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On $α$-monotone operators and their resolvent in Banach spaces

Changchi Huang, Jigen Peng, Yuchao Tang

TL;DR

The paper addresses extending α-monotone operator theory from Hilbert spaces to real Banach spaces by introducing a new definition that uses the normalized duality mapping $J$, aligning monotonicity with Banach-space geometry and recovering the classical case when $J$ is the identity. It develops resolvent properties in real $2$-uniformly convex and uniformly smooth Banach spaces that mirror the Hilbert-space firmly nonexpansive behavior, and establishes a systematic link between α-monotone operators and resolvent-type mappings. It proves the density of strong monotone operators under the new definition relative to the classical notion, and demonstrates strong convergence with $R$-linear rate for the forward-reflected-backward splitting algorithm under suitable curvature conditions. These results provide a unified framework for monotone-operator theory and convergent operator-splitting methods beyond Hilbert spaces, with potential applicability to broader classes of algorithms in Banach spaces.

Abstract

This paper introduces a new definition of $α$-monotone operators in real 2-uniformly convex and smooth Banach spaces. Based on this new definition, we establish several novel structural and analytical properties of such operators, which not only extend classical results from Hilbert spaces but also reveal new insights into the geometry of Banach spaces. In particular, we examine the resolvent of $α$-maximal monotone operators and demonstrate how its behavior is consistent with, and generalizes, the well-known firmly nonexpansive property in the Hilbert space setting. Building upon this theoretical framework, we further investigate algorithmic applications. Specifically, we analyze the forward-reflected-backward splitting algorithm under the new $α$-monotonicity assumption and prove its strong convergence as well as its $R$-linear convergence rate in real 2-uniformly convex and smooth Banach spaces.

On $α$-monotone operators and their resolvent in Banach spaces

TL;DR

The paper addresses extending α-monotone operator theory from Hilbert spaces to real Banach spaces by introducing a new definition that uses the normalized duality mapping , aligning monotonicity with Banach-space geometry and recovering the classical case when is the identity. It develops resolvent properties in real -uniformly convex and uniformly smooth Banach spaces that mirror the Hilbert-space firmly nonexpansive behavior, and establishes a systematic link between α-monotone operators and resolvent-type mappings. It proves the density of strong monotone operators under the new definition relative to the classical notion, and demonstrates strong convergence with -linear rate for the forward-reflected-backward splitting algorithm under suitable curvature conditions. These results provide a unified framework for monotone-operator theory and convergent operator-splitting methods beyond Hilbert spaces, with potential applicability to broader classes of algorithms in Banach spaces.

Abstract

This paper introduces a new definition of -monotone operators in real 2-uniformly convex and smooth Banach spaces. Based on this new definition, we establish several novel structural and analytical properties of such operators, which not only extend classical results from Hilbert spaces but also reveal new insights into the geometry of Banach spaces. In particular, we examine the resolvent of -maximal monotone operators and demonstrate how its behavior is consistent with, and generalizes, the well-known firmly nonexpansive property in the Hilbert space setting. Building upon this theoretical framework, we further investigate algorithmic applications. Specifically, we analyze the forward-reflected-backward splitting algorithm under the new -monotonicity assumption and prove its strong convergence as well as its -linear convergence rate in real 2-uniformly convex and smooth Banach spaces.
Paper Structure (9 sections, 26 theorems, 77 equations)

This paper contains 9 sections, 26 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\alpha$-monotone operators in Banach spaces
  4. New definition of monotone operators in Banach spaces
  5. Basic properties of new monotone operators in Banach spaces
  6. Strong monotone operators under Definition \ref{['our def']} are dense in strong monotone operators under Definition \ref{['classical def']}
  7. The class of nonlinear mappings related to maximal monotone operators in Banach spaces
  8. R-linear convergence of the forward-reflected-backward splitting in Banach spaces
  9. Conclusions

Key Result

Lemma 2.1

AYAKKT2 Let $X$ be a real smooth Banach space. Then the following identities hold: (i)$\phi(x,y) = \phi(x,z) + \phi(z,y) + 2\langle x-z, Jz-Jy\rangle, \forall x,y,z\in X$. (ii)$\phi(x,y) + \phi(y,x) = 2\langle x-y, Jx-Jy\rangle, \forall x,y\in X$. (iii)$\langle x-y, Jz-Jw \rangle = \frac{1}{2}\{\phi

Theorems & Definitions (53)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 3.1: Classical $\alpha$-monotonicity operators in Banach spaces
  • Definition 3.2: $\alpha$-monotonicity operators in smooth Banach spaces
  • Remark 3.1
  • Example 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 43 more