Fixed point approximation under Mann iteration beyond Ishikawa

Anthony Hester; Claudio H. Morales

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 3, page 265-275
  • ISSN: 0010-2628

Abstract

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Consider the Mann iteration x n + 1 = ( 1 - α n ) x n + α n T x n for a nonexpansive mapping T : K → K defined on some subset K of the normed space X . We present an innovative proof of the Ishikawa almost fixed point principle for nonexpansive mapping that reveals deeper aspects of the behavior of the process. This fact allows us, among other results, to derive convergence of the process under the assumption of existence of an accumulation point of { x n } .

How to cite

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Hester, Anthony, and Morales, Claudio H.. "Fixed point approximation under Mann iteration beyond Ishikawa." Commentationes Mathematicae Universitatis Carolinae 61.3 (2020): 265-275. <http://eudml.org/doc/297052>.

@article{Hester2020,
abstract = {Consider the Mann iteration $x_\{n+1\} = ( 1 - \alpha _n ) x_n + \alpha _n Tx_n$ for a nonexpansive mapping $T\colon K \rightarrow K$ defined on some subset $K$ of the normed space $X$. We present an innovative proof of the Ishikawa almost fixed point principle for nonexpansive mapping that reveals deeper aspects of the behavior of the process. This fact allows us, among other results, to derive convergence of the process under the assumption of existence of an accumulation point of $\lbrace x_n \rbrace $.},
author = {Hester, Anthony, Morales, Claudio H.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Mann iteration; fixed point; nonexpansive mapping},
language = {eng},
number = {3},
pages = {265-275},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Fixed point approximation under Mann iteration beyond Ishikawa},
url = {http://eudml.org/doc/297052},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Hester, Anthony
AU - Morales, Claudio H.
TI - Fixed point approximation under Mann iteration beyond Ishikawa
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 3
SP - 265
EP - 275
AB - Consider the Mann iteration $x_{n+1} = ( 1 - \alpha _n ) x_n + \alpha _n Tx_n$ for a nonexpansive mapping $T\colon K \rightarrow K$ defined on some subset $K$ of the normed space $X$. We present an innovative proof of the Ishikawa almost fixed point principle for nonexpansive mapping that reveals deeper aspects of the behavior of the process. This fact allows us, among other results, to derive convergence of the process under the assumption of existence of an accumulation point of $\lbrace x_n \rbrace $.
LA - eng
KW - Mann iteration; fixed point; nonexpansive mapping
UR - http://eudml.org/doc/297052
ER -

References

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  13. Reich S., Shafrir I., On the method of successive approximations for nonexpansive mappings, Nonlinear and Convex Analysis, Santa Barbara, 1985, Lecture Notes in Pure and Appl. Math., 107, Dekker, New York, 1987, 193–201. MR0892792
  14. Schaefer H., Über die Methode sukzessiver Approximationen, Jber. Deutsch. Math.-Verein. 59 (1957), Abt. 1, 131–140 (German). MR0084116

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