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

In this paper we prove a result of the Trotter–Kato type in the weak topology. Let {Aε}ε>0 be a family of quasi m-accretive linear operators on a Hilbert space X and let us denote by the resolvent of Aε. Under certain conditions, the result states that if for any x∈X and k=1,2,…, the sequence converges weakly to k(Jλ)x as ε→0, where Jλ is the resolvent of a linear quasi m-accretive operator A on X, then the sequence of the semigroups generated by −Aε 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.