×
Yatakoat, Pornsak; Suantai, Suthep

A new approximation method for common fixed points of a finite family of generalized asymptotically quasinonexpansive mappings in Banach spaces. (English) Zbl 1206.47092

Abstr. Appl. Anal. 2010, Article ID 706587, 15 p. (2010).
Summary: We introduce a new iterative scheme to approximate a common fixed point for a finite family of generalized asymptotically quasinonexpansive mappings. Several strong and weak convergence theorems of the proposed iteration in Banach spaces are established. The main results obtained in this paper generalize and refine many known results in the current literature.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

generalized asymptotically quasinonexpansive mappings; strong convergence; Banach spaces
Cite Review PDF
Full Text: DOI EuDML
Creative Commons CC license icon Creative Commons BY license icon

References:

[1] 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. · Zbl 1137.47053 · doi:10.1016/j.jmaa.2007.06.051
[2] J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153-159, 1991. · Zbl 0709.47051 · doi:10.1017/S0004972700028884
[3] K.-K. Tan and H. K. Xu, “Fixed point iteration processes for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 122, no. 3, pp. 733-739, 1994. · Zbl 0820.47071 · doi:10.2307/2160748
[4] B. Xu and M. A. Noor, “Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 267, no. 2, pp. 444-453, 2002. · Zbl 1011.47039 · doi:10.1006/jmaa.2001.7649
[5] A. Kettapun, A. Kananthai, and S. Suantai, “A new approximation method for common fixed points of a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces,” Computers and Mathematics with Applications, vol. 60, no. 5, pp. 1430-1439, 2010. · Zbl 1201.65089 · doi:10.1016/j.camwa.2010.06.025
[6] S. Imnang and S. Suantai, “Common fixed points of multistep Noor iterations with errors for a finite family of generalized asymptotically quasi-nonexpansive mappings,” Abstract and Applied Analysis, vol. 2009, Article ID 728510, 14 pages, 2009. · Zbl 1185.47076 · doi:10.1155/2009/728510
[7] S. Saejung, S. Suantai, and P. Yodkaew, “A note on “Common fixed points of multistep Noor iterations with errors for a finite family of generalized asymptotically quasi-nonexpansive mappings”,” Abstract and Applied Analysis, vol. 2009, Article ID 283461, 9 pages, 2009. · Zbl 1185.47081 · doi:10.1155/2009/283461
[8] S. H. Khan and W. Takahashi, “Approximating common fixed points of two asymptotically nonexpansive mappings,” Scientiae Mathematicae Japonicae, vol. 53, no. 1, pp. 143-148, 2001. · Zbl 0985.47042
[9] L.-C. Zeng and J.-C. Yao, “Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 10, no. 5, pp. 1293-1303, 2006. · Zbl 1110.49013
[10] 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. · Zbl 1118.47053
[11] 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. · Zbl 1101.47058 · doi:10.1016/j.jmaa.2005.07.079
[12] L.-C. Zeng and J.-C. Yao, “Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2507-2515, 2006. · Zbl 1105.47061 · doi:10.1016/j.na.2005.08.028
[13] L. C. Ceng, P. Cubiotti, and J. C. Yao, “Approximation of common fixed points of families of nonexpansive mappings,” Taiwanese Journal of Mathematics, vol. 12, no. 2, pp. 487-500, 2008. · Zbl 1220.47084
[14] L. C. Ceng, P. Cubiotti, and J. C. Yao, “Strong convergence theorems for finitely many nonexpansive mappings and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1464-1473, 2007. · Zbl 1123.47044 · doi:10.1016/j.na.2006.06.055
[15] Y. Yao, J.-C. Yao, and H. Zhou, “Approximation methods for common fixed points of infinite countable family of nonexpansive mappings,” Computers & Mathematics with Applications, vol. 53, no. 9, pp. 1380-1389, 2007. · Zbl 1140.47057 · doi:10.1016/j.camwa.2006.10.021
[16] A. Petru\csel and J.-C. Yao, “Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 4, pp. 1100-1111, 2008. · Zbl 1142.47329 · doi:10.1016/j.na.2007.06.016
[17] 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. · Zbl 1095.47046 · doi:10.1016/S0022-247X(03)00537-7
[18] 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, 10 pages, 2006. · Zbl 1102.47055 · doi:10.1155/FPTA/2006/18909
[19] 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. · Zbl 1126.65054 · doi:10.1016/j.amc.2006.11.152
[20] L.-C. Ceng, H.-K. Xu, and J.-C. Yao, “The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 4, pp. 1402-1412, 2008. · Zbl 1142.47326 · doi:10.1016/j.na.2007.06.040
[21] L.-C. Ceng, H.-K. Xu, and J.-C. Yao, “Strong convergence of an iterative method with perturbed mappings for nonexpansive and accretive operators,” Numerical Functional Analysis and Optimization, vol. 29, no. 3-4, pp. 324-345, 2008. · Zbl 1140.47050 · doi:10.1080/01630560801998203
[22] 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. · Zbl 0999.47043 · doi:10.1081/NFA-100105317
[23] K. Nammanee, M. A. Noor, and S. Suantai, “Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 314, no. 1, pp. 320-334, 2006. · Zbl 1087.47054 · doi:10.1016/j.jmaa.2005.03.094
[24] H. Fukhar-ud-din and A. R. Khan, “Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces,” Computers & Mathematics with Applications, vol. 53, no. 9, pp. 1349-1360, 2007. · Zbl 1134.47049 · doi:10.1016/j.camwa.2007.01.008
[25] K.-K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301-308, 1993. · Zbl 0895.47048 · doi:10.1006/jmaa.1993.1309
[26] H. Fukhar-ud-din and S. H. Khan, “Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 821-829, 2007. · Zbl 1113.47055 · doi:10.1016/j.jmaa.2006.05.068
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.