Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. (English) Zbl 1081.47063

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.


47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
65J15 Numerical solutions to equations with nonlinear operators
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[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
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