×

Monotone hybrid projection algorithms for an infinitely countable family of Lipschitz generalized asymptotically quasi-nonexpansive mappings. (English) Zbl 1195.47042

Summary: We prove a weak convergence theorem of the modified Mann iteration process for a uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space. We also introduce two kinds of new monotone hybrid methods and obtain strong convergence theorems for an infinitely countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings in a Hilbert space. The results improve and extend the corresponding ones announced by Kim and Xu (2006) and Nakajo and Takahashi (2003).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] 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
[2] W. A. Kirk, “Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type,” Israel Journal of Mathematics, vol. 17, pp. 339-346, 1974. · Zbl 0286.47034 · doi:10.1007/BF02757136
[3] R. Bruck, T. Kuczumow, and S. Reich, “Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property,” Colloquium Mathematicum, vol. 65, no. 2, pp. 169-179, 1993. · Zbl 0849.47030
[4] Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591-597, 1967. · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[5] W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506-510, 1953. · Zbl 0050.11603 · doi:10.2307/2032162
[6] H. H. Bauschke, E. Matou, and S. Reich, “Projection and proximal point methods: convergence results and counterexamples,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 5, pp. 715-738, 2004. · Zbl 1059.47060 · doi:10.1016/j.na.2003.10.010
[7] A. Genel and J. Lindenstrauss, “An example concerning fixed points,” Israel Journal of Mathematics, vol. 22, no. 1, pp. 81-86, 1975. · Zbl 0314.47031 · doi:10.1007/BF02757276
[8] 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. · Zbl 0423.47026 · doi:10.1016/0022-247X(79)90024-6
[9] 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. · Zbl 1035.47048 · doi:10.1016/S0022-247X(02)00458-4
[10] T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 5, pp. 1140-1152, 2006. · Zbl 1090.47059 · doi:10.1016/j.na.2005.05.059
[11] P. Cholamjiak, “A hybrid iterative scheme for equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 719360, 18 pages, 2009. · Zbl 1167.65379 · doi:10.1155/2009/719360
[12] 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. · Zbl 1116.47053 · doi:10.1016/j.jmaa.2006.06.055
[13] 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. · Zbl 1105.47060 · doi:10.1016/j.na.2005.08.018
[14] W. Nilsrakoo and S. Saejung, “Weak and strong convergence theorems for countable Lipschitzian mappings and its applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2695-2708, 2008. · Zbl 1170.47041 · doi:10.1016/j.na.2007.08.044
[15] W. Nilsrakoo and S. Saejung, “Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications,” Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 154-167, 2009. · Zbl 1162.47051 · doi:10.1016/j.jmaa.2009.03.002
[16] D. R. Sahu, H.-K. Xu, and J.-C. Yao, “Asymptotically strict pseudocontractive mappings in the intermediate sense,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 10, pp. 3502-3511, 2009. · Zbl 1221.47122 · doi:10.1016/j.na.2008.07.007
[17] A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359-370, 2007. · Zbl 1147.47052 · doi:10.1007/s10957-007-9187-z
[18] W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276-286, 2008. · Zbl 1134.47052 · doi:10.1016/j.jmaa.2007.09.062
[19] H. Zhou, “Strong convergence theorems for a family of Lipschitz quasi-pseudo-contractions in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 1-2, pp. 120-125, 2009. · Zbl 1225.47123 · doi:10.1016/j.na.2008.10.059
[20] H. Zhou and Y. Su, “Strong convergence theorems for a family of quasi-asymptotic pseudo-contractions in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 11, pp. 4047-4052, 2009. · Zbl 1223.47111 · doi:10.1016/j.na.2008.08.013
[21] M. O. Osilike and S. C. Aniagbosor, “Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings,” Mathematical and Computer Modelling, vol. 32, no. 10, pp. 1181-1191, 2000. · Zbl 0971.47038 · doi:10.1016/S0895-7177(00)00199-0
[22] H. K. Xu, “Inequality in Banach spacees with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, pp. 1127-1138, 1991. · Zbl 0757.46033
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.