TWO-PARAMETER SECOND-ORDER DIFFERENTIAL INCLUSIONS IN HILBERT SPACES.

In a real Hilbert space H, let us consider the boundary-value problem -εu"(t) + βu'(t) + Au(t) + Bu(t) ∋ f(t), t ∈ [0; T]; u(0) = u0, u'(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 u0(t) + Au(t) + Bu(t) ∋ f(t); t 2 [0; T]; u(0) = u0, which was recently studied by L. Barbu and G. Moroşanu [Commun. Contemp. Math. 19 (2017)]. Our abstract results are illustrated with examples related to the heat equation and the telegraph differential system. [ABSTRACT FROM AUTHOR]