An interpolation inequality and its applications to stability of fractional resolvent families

In this paper, we prove an interpolation inequality on Riemann–Liouville fractional integrals and then use it to study the strong stability and semi-uniform stability of fractional resolvent families of order \(0<\alpha <2\) $0<\alpha <2$ . Let A denote the generator of a bounded fractional resolvent family. We show that if \(\sigma (A)\cap (\textrm{i}{\mathbb {R}})^\alpha \) $\sigma (A)\cap {(\text{i}\mathbb{R})}^{\alpha }$ is countable and \(\sigma _r(A) \cap (\textrm{i}{\mathbb {R}})^\alpha =\varnothing \) ${\sigma }_r(A)\cap {(\text{i}\mathbb{R})}^{\alpha }=\varnothing$ , then the bounded fractional resolvent family is strongly stable. And the semi-uniform stability of the fractional resolvent family is equivalent to \(\sigma (A)\cap (\textrm{i}{\mathbb {R}})^\alpha =\varnothing \) $\sigma (A)\cap {(\text{i}\mathbb{R})}^{\alpha }=\varnothing$ . Moreover, the relation between decay rates of semi-uniform stability and growth of the resolvent of A along \((\textrm{i}{\mathbb {R}})^\alpha \) ${(\text{i}\mathbb{R})}^{\alpha }$ is given.