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Representation formulas for maximal monotone operators of type (D) in Banach spaces whose dual spaces are strictly convex

Nguyen B. Tran, Tran N. Nguyen, Huynh M. Hien

Abstract

This work deals with a maximal monotone operator $A$ of type (D) in a Banach space whose dual space is strictly convex. We establish some representations for the value $Ax$ at a given point $x$ via its values at nearby points of $x$. We show that the faces of $Ax$ are contained in the set of all weak$^*$ convergent limits of bounded nets of the operator at nearby points of $x$, then we obtain a representation for $Ax$ by use of this set. In addition, representations for the support function of $Ax$ based on the minimal-norm selection of the operator in certain Banach spaces are given.

Representation formulas for maximal monotone operators of type (D) in Banach spaces whose dual spaces are strictly convex

Abstract

This work deals with a maximal monotone operator of type (D) in a Banach space whose dual space is strictly convex. We establish some representations for the value at a given point via its values at nearby points of . We show that the faces of are contained in the set of all weak convergent limits of bounded nets of the operator at nearby points of , then we obtain a representation for by use of this set. In addition, representations for the support function of based on the minimal-norm selection of the operator in certain Banach spaces are given.
Paper Structure (4 sections, 14 theorems, 99 equations)

This paper contains 4 sections, 14 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation formulas for maximal monotone operators of type (D)
  4. A representation formula for the support functions of the values of maximal monotone operators

Key Result

Theorem 2.1

Let $X$ be a Banach space and $A: X \rightrightarrows X^{*}$ be a maximal monotone operator. Then $A$ is of type (D) if and only if

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.1: MaSv1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 21 more