Convergence of a One-step Iteration Scheme for Quasi-asymptotically Nonexpansive Mappings
Authors: Safeer Hussain Khan
In this paper, we use a one-step iteration scheme to approximate common fixed points of two quasi-asymptotically nonexpansive mappings. We prove weak and strong convergence theorems in a uniformly convex Banach space. Our results generalize the corresponding results of Yao and Chen  to a wider class of mappings while extend those of Khan, Abbas and Khan  to an improved one-step iteration scheme without any condition and improve upon many others in the literature.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1073427Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF
 M. Abbas, S. H. Khan, A. R. Khan and R.P. Agarwal , Common fixed points of two multivalued nonexpansive mappings by one-step iterative scheme, Applied Mathematics Letters 24 (2011) 97-102.
 G. Das and J. P. Debata, Fixed points of quasi-nonexpansive mappings, Indian J. Pure Appl. Math., 17 (1986), 1263-1269.
 H. Fukhar-ud-din and S. H. Khan, Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl. 328 (2007), 821-829.
 S. H. Khan, M. Abbas and A. R. Khan, Common fixed points of two nonexpansive mappings by a new one step iterative scheme, Iran. J. Sci. Technol. Trans. A Sci., Vol. 33, No. A3 (2009), 249-257.
 S. H. Khan andW. Takahashi, Approximating common fixed points of two asymptotically nonexpansive mappings, Sci. Math. Jpn., 53(1) (2001), 143-148.
 L. S. Liu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194(1)(1995),114-125.
 Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597.
 S. Plubtieng, and K. Ungchittrakool, Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semi groups, Nonlinear Anal. TMA., 67(7)(2007), 2306-2315.
 H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44(2) (1974), 375-380.
 W. Takahashi, Iterative methods for approximation of fixed points and their applications, J.Oper.Res.Soc. Jpn., 43(1) (2000), 87 -108.
 W. Takahashi and T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Analysis, 5(1) (1995), 45-58.
 K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301-308.
 Y. Xu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998), 91-101.
 B. Xu and M. A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002), 444-453.
 Y. Yao, Y.Chen, Weak and strong convergence of a modified Mann iteration for asymptotically nonexpansive mappings, Nonlin. Funct. Anal. Appl., 12(2007), 307-315.
 Y. Yao, M. A. Noor, Convergence of three-step iterations for asymptotically nonexpansive mappings, Appl. Math. Comput., 187(2007), 883- 892.