Uniform stability of a family of resolvent operators in Hilbert spaces

Abstract

In this paper, two main results on the uniform stability of a resolvent family $$\{R_{h}(t)\}_{t\ge 0}$$ , depending on a parameter h are presented. First, we discuss a GGP type theorem on the resolvent family $$\{R_{h}(t)\}_{t\ge 0}$$ and give some sufficient conditions on the uniform stability of $$\{R_{h}(t)\}_{t\ge 0}$$ . Then we prove that under some suitable conditions, the weak $$L^{p}$$ -stability of $$\{R_{h}(t)\}_{t\ge 0}$$ implies its uniform stability. Our results both essentially generalize previous work on the uniform stability of a family of $$C_{0}$$ -semigroups $$\{T_{h}(t)\}_{t\ge 0}$$ and a resolvent family $$\{R(t)\}_{t\ge 0}$$ , without depending on the parameter h. Examples are also given to illustrate our results.