Cho, Yeol Je; Zhou, Haiyun; Guo, Ginti Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. (English) Zbl 1081.47063 Comput. Math. Appl. 47, No. 4-5, 707-717 (2004). In the present paper, several weak and strong convergence theorems are established for three-step iterative schemes with errors for asymptotically nonexpansive mappings. The results presented extend and improve the recent ones announced by K.–K. Tan and H. K. Xu [Proc. Am. Math. Soc. 122, No. 3, 733–739 (1994; Zbl 0820.47071)], B.–L. Xu and M. A. Noor [J. Math. Anal. Appl. 267, No. 2, 444–453 (2002; Zbl 1011.47039)], and others. Reviewer: Zhilin Yang (Qingdao) Cited in 6 ReviewsCited in 131 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 65J15 Numerical solutions to equations with nonlinear operators Keywords:asymptotically nonexpansive mapping; three-step iterative scheme with errors; weak convergence; Xu’s inequality; mixed monotone operator Citations:Zbl 0820.47071; Zbl 1011.47039 PDFBibTeX XMLCite \textit{Y. J. Cho} et al., Comput. Math. Appl. 47, No. 4--5, 707--717 (2004; Zbl 1081.47063) Full Text: DOI References: [1] Schu, J., Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158, 407-413 (1991) · Zbl 0734.47036 [2] Schu, J., Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43, 153-159 (1991) · Zbl 0709.47051 [3] Liu, Q. H., Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member in a uniformly convex Banach space, J. Math. Anal. Appl., 266, 468-471 (2002) · Zbl 1057.47057 [4] Kruppel, M., On an inequality for nonexpansive mappings in uniformly convex Banach spaces, Rostock. Math. Kolloq., 51, 25-32 (1997) · Zbl 0891.47037 [5] Rhoades, B. E., Fixed point iterations for certain nonlinear mappings, J. Math. Anal. Appl., 183, 118-120 (1994) · Zbl 0807.47045 [6] Xu, H. K., Inequalities in Banach spaces with applications, Nonlinear Anal., 16, 1127-1138 (1991) · Zbl 0757.46033 [7] Tan, K. K.; Xu, H. K., Fixed point iteration processes for asymptotically nonexpansive mappings, (Proc. Amer. Math. Soc., 122 (1994)), 733-739 · Zbl 0820.47071 [8] Tan, K. K.; Xu, H. K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178, 301-308 (1993) · Zbl 0895.47048 [9] Xu, H. K., Existence and convergence for fixed points of mappings of asymptotically nonexpansive type, Nonlinear Anal., 16, 1139-1146 (1991) · Zbl 0747.47041 [10] Zhou, H. Y.; Gao, G. L.; Guo, J. T.; Cho, Y. J., Some general convergence principles with applications, Bull. Korean Math. Soc., 40, 351-363 (2003) · Zbl 1067.47084 [11] Xu, B. L.; Noor, M. A., Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 267, 444-453 (2002) · Zbl 1011.47039 [12] Bruck, R. E., A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math., 32, 107-116 (1979) · Zbl 0423.47024 [13] Opial, Z., Weak convergence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73, 591-597 (1967) · Zbl 0179.19902 [14] Senter, H. F.; Dotson, W. G., Approximating fixed points of non-expansive mappings, (Proc. Amer. Math. Soc., 44 (1974)), 375-380 · Zbl 0299.47032 [15] Goebel, K.; Kirk, W. A., A fixed point theorem for asymptotically non-expansive mappings, (Proc. Amer. Math. Soc., 35 (1972)), 171-174 · Zbl 0256.47045 [16] Xu, Y. G., Ishikawa and Mann iterative methods with errors for nonlinear accretive operator equations, J. Math. Anal. Appl., 224, 91-101 (1998) · Zbl 0936.47041 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.