# A New Iterative Algorithm for Approximating Common Fixed Points for Asymptotically Nonexpansive Mappings

- HY Zhou
^{1}Email author, - YJ Cho
^{2}and - SM Kang
^{2}

**2007**:064874

**DOI: **10.1155/2007/64874

© H. Y. Zhou et al. 2007

**Received: **28 February 2007

**Accepted: **13 April 2007

**Published: **21 May 2007

## Abstract

Suppose that is a nonempty closed convex subset of a real uniformly convex and smooth Banach space with as a sunny nonexpansive retraction. Let be two weakly inward and asymptotically nonexpansive mappings with respect to with sequences , , respectively. Suppose that is a sequence in generated iteratively by , , for all , where , , and are three real sequences in for some which satisfy condition . Then, we have the following. (1) If one of and is completely continuous or demicompact and , then the strong convergence of to some is established. (2) If is a real uniformly convex Banach space satisfying Opial's condition or whose norm is Fréchet differentiable, then the weak convergence of to some is proved.

## Authors’ Affiliations

## References

- Goebel K, Kirk WA:
**A fixed point theorem for asymptotically nonexpansive mappings.***Proceedings of the American Mathematical Society*1972,**35**(1):171–174. 10.1090/S0002-9939-1972-0298500-3MATHMathSciNetView ArticleGoogle Scholar - Bose SC:
**Weak convergence to the fixed point of an asymptotically nonexpansive map.***Proceedings of the American Mathematical Society*1978,**68**(3):305–308. 10.1090/S0002-9939-1978-0493543-4MATHMathSciNetView ArticleGoogle Scholar - Passty GB:
**Construction of fixed points for asymptotically nonexpansive mappings.***Proceedings of the American Mathematical Society*1982,**84**(2):212–216. 10.1090/S0002-9939-1982-0637171-7MATHMathSciNetView ArticleGoogle Scholar - Tan K-K, Xu HK:
**The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces.***Proceedings of the American Mathematical Society*1992,**114**(2):399–404. 10.1090/S0002-9939-1992-1068133-2MATHMathSciNetView ArticleGoogle Scholar - Tan K-K, Xu HK:
**Fixed point iteration processes for asymptotically nonexpansive mappings.***Proceedings of the American Mathematical Society*1994,**122**(3):733–739. 10.1090/S0002-9939-1994-1203993-5MATHMathSciNetView ArticleGoogle Scholar - Schu J:
**Iterative construction of fixed points of asymptotically nonexpansive mappings.***Journal of Mathematical Analysis and Applications*1991,**158**(2):407–413. 10.1016/0022-247X(91)90245-UMATHMathSciNetView ArticleGoogle Scholar - Schu J:
**Weak and strong convergence to fixed points of asymptotically nonexpansive mappings.***Bulletin of the Australian Mathematical Society*1991,**43**(1):153–159. 10.1017/S0004972700028884MATHMathSciNetView ArticleGoogle Scholar - Rhoades BE:
**Fixed point iterations for certain nonlinear mappings.***Journal of Mathematical Analysis and Applications*1994,**183**(1):118–120. 10.1006/jmaa.1994.1135MATHMathSciNetView ArticleGoogle Scholar - Osilike MO, Aniagbosor SC:
**Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings.***Mathematical and Computer Modelling*2000,**32**(10):1181–1191. 10.1016/S0895-7177(00)00199-0MATHMathSciNetView ArticleGoogle Scholar - Chang S-S, Cho YJ, Zhou HY:
**Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings.***Journal of the Korean Mathematical Society*2001,**38**(6):1245–1260.MATHMathSciNetGoogle Scholar - Górnicki J:
**Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces.***Commentationes Mathematicae Universitatis Carolinae*1989,**30**(2):249–252.MATHMathSciNetGoogle Scholar - Reich S:
**Weak convergence theorems for nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*1979,**67**(2):274–276. 10.1016/0022-247X(79)90024-6MATHMathSciNetView ArticleGoogle Scholar - Ishikawa S:
**Fixed points and iteration of a nonexpansive mapping in a Banach space.***Proceedings of the American Mathematical Society*1976,**59**(1):65–71. 10.1090/S0002-9939-1976-0412909-XMATHMathSciNetView ArticleGoogle Scholar - Jung JS, Kim SS:
**Strong convergence theorems for nonexpansive nonself-mappings in Banach spaces.***Nonlinear Analysis*1998,**33**(3):321–329. 10.1016/S0362-546X(97)00526-9MATHMathSciNetView ArticleGoogle Scholar - Khan SH, Fukhar-ud-din H:
**Weak and strong convergence of a scheme with errors for two nonexpansive mappings.***Nonlinear Analysis*2005,**61**(8):1295–1301. 10.1016/j.na.2005.01.081MATHMathSciNetView ArticleGoogle Scholar - Matsushita S-Y, Kuroiwa D:
**Strong convergence of averaging iterations of nonexpansive nonself-mappings.***Journal of Mathematical Analysis and Applications*2004,**294**(1):206–214. 10.1016/j.jmaa.2004.02.010MATHMathSciNetView ArticleGoogle Scholar - Shahzad N:
**Approximating fixed points of non-self nonexpansive mappings in Banach spaces.***Nonlinear Analysis*2005,**61**(6):1031–1039. 10.1016/j.na.2005.01.092MATHMathSciNetView ArticleGoogle Scholar - Song Y, Chen R:
**Viscosity approximation methods for nonexpansive nonself-mappings.***Journal of Mathematical Analysis and Applications*2006,**321**(1):316–326. 10.1016/j.jmaa.2005.07.025MATHMathSciNetView ArticleGoogle Scholar - Chidume CE, Ofoedu EU, Zegeye H:
**Strong and weak convergence theorems for asymptotically nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2003,**280**(2):364–374. 10.1016/S0022-247X(03)00061-1MATHMathSciNetView ArticleGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis. Fixed Point Theory and Its Applications*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Tan K-K, Xu HK:
**Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.***Journal of Mathematical Analysis and Applications*1993,**178**(2):301–308. 10.1006/jmaa.1993.1309MATHMathSciNetView ArticleGoogle Scholar - Zhou HY, Guo GT, Hwang HJ, Cho YJ:
**On the iterative methods for nonlinear operator equations in Banach spaces.***PanAmerican Mathematical Journal*2004,**14**(4):61–68.MATHMathSciNetGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.