×

On the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings. (English) Zbl 1097.47057

Summary: Several weak and strong convergence theorems are established for a modified Noor iterative scheme with errors for three asymptotically nonexpansive mappings in Banach spaces. Mann-type, Ishikawa-type, and Noor-type iterations are covered by the new iteration scheme. Our results extend and improve the recently announced ones [B.–L. Xu and M. A. Noor, J. Math. Anal. Appl. 267, No. 2, 444–453 (2002; Zbl 1011.47039); Y. J. Cho, H.–Y. Zhou and G.–T. Guo, Comput. Math. Appl. 47, No. 4–5, 707–717 (2004; Zbl 1081.47063)], and many others.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Chidume, C.E.; Moore, C., Fixed points iteration for pseudocontractive maps, Proc. amer. math. soc., 127, 4, 1163-1170, (1999) · Zbl 0913.47052
[2] Cho, Y.J.; Zhou, H.; Guo, G., Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. math. appl., 47, 707-717, (2004) · Zbl 1081.47063
[3] Das, G.; Debata, J.P., Fixed points of quasi-nonexpansive mappings, Indian J. pure appl. math., 17, 1263-1269, (1986) · Zbl 0605.47054
[4] Khan, S.H.; Un-din, H.F., Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear anal., 61, 1295-1301, (2005) · Zbl 1086.47050
[5] Maiti, M.; Gosh, M.K., Approximating fixed points by Ishikawa iterates, Bull. austral. math. soc., 40, 113-117, (1989) · Zbl 0667.47030
[6] Mann, W.R., Mean value methods in iterations, Proc. amer. math. soc., 4, 506-510, (1953) · Zbl 0050.11603
[7] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. math. soc., 733, 591-597, (1967) · Zbl 0179.19902
[8] Schu, J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. austral. math. soc., 43, 153-159, (1991) · Zbl 0709.47051
[9] Senter, H.F.; Dotson, W.G., Approximating fixed points of nonexpansive mappings, Proc. amer. math. soc., 44, 2, 375-380, (1974) · Zbl 0299.47032
[10] S. Suantai, Weak and strong convergence criteria of Noor iteration for asymptotically nonexpansive mappings, J. Math. Anal. Appl., in press · Zbl 1086.47057
[11] Takahashi, W.; Tamura, T., Convergence theorems for a pair of nonexpansive mappings, J. convex anal., 5, 1, 45-58, (1998) · Zbl 0916.47042
[12] 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
[13] Xu, Y., Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. math. anal. appl., 224, 91-101, (1998) · Zbl 0936.47041
[14] 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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.