebraic equation (inclusion), Au(t) + Bu(t) (sic) f(t), 0 <= t <= T. (E-00) In this paper we are concerned with the following topics: (a) existence and uniqueness of solutions to the above problems and to equation (E-00); (b) continuity of the solution to problem (P-epsilon mu) with respect to epsilon > 0 and mu >= 0; (c) convergence of the solution of problem (P-epsilon mu) to the solution of problem (P-mu 0), as epsilon -> 0(+) and mu -> mu(0), where mu(0) is a fixed positive number; (d) convergence of the solution of problem (P-epsilon mu) to the solution of the equation Au + Bu (sic) f (t) as epsilon -> 0(+) and mu -> 0(+); (e) applications.