On the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings

Abstract In this paper, 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, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444–453; Y.J. Cho, H. Zhou, G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47 (2004) 707–717] and many others.

[1]  Safeer Hussain Khan,et al.  Weak and strong convergence of a scheme with errors for two nonexpansive mappings , 2005 .

[2]  Hong-Kun Xu,et al.  Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process , 1993 .

[3]  Yuguang Xu,et al.  Ishikawa and Mann Iterative Processes with Errors for Nonlinear Strongly Accretive Operator Equations , 1998 .

[4]  Suthep Suantai,et al.  Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings , 2005 .

[5]  W. G. Dotson,et al.  Approximating fixed points of nonexpansive mappings , 1974 .

[6]  W. G. Dotson,et al.  Fixed points of quasi-nonexpansive mappings , 1972, Journal of the Australian Mathematical Society.

[7]  J. Schu,et al.  Weak and strong convergence to fixed points of asymptotically nonexpansive mappings , 1991, Bulletin of the Australian Mathematical Society.

[8]  M. Maiti,et al.  Approximating fixed points by ishikawa iterates , 1989, Bulletin of the Australian Mathematical Society.

[9]  W. R. Mann,et al.  Mean value methods in iteration , 1953 .

[10]  Muhammad Aslam Noor,et al.  Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces , 2002 .

[11]  C. E. Chidume,et al.  FIXED POINT ITERATION FOR PSEUDOCONTRACTIVE MAPS , 1999 .

[12]  Yeol Je Cho,et al.  Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings☆ , 2004 .

[13]  Z. Opial.  Weak convergence of the sequence of successive approximations for nonexpansive mappings , 1967 .