Uniform stability of a family of resolvent operators in Hilbert spaces (Q2031450)

In this paper, the authors consider two results on the uniform stability of a resolvent family \(\{R_h(t)\}_{t\geq 0}\), depending on a parameter \(h\). The first result is an extension of the Gearhart-Greiner-Pruss theorem on the resolvent family \(\{R_h(t)\}_{t\geq 0}\) and some sufficient conditions on the uniform stability of \(\{R_h(t)\}_{t\geq 0}\). Then they prove that under some suitable conditions, the weak \(L^p\)-stability of \(\{R_h(t)\}_{t\geq 0}\) implies its uniform stability. The results both essentially generalize previous work on the uniform stability of a family of \(C_0\)-semigroups \(\{T_h(t)\}_{t\geq 0}\) and a resolvent family \(\{R_h(t)\}_{t\geq 0}\), without depending on the parameter \(h\). Examples are also given to illustrate our results. Referee's remark: The concept of uniform stability used in this paper is different from that of uniform stabilty as it is widely used in Stability Theory.