Abdul Rahim Khan

Coupled common fixed point results in two generalized metric spaces

In this paper, study of necessary conditions for the existence of unique coupled common fixed point of contractive type mappings in the context of two generalized metric spaces is initiated. These results generalize several comparable results from the current literature. We also provide illustrative examples in support of our new results.

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.

Common fixed points of two multivalued nonexpansive mappings by one-step iterative scheme

In this paper, we introduce a new one-step iterative process to approximate common fixed points of two multivalued nonexpansive mappings in a real uniformly convex Banach space. We establish weak and strong convergence theorems for the proposed process under some basic boundary conditions.

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].

Common fixed-point results in best approximation theory

A common fixed-point generalization of the results of Dotson, Tarafdar, and Taylor is obtained which in turn extends a recent theorem by Jungck and Sessa to locally convex spaces. As applications of our work, we improve and unify well-known results on fixed points and common fixed points of best approximation.

Applications of the Best Approximation Operator to *-Nonexpansive Maps in Hilbert Spaces

Abstract The notion of a *-nonexpansive multivalued map is different from that of a continuous map. We give some Ky Fan type best approximation theorems for *-nonexpansive mappings defined on closed convex unbounded subsets of a Hilbert space. As applications of our theorems, we derive fixed point results under many boundary conditions. Approximating sequences to the fixed points are also constructed.

APPROXIMATION OF *-NONEXPANSIVE RANDOM MULTIVALUED OPERATORS ON BANACH SPACES

We establish the existence and approximation of solutions to the operator inclusion y 2 Ty for deterministic and random cases for a nonexpansive and *-nonexpansive multivalued mapping T defined on a closed bounded (not necessarily convex) subset C of a Banach space. We also prove random fixed points and approximation results for *-nonexpansive random operators defined on an unbounded subset C of a uniformly convex Banach space.

Analytical and numerical treatment of Jungck-type iterative schemes

In this paper, we introduce a new and general Jungck-type iterative scheme for a pair of nonself mappings and study its strong convergence, stability and data dependence. It is exhibited that our iterative scheme has much better convergence rate than those of Jungck-Mann, Jungck-Ishikawa, Jungck-Noor and Jungck-CR iterative schemes. Numerical examples in support of validity and applications of our results are provided. Our results are extension, improvement and generalization of many known results in the literature of fixed point theory.

Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications

In this paper, we introduce the iterative scheme due to Khan, Domlo and Fukhar-ud-din (2008) [8] in convex metric spaces and establish strong convergence of this scheme to a unique common fixed point of a finite family of asymptotically quasi-nonexpansive mappings. As a consequence of our result, we obtain some related convergence theorems. Our results generalize some recent results obtained in [8].

Common fixed point and invariant approximation in hyperbolic ordered metric spaces

We prove a common fixed point theorem for four mappings defined on an ordered metric space and apply it to find new common fixed point results. The existence of common fixed points is established for two or three noncommuting mappings where T is either ordered S-contraction or ordered asymptotically S-nonexpansive on a nonempty ordered starshaped subset of a hyperbolic ordered metric space. As applications, related invariant approximation results are derived. Our results unify, generalize, and complement various known comparable results from the current literature.2010 Mathematics Subject Classification:47H09, 47H10, 47H19, 54H25.