A Trotter–Kato type theorem in the weak topology and an application to a singular perturbed problem

Abstract

In this paper we prove a result of the Trotter–Kato type in the weak topology. Let { A ε } ε > 0 be a family of quasi m-accretive linear operators on a Hilbert space X and let us denote by J λ ε the resolvent of A ε . Under certain conditions, the result states that if for any x ∈ X and k = 1 , 2 , … , the sequence ( J λ ε ) k x converges weakly to ( J λ ) k x as ε → 0 , where J λ is the resolvent of a linear quasi m-accretive operator A on X, then the sequence of the semigroups generated by − A ε tends weakly to the semigroup generated by − A, uniformly with respect to t on compact intervals. The result is different from other results of the same type (see e.g., Yosida (1980) [9, p. 269]) and gives an answer to an open problem put in Eisner and Serény (2010) [3]. It is finally applied to compare the asymptotic behavior of a singular perturbation problem associated to a first order hyperbolic problem with diffusion.