## Abstract and Applied Analysis

### A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings

#### Abstract

Let $H$ be a real Hilbert space and $C\subset \mathrm{H }$a closed convex subset. Let $T:C\to C$ be a nonexpansive mapping with the nonempty set of fixed points $\text{F}\text{i}\text{x}\left(T\right)$. Kim and Xu (2005) introduced a modified Mann iteration ${x}_{0}=x\in C$, ${y}_{n}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right)T{x}_{n}$, ${x}_{n+1}={\beta }_{n}u+\left(1-{\beta }_{n}\right){y}_{n}$, where $u\in C$ is an arbitrary (but fixed) element, and $\left\{{\alpha }_{n}\right\}$ and $\left\{{\beta }_{n}\right\}$ are two sequences in $\left(0,1\right)$. In the case where $0\in C$, the minimum-norm fixed point of $T$ can be obtained by taking $u=0$. But in the case where $0\notin C$, this iteration process becomes invalid because ${x}_{n}$ may not belong to $C$. In order to overcome this weakness, we introduce a new modified Mann iteration by boundary point method (see Section 3 for details) for finding the minimum norm fixed point of $\mathrm{ T}$ and prove its strong convergence under some assumptions. Since our algorithm does not involve the computation of the metric projection ${P}_{C}$, which is often used so that the strong convergence is guaranteed, it is easy implementable. Our results improve and extend the results of Kim, Xu, and some others.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 768595, 6 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511808

Digital Object Identifier
doi:10.1155/2013/768595

Mathematical Reviews number (MathSciNet)
MR3039125

Zentralblatt MATH identifier
06209437

#### Citation

He, Songnian; Zhu, Wenlong. A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings. Abstr. Appl. Anal. 2013 (2013), Article ID 768595, 6 pages. doi:10.1155/2013/768595. https://projecteuclid.org/euclid.aaa/1393511808