TWO-PARAMETER SECOND-ORDER DIFFERENTIAL INCLUSIONS IN HILBERT SPACES

Abstract

In a real Hilbert space H, let us consider the boundary-value problem −εu00(t) + µu0 (t) + Au(t) + Bu(t) 3 f(t), t ∈ [0, T]; u(0) = u0, u0 (T) = 0, where T > 0 is a given time instant, ε, µ are positive parameters, A : D(A) ⊂ H → H is a (possibly set-valued) maximal monotone operator, and B : H → H is a Lipschitz operator. In this paper, we investigate the behavior of the solutions to this problem in two cases: (i) µ > 0 fixed, 0 < ε → 0, and (ii) ε > 0 fixed and 0 < µ → 0. Notice that if µ = 1 and ε is a positive small parameter, the above problem is a Lions-type regularization of the Cauchy problem u 0 (t) + Au(t) + Bu(t) 3 f(t), t ∈ [0, T]; u(0) = u0, which was recently studied by L. Barbu and G. Moro¸sanu [Commun. Contemp. Math. 19 (2017)]. Our abstract results are illustrated with examples related to the heat equation and the telegraph differential system.