# Iterative scheme based on boundary point method for common fixed‎ ‎point of strongly nonexpansive sequences

Document Type: Research Paper

Authors

College of Management and Economics‎, ‎Tianjin University‎, ‎Tianjin 300072‎, ‎China.

Abstract

Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$. Let ${S_n}$ and ${T_n}$ be sequences of nonexpansive self-mappings of $C$, where one of them is a strongly nonexpansive sequence. K. Aoyama and Y. Kimura introduced the iteration process $x_{n+1}=\beta_nx_n+(1-\beta_n)S_n(\alpha_nu+(1-\alpha_n)T_nx_n)$ for finding the common fixed point of ${S_n}$ and ${T_n}$, where $uin C$ is an arbitrarily (but fixed) element in $C$, $x_0\in C$ arbitrarily, ${\alpha_n}$ and ${\beta_n}$ are sequences in $[0,1]$. But in the case where $u\notin C$, the iterative scheme above becomes invalid because $x_n$ may not belong to $C$. To overcome this weakness, a new iterative scheme based on the thought of boundary point method is proposed and the strong convergence theorem is proved. As a special case, we can find the minimum-norm common fixed point of ${S_n}$ and ${T_n}$ whether $0\in C$ or $0\notin C$.

Keywords

Main Subjects

### References

K. Aoyama and Y. Kimura, Strong convergence theorem for strongly nonexpansive sequences, Appl. Math. Comput. 217 (2011), no. 19, 7537--7545.

W. Takahashi and T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal. 5 (1998), no. 1, 45--56.

S. Takahashi, W. Takahashi and M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl. 147 (2010), no. 1, 27--41.

L. C. Zeng, N. C. Wong and J. C. Yao, Convergence analysis of iterative sequences for a pair of mappings in Banach spaces, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 3, 463--470.

Y. Yao and J. C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2007), no. 2, 1551--1558.

K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, On a strongly nonexpansive sequence in Hilbert spaces, J. Nonlinear Convex Anal. 8 (2007), no. 3, 471--489.

S. Chandok, W. Sintunavarat and P. Kumam, Some coupled common fixed points for a pair of mappings in partially ordered G-metric spaces, Math. Sci. (Springer) 7 (2013), 7 pages.

W. Shatanawi and M. Postolache, Common fixed point results for mappings under non- linear contraction of cyclic form in ordered metric spaces, Fixed Point Theory Appl. 2013 (2013) 13 pages.

D. K. Patel, P. Kumam and D. Gopal, Some discussion on the existence of common fixed points for a pair of maps, Fixed Point Theory Appl. 2013 (2013) 17 pages.

H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), no. 1, 279--291.

Hong-Kun Xu, Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math. 14 (2010), no. 2, 463--478.

A. Mouda, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000), no. 1, 46--55.

R. Chen and Z. Zhu, Viscosity approximation method for accretive operator in Banach space, Nonlinear Anal. 69 (2008), no. 4, 1356--1363.

Rabian Wangkeeree, and Pakkapon Preechasilp, Viscosity approximation methods for nonexpansive semigroups in CAT(0) spaces, Fixed Point Theory Appl. 2013 (2013) 16 pages.

P. E. Mainge, The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces, Comput. Math. Appl. 59 (2010), no. 1, 74--79.

J. S. Jung, Strong convergence of viscosity approximation methods for finding zeros of accretive operators in Banach spaces, Nonlinear Anal. 72 (2010), no. 1, 449--459.

J. S. Jung, Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces, Nonlinear Anal. 64 (2006), no. 11, 2536--2552.

S. S. Chang, Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 323 (2006), no. 2, 1402--1416.

L. C. Ceng, H. K. Xu and J. C. Yao, The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Anal. 69 (2008), no. 4, 1402--1412.

J. Lou, L. J. Zhang and Z. He, Viscosity approximation methods for asymptotically nonexpansive mappings, Appl. Math. Comput. 203 (2008), no. 1, 171--177.

S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), no. 1, 506--515.

S. Plubtieng and T. Thammathiwat, A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities, J. Global Optim. 46 (2010), no. 3, 447--464.

K. Aoyama, An iterative method for a variational inequality problem over the common fixed point set of nonexpansive mappings, Nonlinear analysis and convex analysis, 21--28, Yokohama Publ., Yokohama, 2010.

C. M. Yanes and H. K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), no. 11, 2400--2411.

S. Wang, A general iterative method for an infinite family of strictly pseudo-contractive mappings in Hilbert spaces, Appl. Math. Lett. 24 (2011), no. 6, 901--907.

T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappingsin general Banach spaces, Fixed Point Theory Appl. (2005), no. 1, 103--123.

S. Wang, Convergence and certain control conditions for hybrid iterative algorithms, Appl. Math. Comput. 219 (2013), no. 20, 10325--10332.

Y. Yao, Y. J. Cho and Y. C. Liou, Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems, European J. Oper. Res. 212 (2011), no. 2, 242--250.

H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2) 66 (2002), no. 1, 240--256.

S. He and W. Zhu, 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.

### History

• Receive Date: 12 December 2014
• Revise Date: 15 April 2015
• Accept Date: 17 April 2015