Abstract:
We introduce a new iterative sequence for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then, we study the strong convergence of the sequences. With an appropriate setting, we obtain the corresponding results due to Takahashi-Takahashi and Takahashi-Zembayashi. Some of our results are established with weaker assumptions.

Abstract:
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. 1. Introduction Let be a nonempty closed convex subset of a real Hilbert space . A mapping is said to be Lipschitzian if there exists a positive constant such that In this case, is also said to be -Lipschitzian. Clearly, if is -Lipschitzian and , then is -Lipschitzian. Throughout the paper, we assume that every Lipschitzian mapping is -Lipschitzian with . If , then is known as a nonexpansive mapping. We denote by the set of fixed points of . If is nonempty bounded closed convex and is a nonexpansive of into itself, then (see [1]). There are many methods for approximating fixed points of a nonexpansive mapping. In 1953, Mann [2] introduced the iteration as follows: a sequence defined by where the initial guess element is arbitrary and is a real sequence in . Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results is proved by Reich [3]. In an infinite-dimensional Hilbert space, Mann iteration can conclude only weak convergence [4]. Attempts to modify the Mann iteration method (1.2) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [5] proposed the following modification of Mann iteration method (1.2): where denotes the metric projection from onto a closed convex subset of . They prove that if the sequence bounded above from one, then defined by (1.3) converges strongly to . Takahashi et al. [6] modified (1.3) so-called the shrinking projection method for a countable family of nonexpansive mappings as follows: and prove that if the sequence bounded above from one, then defined by (1.4) converges strongly to . Recently, the present authors [7] extended (1.3) to obtain a strong convergence theorem for common fixed points of a countable family of -Lipschitzian mappings by where as and prove that defined by (1.5) converges strongly to . In this paper, we establish strong convergence theorems for finding common fixed points of a countable family of Lipschitzian mappings in a real Hilbert space. Moreover, we also apply our results for asymptotically nonexpansive mappings. 2. Preliminaries Let be a real Hilbert space with inner product and norm . Then, for all and . For any points in , the following generalized identity holds: where

Abstract:
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.

Abstract:
We introduce a new class of normalized norms on 2 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.

Abstract:
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.

Abstract:
We prove that a sequence generated by the monotone CQ-method converges strongly to a common fixed point of a countable family of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. Our result is applicable to a wide class of mappings.

Abstract:
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 Takahashi's question in positive. We also introduce the concept of strongly generalized nonexpansive mappings and prove the analogue version of the result above for Ibaraki-Takahashi's generalized nonexpansive mappings. The duality theorem for two classes of strongly relatively nonexpansive mappings and of strongly generalized nonexpansive mappings is proved.

Abstract:
We prove that a sequence generated by the monotone CQ-method converges strongly to a common fixed point of a countable family of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. Our result is applicable to a wide class of mappings.

Abstract:
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 Takahashi's question in positive. We also introduce the concept of strongly generalized nonexpansive mappings and prove the analogue version of the result above for Ibaraki-Takahashi's generalized nonexpansive mappings. The duality theorem for two classes of strongly relatively nonexpansive mappings and of strongly generalized nonexpansive mappings is proved.