Uniform stability of a family of resolvent operators in Hilbert spaces.

In this paper, two main results on the uniform stability of a resolvent family { R h (t) } t ≥ 0 , depending on a parameter h are presented. First, we discuss a GGP type theorem on the resolvent family { R h (t) } t ≥ 0 and give some sufficient conditions on the uniform stability of { R h (t) } t ≥ 0 . Then we prove that under some suitable conditions, the weak L p -stability of { R h (t) } t ≥ 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 ≥ 0 and a resolvent family { R (t) } t ≥ 0 , without depending on the parameter h. Examples are also given to illustrate our results. [ABSTRACT FROM AUTHOR]