Hölder property of the resolvent of a monotone operator in Banach spaces

Let $E$ be a Banach space, and let $J:E\to E^{\ast }$ denote the normalized duality mapping. In this paper, we establish an upper bound for $\mathrm{\parallel }Jx-Jy\mathrm{\parallel }$ in $q$-uniformly smooth Banach spaces, where the bound is expressed in terms of a relatively simple function of $\mathrm{\parallel }x-y\mathrm{\parallel }$. Subsequently, we derive the Hölder property of mappings of firmly nonexpansive type in 2-uniformly convex and $q$-uniformly smooth Banach spaces ($1<q\le 2$). As an application, we apply this result to the resolvent of a monotone operator in Banach spaces.