## Abstract and Applied Analysis

### Strong Convergence Theorems for Common Fixed Points of an Infinite Family of Asymptotically Nonexpansive Mappings

Yuanheng Wang

#### Abstract

In the framework of a real Banach space with uniformly Gateaux differentiable norm, some new viscosity iterative sequences $\{{x}_{n}\}$ are introduced for an infinite family of asymptotically nonexpansive mappings ${\{{T}_{i}\}}_{i=1}^{\infty }$ in this paper. Under some appropriate conditions, we prove that the iterative sequences $\{{x}_{n}\}$ converge strongly to a common fixed point of the mappings ${\{{T}_{i}\}}_{i=1}^{\infty }$, which is also a solution of a variational inequality. Our results extend and improve some recent results of other authors.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 809528, 6 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412606022

Digital Object Identifier
doi:10.1155/2014/809528

Mathematical Reviews number (MathSciNet)
MR3226233

Zentralblatt MATH identifier
07023120

#### Citation

Wang, Yuanheng. Strong Convergence Theorems for Common Fixed Points of an Infinite Family of Asymptotically Nonexpansive Mappings. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 809528, 6 pages. doi:10.1155/2014/809528. https://projecteuclid.org/euclid.aaa/1412606022

#### References

• K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.
• S. H. Khan and W. Takahasi, “Approximating common fixed points of two asymptotically nonexpanisve mappings,” Scientiae Mathematicae Japonicae, vol. 53, pp. 143–148, 2001.
• Z.-H. Sun, “Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 351–358, 2003.
• H.-K. Xu and R. G. Ori, “An implicit iteration process for nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 22, no. 5-6, pp. 767–773, 2001.
• N. Shahzad and A. Udomene, “Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2006, Article ID 18909, 2006.
• N. Shahzad and H. Zegeye, “Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1058–1065, 2007.
• A. R. Khan, A.-A. Domlo, and H. Fukhar-ud-din, “Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 1–11, 2008.
• L. C. Zhao, “Viscosity approximation of fixed points for a finite family of nonexpansive mappings,” Journal of Acta Mathematica Sinica, vol. 31, pp. 599–607, 2008.
• Y.-C. Lin, N.-C. Wong, and J.-C. Yao, “Strong convergence theorems of Ishikawa iteration process with errors for fixed points of lipschitz continuous mappings in Banach spaces,” Taiwanese Journal of Mathematics, vol. 10, no. 2, pp. 543–552, 2006.
• L.-C. Zeng and J.-C. Yao, “Stability of iterative procedures with errors for approximating common fixed points of a couple of q-contractive-like mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 661–674, 2006.
• Y. Yao, J.-C. Yao, and H. Zhou, “Approximation methods for common fixed points of infinite countable family of nonexpansive mappings,” Computers and Mathematics with Applications, vol. 53, no. 9, pp. 1380–1389, 2007.
• W. Cholamjiak and S. Suantai, “A new hybrid algorithm for a countable family of quasi-nonexpansive mappings and equilibrium problems,” Journal of Nonlinear and Convex Analysis, vol. 12, pp. 381–398, 2011.
• S. Plubtieng and K. Ungchittrakool, “Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications,” Nonlinear Analysis: Theory, Methods and Applications, vol. 72, no. 6, pp. 2896–2908, 2009.
• L.-C. Zeng and J.-C. Yao, “Implicit iteration scheme with per-turbed mapping for common fixed points of a finite family of nonexpansive mappings,” Nonlinear Analysis: Theory, Methods and Applications, vol. 64, no. 11, pp. 2507–2515, 2006.
• G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
• Y.-H. Wang and Y.-H. Xia, “Strong convergence for asymptotically pseudocontractions with the demiclosedness principle in Banach spaces,” Fixed Point Theory and Applications, vol. 2012, article 45, 2012.
• Y. Wang and W. Xuan, “Convergence theorems for common fixed points of a finite family of relatively nonexpansive mappings in Banach spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 259470, 7 pages, 2013.
• Y. Wang and C. Wang, “Convergence of a new modified ishi-kawa type iteration for common fixed points of total asymptotically strict pseudocontractive semigroups,” Abstract and Applied Analysis, vol. 2013, Article ID 319241, 7 pages, 2013.
• Y. J. Cho, S. M. Kang, and X. Qin, “Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces,” Computers and Mathematics with Applications, vol. 56, no. 8, pp. 2058–2064, 2008.
• H. K. Xu, “Vistosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, pp. 279–291, 2004.
• T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005. \endinput