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A random demiclosedness principle for random asymptotically nonexpansive mappings

Yuanyuan Sun, Tiexin Guo, Qiang Tu

Abstract

By making full use of the inherent connection between the theory of random conjugate spaces and the theory of classical conjugate spaces, in this paper we establish a random demiclosedness principle for a random asymptotically nonexpansive mapping, which generalizes Xu's classical demiclosedness principle from a uniformly convex Banach space to a complete random uniformly convex random normed module: let $(E,\|\cdot\|)$ be a complete random uniformly convex random normed module, $E^{*}$ the random conjugate space of $E$, $G$ an almost surely bounded closed $L^{0}$-convex subset of $E$ and $f: G \rightarrow G$ a random asymptotically nonexpansive mapping, then $(I-f)$ is random demiclosed at $θ$, namely, for each sequence $\{x_{n}, n\in \mathbb{N}\}$ in $G$, if $\{x_{n}, n\in \mathbb{N}\}$ converges in $σ(E, E^{*})$ to $x$ and $\{(I-f)x_{n}, n\in \mathbb{N}\}$ converges to $θ$, then $(I-f)x=θ$, where $I$ denotes the identity operator on $E$ and $σ(E, E^{*})$ the random weak topology on $E$.

A random demiclosedness principle for random asymptotically nonexpansive mappings

Abstract

By making full use of the inherent connection between the theory of random conjugate spaces and the theory of classical conjugate spaces, in this paper we establish a random demiclosedness principle for a random asymptotically nonexpansive mapping, which generalizes Xu's classical demiclosedness principle from a uniformly convex Banach space to a complete random uniformly convex random normed module: let be a complete random uniformly convex random normed module, the random conjugate space of , an almost surely bounded closed -convex subset of and a random asymptotically nonexpansive mapping, then is random demiclosed at , namely, for each sequence in , if converges in to and converges to , then , where denotes the identity operator on and the random weak topology on .
Paper Structure (3 sections, 5 theorems, 21 equations)

This paper contains 3 sections, 5 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Proof of the Theorem 2.4

Key Result

Proposition 2.1

$L^{q}(E^{*})\cong (L^{p}(E))^{'}$ under the canonical mapping $T$, where $p$ and $q$ are a pair of Hölder conjugate numbers with $1\leq p<+\infty$, $(L^{p}(E))^{'}$ denotes the classical conjugate space of $L^{p}(E)$ and for each $F\in L^{q}(E^{*})$, $T_{F}$ (denoting $T(F))~:~L^{p}(E)\rightarrow \

Theorems & Definitions (11)

  • Proposition 2.1: Guo2000
  • Definition 2.2: GZ10
  • Proposition 2.3: GZ10
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • proof : Proof of Theorem \ref{['theorem1.7']}
  • ...and 1 more