Zero-r Law on the Analyticity and the Uniform Continuity of Fractional Resolvent Families

In this paper, we demonstrate some zero-r laws related to the analyticity of fractional resolvent families. Using these findings, we establish a zero-r law for the uniform continuity of fractional resolvent families. Additionally, we provide a similar result to the previous one by utilizing the algebraic characteristic of fractional resolvent families. Here, r is taken to be greater than 1. In particular, we prove a zero-r law on the scalar of fractional resolvent families, which offers a positive response to the question raised by F.L. Schwenninger and H. Zwart.