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

TL;DR

The paper addresses obtaining Hölder-type regularity for resolvents of monotone operators in Banach spaces by bounding the duality mapping difference $||Jx-Jy||$ in $q$-uniformly smooth spaces. The authors derive a sharp bound $||J(x)-J(y)|| \le M ||x-y||^{q-1} \max\{||x||,||y||\}^{2-q}$, then deduce a Hölder property $||Tx-Ty|| \le L ||x-y||^{q-1}$ for mappings of firmly nonexpansive type in $2$-uniformly convex, $q$-uniformly smooth spaces, and apply these results to the resolvent $J_r=(J+rA)^{-1}J$ of maximal monotone operators. When $q=2$, the bound reduces to Lipschitz continuity of $J$, recovering a Hilbert-space-like behavior. Overall, the work extends refined geometric control of resolvent mappings beyond Hilbert spaces, with potential impact on variational inequalities and fixed-point algorithms in Banach spaces.

Abstract

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

Theorems & Definitions (14)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.1