Abstract:
We give the matrix characterizations from Nakano vector-valued sequence space ℓ(X,p) and Fr(X,p) into the sequence spaces Er, ℓ∞, ℓ¯∞(q), bs, and cs, where p=(pk) and q=(qk) are bounded sequences of positive real numbers such that Pk>1 for all k∈ℕ and r≥0.

Abstract:
We prove a weak convergence theorem of the modified Mann iteration process for a uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space. We also introduce two kinds of new monotone hybrid methods and obtain strong convergence theorems for an infinitely countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings in a Hilbert space. The results improve and extend the corresponding ones announced by Kim and Xu (2006) and Nakajo and Takahashi (2003).

Abstract:
We introduce a new iterative scheme to approximate a common fixed point for a finite family of generalized asymptotically quasinonexpansive mappings. Several strong and weak convergence theorems of the proposed iteration in Banach spaces are established. The main results obtianed in this paper generalize and refine many known results in the current literature. 1. Introduction Let be a convex subset of a Banach space , and let be a family of self-mappings of . Suppose that for all and For , let be the sequence generated by the following algorithm: where for all . The iterative process (1.1) for a finite family of mappings introduced by Khan et al. [1], and the iterative process is the generalized form of the modified Mann (one-step) iterative process by Schu [2], the modified Ishikawa (two-step) iterative process by Tan and Xu [3], and the three-step iterative process by Xu and Noor [4]. Common fixed points of nonlinear mappings play an important role in solving systems of equations and inequalities. Many researchers [1, 5–19] are interested in studying approximation method for finding common fixed points of nonlinear mapping. Also, approximation methods for finding fixed points for nonexpansive mappings can be seen in [12–16, 20, 21]. In 2003, Sun [17] studied an implicit iterative scheme initiated by Xu and Ori [22] for a finite family of asymptotically quasinonexpansive mappings. Shahzand and Udomene [18], in 2006, proved some convergence theorems for the modified Ishikawa iterative process of two asymptotically quasinonexpence mappings to a common fixed point. Nammanee et al. [23] introduced a three-step iteration scheme for asymptotically nonexpansive mappings and proved weak and strong convergence theorems of that iteration scheme under some control conditions. In 2007, Fukhar-ud-din and Khan [24] studied a new three-step iteration scheme for approximating a common fixed point of asymptotically nonexpansive mappings in uniformly convex Banach spaces. Shahzad and Zegeye [19] introduced a new concept of generalized asymptotially nonexpansive mappings and proved some strong convergence theorems for fixed points of finite family of this class. Recently, Khan et al. [1] introduced the iterative sequence (1.1) for a finite family of asymptotically quasinonexpansive mappings in Banach spaces. Motivated by Khan et al. [1], we introduce a new iterative scheme for finding a common fixed point of a finite family of generalized asymptotically quasinonexpansive mappings as follows: For , let be the sequence generated by where for all . The aim of this paper is

Abstract:
We introduce a new hybrid iterative scheme for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of relatively quasi-nonexpansive mappings in the framework of Banach spaces. We prove the strong convergence theorem by the shrinking projection method. In addition, the results obtained in this paper can be applied to a system of variational inequality problems and to a system of convex minimization problems in a Banach space. 1. Introduction Let be a real Banach space, and let be the dual of . Let be a closed and convex subset of . Let be bifunctions from to , where is the set of real numbers and is an arbitrary index set. The system of equilibrium problems is to find such that If is a singleton, then problem (1.1) reduces to find such that The set of solutions of the equilibrium problem (1.2) is denoted by . Combettes and Hirstoaga [1] introduced an iterative scheme for finding a common element in the solutions set of problem (1.1) in a Hilbert space and obtained a weak convergence theorem. In 2004, Matsushita and Takahashi [2] introduced the following algorithm for a relatively nonexpansive mapping in a Banach space : for any initial point , define the sequence by where is the duality mapping on , is the generalized projection from onto , and is a sequence in . They proved that the sequence converges weakly to fixed point of under some suitable conditions on . In 2008, Takahashi and Zembayashi [3] introduced the following iterative scheme which is called the shrinking projection method for a relatively nonexpansive mapping and an equilibrium problem in a Banach space : They proved that the sequence converges strongly to under some appropriate conditions. 2. Preliminaries and Lemmas Let be a real Banach space, and let be the unit sphere of . A Banach space is said to be strictly convex if, for any , It is also said to be uniformly convex if, for each , there exists such that, for any , It is known that a uniformly convex Banach space is reflexive and strictly convex. The function which is called the modulus of convexity of is defined as follows: The space is uniformly convex if and only if for all . A Banach space is said to be smooth if the limit exists for all . It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for . The duality mapping is defined by for all . If is a Hilbert space, then , where is the identity operator. It is also known that, if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subset

Abstract:
We introduce an iterative method for finding a common fixedpoint of a countable family of multivalued quasi-nonexpansive mapping {} in auniformly convex Banach space. We prove that under certain control conditions, the iterative sequence generated by our method is an approximating fixed pointsequence of each . Some strong convergence theorems of the proposed methodare also obtained for the following cases: all are continuous and one of ishemicompact, and the domain is compact.

Abstract:
The β-dual of a vector-valued sequence space is defined and studied. We show that if an X-valued sequence space E is a BK-space having AK property, then the dual space of E and its β-dual are isometrically isomorphic. We also give characterizations of β-dual of vector-valued sequence spaces of Maddox ℓ(X,p), ℓ∞(X,p), c0(X,p), and c(X,p).

Abstract:
We consider the generalized Cesàro sequence spaces defined by Suantai (2003) and consider it equipped with the Amemiya norm. The main purpose of this paper is to show that ces(p) equipped with the Amemiya norm is rotund and has uniform Kadec-Klee property.

Abstract:
We define a generalized Cesàro sequence space ces(p), where p=(pk) is a bounded sequence of positive real numbers, and consider it equipped with the Luxemburg norm. The main purpose of this paper is to show that ces(p) is k-nearly uniform convex (k-NUC) for k≥2 when limn→∞infpn>1. Moreover, we also obtain that the Cesàro sequence space cesp(where 1

Abstract:
We introduce a new modified Halpern iteration for a countable infinite family of nonexpansive mappings {} in convex metric spaces. We prove that the sequence {} generated by the proposed iteration is an approximating fixed point sequence of a nonexpansive mapping when {} satisfies the AKTT-condition, and strong convergence theorems of the proposed iteration to a common fixed point of a countable infinite family of nonexpansive mappings in CAT(0) spaces are established under AKTT-condition and the SZ-condition. We also generalize the concept of W-mapping for a countable infinite family of nonexpansive mappings from a Banach space setting to a convex metric space and give some properties concerning the common fixed point set of this family in convex metric spaces. Moreover, by using the concept of W-mappings, we give an example of a sequence of nonexpansive mappings defined on a convex metric space which satisfies the AKTT-condition. Our results generalize and refine many known results in the current literature.

Abstract:
Let be a real uniformly convex Banach space and a closed convex nonempty subset of . Let {}=1 be a finite family of nonexpansive self-mappings of . For a given 1∈, let {} and {()}, =1,2,…,, be sequences defined (0)=,(1)=(1)11(0)