Hafiz Fukhar-ud-din

An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces

In this article, we propose and analyze an implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces. Results concerning Δ-convergence as well as strong convergence of the proposed algorithm are proved. Our results are refinement and generalization of several recent results in CAT(0) spaces and uniformly convex Banach spaces.Mathematics Subject Classification (2010): Primary: 47H09; 47H10; Secondary: 49M05.

Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces

We introduce three-step iterative schemes with errors for two and three nonexpansive maps and establish weak and strong convergence theorems for these schemes. Mann-type and Ishikawa-type convergence results are included in the analysis of these new iteration schemes. The results presented in this paper substantially improve and extend the results due to [S.H. Khan, H. Fukhar-ud-din, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal. 8 (2005) 1295-1301], [N. Shahzad, Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Anal. 61 (2005) 1031-1039], [W. Takahashi, T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal. 5 (1995) 45-58], [K.K. Tan, H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993) 301-308] and [H.F. Senter, W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974) 375-380].

CONVERGENCE OF TWO-STEP ITERATIVE SCHEME WITH ERRORS FOR TWO ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

A two-step iterative scheme with errors has been studied to approximate the common fixed points of two asymptotically nonexpansive mappings through weak and strong convergence in Banach spaces.

Convergence of implicit iterates with errors for mappings with unbounded domain in Banach spaces

We prove that an implicit iterative process with errors converges weakly and strongly to a common fixed point of a finite family of asymptotically quasi-nonexpansive mappings on unbounded sets in a uniformly convex Banach space. Our results generalize and improve upon, among others, the corresponding recent results of Sun (2003) in the following two different directions: (i) domain of the mappings is unbounded, (ii) the iterative sequence contains an error term.

Approximating Fixed Points of Some Maps in Uniformly Convex Metric Spaces

We study strong convergence of the Ishikawa iterates of qasi-nonexpansive (generalized nonexpansive) maps and some related results in uniformly convex metric spaces. Our work improves and generalizes the corresponding results existing in the literature for uniformly convex Banach spaces.

Convergence analysis of a general iteration schema of nonlinear mappings in hyperbolic spaces

Iterative schemas are ubiquitous in the area of abstract nonlinear analysis and still remain as a main tool for approximation of fixed points of generalizations of nonexpansive maps. The analysis of general iterative schemas, in a more general setup, is a problem of interest in theoretical numerical analysis. Therefore, we propose and analyze a general iterative schema for two finite families of asymptotically quasi-nonexpansive maps in hyperbolic spaces. Results concerning △-convergence as well as strong convergence of the proposed iteration are proved. It is instructive to compare the proposed general iteration schema and the consequent convergence results with that of several recent results in CAT(0) spaces and uniformly convex Banach spaces.MSC:47H09, 47H10, 49M05.

Approximating common fixed points in hyperbolic spaces

AbstractWe establish strong convergence and Δ-convergence theorems of an iteration scheme associated to a pair of nonexpansive mappings on a nonlinear domain. In particular we prove that such a scheme converges to a common fixed point of both mappings. Our results are a generalization of well-known similar results in the linear setting. In particular, we avoid assumptions such as smoothness of the norm, necessary in the linear case. MSC:47H09, 46B20, 47H10, 47E10.

Strong Convergence of an Implicit Algorithm in CAT(0) Spaces

We establish strong convergence of an implicit algorithm to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive maps in CAT spaces. Our work improves and extends several recent results from the current literature.

Convergence of a general algorithm of asymptotically nonexpansive maps in uniformly convex hyperbolic spaces

Abstract In this paper, we establish convergence theorems for a general algorithm of an asymptotically nonexpansive map in a uniformly convex hyperbolic space. Our results generalize simultaneously the approximation results of Rhoades (1994) [18], Suantai (2005) [20] and Xu and Noor (2002) [26] on a nonlinear domain. Our results are refinements and generalizations of the corresponding ones in uniformly convex Banach spaces and CAT ( 0 ) spaces.

Fixed point approximation of asymptotically nonexpansive mappings in hyperbolic spaces

Convergence theorems are established in a hyperbolic space for the modified Noor iterations with errors of asymptotically nonexpansive mappings. The obtained results extend and improve the several known results in Banach spaces and CAT(0) spaces simultaneously.