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Fixed points of asymptotically nonexpansive mappings with center 0 and applications

Abdelkader Dehici, Sami Atailia, Najeh Redjel

Abstract

In this paper, we investigate the existence of fixed points for asymptotically nonexpansive mappings with center 0 defined on closed convex subsets of various Banach spaces. Three applications are given. Firstly, we prove that our results refine those concerning alternate convexically nonexpansive (in short; ACN) mappings studied by P. N. Dowling in " On a fixed point result of Amini-Harandi in strictly convex Banach spaces, Acta. Math. Hungar., 112 (1-2), (2006), 85-88" . Secondly, by using Lau's result in " Closed convex invariant subsets of $L_p(G)$, Trans. Amer. Math. Soc., {\bf 232}, (1977), 131-142", we give another characterization for the noncompactness of locally compact groups $G$. Finally, we discuss the existence of a solution for a nonlinear transport equation without using compactness results.

Fixed points of asymptotically nonexpansive mappings with center 0 and applications

Abstract

In this paper, we investigate the existence of fixed points for asymptotically nonexpansive mappings with center 0 defined on closed convex subsets of various Banach spaces. Three applications are given. Firstly, we prove that our results refine those concerning alternate convexically nonexpansive (in short; ACN) mappings studied by P. N. Dowling in " On a fixed point result of Amini-Harandi in strictly convex Banach spaces, Acta. Math. Hungar., 112 (1-2), (2006), 85-88" . Secondly, by using Lau's result in " Closed convex invariant subsets of , Trans. Amer. Math. Soc., {\bf 232}, (1977), 131-142", we give another characterization for the noncompactness of locally compact groups . Finally, we discuss the existence of a solution for a nonlinear transport equation without using compactness results.

Paper Structure

This paper contains 7 sections, 19 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Preparatory Results
  3. Main results
  4. Applications
  5. The case of alternate convexically nonexpansive mappings
  6. The linear isometries $l_g$ and $r_g$
  7. The case of a nonlinear transport equation

Key Result

Lemma 2.1

Let $C$ be a bounded subset of a Banach space $X$ and let $T: C \longrightarrow C$ be a $(c)$-mapping. Then $T$ is asymptotically regular i.e., $\lim_{n \longrightarrow +\infty}\|T^{n+1}x- T^{n}x\|= 0 \hbox{for all} \ x \in C.$

Theorems & Definitions (71)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Example 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.3
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • ...and 61 more