Translator Disclaimer
2012 Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings
Chang-He Xiang, Jiang-Hua Zhang, Zhe Chen
J. Appl. Math. 2012(SI15): 1-9 (2012). DOI: 10.1155/2012/327878

## Abstract

Suppose that $E$ is a real normed linear space, $C$ is a nonempty convex subset of $E$, $T:C\to C$ is a Lipschitzian mapping, and ${x}^{*}\in C$ is a fixed point of $T$. For given ${x}_{0}\in C$, suppose that the sequence $\{{x}_{n}\}\subset C$ is the Mann iterative sequence defined by ${x}_{n+1}=(1-{\alpha }_{n}){x}_{n}+{\alpha }_{n}T{x}_{n},n\ge 0$, where $\{{\alpha }_{n}\}$ is a sequence in [0, 1], ${\sum }_{n=0}^{\infty }{\alpha }_{n}^{2}<\infty$, ${\sum }_{n=0}^{\infty }{\alpha }_{n}=\infty$. We prove that the sequence $\{{x}_{n}\}$ strongly converges to ${x}^{*}$ if and only if there exists a strictly increasing function $\mathrm{\Phi }:[0,\infty )\to [0,\infty )$ with $\mathrm{\Phi }(0)=0$ such that ${\mathrm{limsup} }_{n\to \infty }{\mathrm{inf} }_{j({x}_{n}-{x}^{*})\in J({x}_{n}-{x}^{*})}\{〈T{x}_{n}-{x}^{*},j({x}_{n}-{x}^{*})〉-\parallel {x}_{n}-{x}^{*}{\parallel }^{2}+\mathrm{\Phi }(\parallel {x}_{n}-{x}^{*}\parallel )\}\le 0$.

## Citation

Chang-He Xiang. Jiang-Hua Zhang. Zhe Chen. "Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings." J. Appl. Math. 2012 (SI15) 1 - 9, 2012. https://doi.org/10.1155/2012/327878

## Information

Published: 2012
First available in Project Euclid: 3 January 2013

zbMATH: 1325.47135
MathSciNet: MR2979447
Digital Object Identifier: 10.1155/2012/327878