## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2012, Special Issue (2012), Article ID 327878, 9 pages.

### Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings

#### 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 ${\mathrm{limsup} }_{n\to \infty }{\mathrm{inf} }_{j({x}_{n}-{x}^{*})\in J({x}_{n}-{x}^{*})}\{〈T{x}_{n}-{x}^{*},j({x}_{n}-{x}^{*})〉-\parallel {x}_{n}-{x}^{*}{\parallel }^{2}+\mathrm{\Phi }(\parallel {x}_{n}-{x}^{*}\parallel )\}\le 0$.

#### Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 327878, 9 pages.

Dates
First available in Project Euclid: 3 January 2013

https://projecteuclid.org/euclid.jam/1357178244

Digital Object Identifier
doi:10.1155/2012/327878

Mathematical Reviews number (MathSciNet)
MR2979447

Zentralblatt MATH identifier
1325.47135

#### Citation

Xiang, Chang-He; Zhang, Jiang-Hua; Chen, Zhe. Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings. J. Appl. Math. 2012, Special Issue (2012), Article ID 327878, 9 pages. doi:10.1155/2012/327878. https://projecteuclid.org/euclid.jam/1357178244