Weerayuth Nilsrakoo

Strong Convergence to Common Fixed Points of Countable Relatively Quasi-Nonexpansive Mappings

Correspondence should be addressed to Satit Saejung,saejung@kku.ac.thReceived 30 August 2007; Accepted 24 December 2007Recommended by Simeon ReichWe prove that a sequence generated by the monotone CQ-method converges strongly to a commonfixed point of a countable family of relatively quasi-nonexpansive mappings in a uniformly convexand uniformly smooth Banach space. Our result is applicable to a wide class of mappings.Copyrightq2008 W. Nilsrakoo and S. Saejung. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings

In the present paper, we define and study a new three-step iterative scheme inspired by Suantai [J. Math. Anal. Appl. 311 (2005) 506-517]. This scheme includes many well-known iterations, for examples, modified Mann-type, modified Ishikawa-type iterative schemes, and the three-step iterative scheme of Xu and Noor. Several convergence theorems of this scheme are established for asymptotically nonexpansive mappings. Our results extend and improve the recent ones announced by Schu [J. Math. Anal. Appl. 158 (1991) 407-413; Bull. Aust. Math. Soc. 43 (1991) 153-159], Xu and Noor [J. Math. Anal. Appl. 267 (2002) 444-453], Suantai [J. Math. Anal. Appl. 311 (2005) 506-517], and many others. A misleading conclusion of Theorem 2.6 in Suantais paper is also corrected.

A reconsideration on convergence of three-step iterations for asymptotically nonexpansive mappings

Abstract We illustrate that the control conditions of the main convergence theorems of Yao and Noor [Convergence of three-step iterations for asymptotically nonexpansive mappings, Appl. Math. Comput. in press] are incorrect. We also provide new control conditions which are complementary to Nilsrakoo and Saejung’s results [W. Nilsrakoo, S. Saejung, A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings, Appl. Math. Comput. 181 (2006) 1026–1034].

The James constant of normalized norms on

We introduce a new class of normalized norms on which properly contains all absolute normalized norms. We also give a criterion for deciding whether a given norm in this class is uniformly nonsquare. Moreover, an estimate for the James constant is presented and the exact value of some certain norms is computed. This gives a partial answer to the question raised by Kato et al.

On the Fixed-Point Set of a Family of Relatively Nonexpansive and Generalized Nonexpansive Mappings

We prove that the set of common fixed points of a given countable family of relatively nonexpansive mappings is identical to the fixed-point set of a single strongly relatively nonexpansive mapping. This answers Kohsaka and Takahashis question in positive. We also introduce the concept of strongly generalized nonexpansive mappings and prove the analogue version of the result above for Ibaraki-Takahashis generalized nonexpansive mappings. The duality theorem for two classes of strongly relatively nonexpansive mappings and of strongly generalized nonexpansive mappings is proved.

Weak and Strong Convergence Theorems of an Implicit Iteration Process for a Countable Family of Nonexpansive Mappings

Using the implicit iteration and the hybrid method in mathematical programming, we prove weak and strong convergence theorems for finding common fixed points of a countable family of nonexpansive mappings in a real Hilbert space. Our results include many convergence theorems by Xu and Ori (2001) and Zhang and Su (2007) as special cases. We also apply our method to find a common element to the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem. Finally, we propose an iteration to obtain convergence theorems for a continuous monotone mapping.

Strong Convergence Theorems for a Countable Family of Lipschitzian Mappings

We modify the iterative method introduced by Kim and Xu (2006) for a countable family of Lipschitzian mappings by the hybrid method of Takahashi et al. (2008). Our results include recent ones concerning asymptotically nonexpansive mappings due to Plubtieng and Ungchittrakool (2007) and Zegeye and Shahzad (2008, 2010) as special cases.

Convergence theorems for a countable family of Lipschitzian mappings

This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We introduce a new condition for a class of mappings to obtain several weak and strong convergence theorems. This new condition is implied by many previous known conditions introduced by many authors. We also apply our results for a class of nonexpansive mappings and asymptotically nonexpansive mappings and we immediately obtain convergence theorems proved by Song-Chen, Kimura-Takahashi, Tan-Xu, and many others.

A Halpern-Mann Type Iteration for Fixed Point Problems of a Relatively Nonexpansive Mapping and a System of Equilibrium Problems

A new modified Halpern-Mann type iterative method is constructed. Strong convergence of the scheme to a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth is proved. The results presented in this work improve on the corresponding ones announced by many others.

An Implicit Iteration for a Countable Family of Nonexpansive Mappings in Banach Spaces

We introduce an implicit sequence for an infinite family of nonexpansive mappings in a uniformly convex Banach space. We prove weak and strong convergence theorems for finding a common fixed point of the mappings. Our results not only include Plubtieng et al. (Numer. Funct. Anal. Optim. 2007; 28:737–749), Kikkawa and Takahashi (Ann. Univ. Mariae Curie-Skłodowska Sect. A 2004; 58:69–78), Kimura and Takahashi (Set-Valued Anal. 2008; 16:597–619) as special cases but also are established under the weaker assumptions.