Abstract:
We prove that the moduli of U-convexity, introduced by Gao (1995), of the ultrapower X˜ of a Banach space X and of X itself coincide whenever X is super-reflexive. As a consequence, some known results have beenproved and improved. More precisely, we prove that uX(1)>0 implies that both X and the dual space X∗ of X have uniform normal structure and hence the “worth” property in Corollary 7 of Mazcuñán-Navarro (2003) can be discarded.

Abstract:
We prove a convergence theorem by the new iterative method introduced by Takahashi et al. (2007). Our result does not use Bochner integrals so it is different from that by Takahashi et al. We also correct the strong convergence theorem recently proved by He and Chen (2007).

Abstract:
Motivated by Halpern's result, we prove strong convergence theorem of an iterative sequence in CAT(0) spaces. We apply our result to find a common fixed point of a family of nonexpansive mappings. A convergence theorem for nonself mappings is also discussed.

Abstract:
We prove that every firmly nonexpansive-like mapping from a closed convex subset of a smooth, strictly convex and reflexive Banach pace into itself has a fixed point if and only if is bounded. We obtain a necessary and sufficient condition for the existence of solutions of an equilibrium problem and of a variational inequality problem defined in a Banach space.

Abstract:
Motivated by Halpern's result, we prove strong convergence theorem of an iterative sequence in CAT(0) spaces. We apply our result to find a common fixed point of a family of nonexpansive mappings. A convergence theorem for nonself mappings is also discussed.

Abstract:
We prove a convergence theorem by the new iterative method introduced by Takahashi et al. (2007). Our result does not use Bochner integrals so it is different from that by Takahashi et al. We also correct the strong convergence theorem recently proved by He and Chen (2007).

Abstract:
We improve the viscosity approximation process for approximation of a fixed point of a quasi-nonexpansive mapping in a Hilbert space proposed by Maingé (2010). An example beyond the scope of the previously known result is given.

Abstract:
Inspired by the concept of U-spaces introduced by Lau, (1978), we introduced the class of semi-uniform Kadec-Klee spaces, which is a uniform version of semi-Kadec-Klee spaces studied by Vlasov, (1972). This class of spaces is a wider subclass of spaces with weak normal structure and hence generalizes many known results in the literature. We give a characterization for a certain direct sum of Banach spaces to be semi-uniform Kadec-Klee and use this result to construct a semi-uniform Kadec-Klee space which is not uniform Kadec-Klee. At the end of the paper, we give a remark concerning the uniformly alternative convexity or smoothness introduced by Kadets et al., (1997). 1. Introduction Let be a real Banach space with the unit sphere and the closed unit ball . In this paper, the strong and weak convergences of a sequence in to an element are denoted by and , respectively. We also let Definition 1.1 (see [1]). We say that a Banach space is a Kadec-Klee space if A uniform version of the KK property is given in the following definition. Definition 1.2 (see [2]). We say that a Banach space is uniform Kadec-Klee if for every there exists a such that Two properties above are weaker than the following one. Definition 1.3 (see [3]). We say that a Banach space is uniformly convex if for every there exists a such that Let us summarize a relationship between these properties in the following implication diagram: In the literature, there are some generalizations of UC and KK. Definition 1.4 (see [4]). We say that a Banach space is a -space if for every there exists a such that Here . Definition 1.5 (see [5]). We say that a Banach space is semi-Kadec-Klee if Some interesting results concerning semi-KK property are studied by Megginson [6]. Remark 1.6. It is clear that Remark 1.7. A Banach space is semi-KK if and only if We now introduce a property lying between -space and semi-KK. Definition 1.8. We say that a Banach space is semi-uniform Kadec-Klee if for every there exists a such that In this paper, we prove that semi-UKK property is a nice generalization of -space and semi-KK property. Moreover, every semi-UKK space has weak normal structure. We also give a characterization of the direct sum of finitely many Banach spaces which is semi-KK and semi-UKK. We use such a characterization to construct a Banach space which is semi-UKK but not UKK. Finally we give a remark concerning the uniformly alternative convexity or smoothness introduced by Kadets et al. [7]. 2. Results 2.1. Some Implications For a sequence and satisfying for all , we let It is clear that . Theorem

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