## 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

#### References

• K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003.
• J. S. Jung, “Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 509–520, 2005.
• H. H. Bauschke, “The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 202, no. 1, pp. 150–159, 1996.
• S. S. Chang, “Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1402–1416, 2006.
• F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.
• G. Marino and H. K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007.
• G. Marino and H. K. Xu, “Convergence of generalized proximal point algorithms,” Communications on Pure and Applied Analysis, vol. 3, no. 4, pp. 791–808, 2004.
• A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
• B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967.
• S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974.
• S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979.
• N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641–3645, 1997.
• W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.
• A. Genel and J. Lindenstrauss, “An example concerning fixed points,” Israel Journal of Mathematics, vol. 22, no. 1, pp. 81–86, 1975.
• T. H. Kim and H. K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51–60, 2005.
• C. Martinez-Yanes and H. K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 2006.
• M. Li and Y. Yao, “Strong convergence of an iterative algorithm for $\lambda$-strictly pseudo-contractive mappings in Hilbert spaces,” Analele stiintifice ale Universitatii Ovidius Constanta, vol. 18, no. 1, pp. 219–228, 2010.
• B. Beauzamy, Introduction to Banach Spaces and Their Geometry, vol. 68 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1982.
• J. Diestel, Geometry of Banach Spaces–-Selected Topics, vol. 485 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.
• K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
• F. Wang and H. K. Xu, “Approximating curve and strong convergence of the $CQ$ algorithm for the split feasibility problem,” Journal of Inequalities and Applications, vol. 2010, Article ID 102085, 13 pages, 2010.
• L. S. Liu, “Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114–125, 1995.