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Convergence of Implicit and Explicit Schemes for an Asymptotically Nonexpansive Mapping in -Uniformly Smooth and Strictly Convex Banach Spaces
Meng Wen,Changsong Hu,Zhiyu Wu
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/474031
Abstract: We introduce a new iterative scheme with Meir-Keeler contractions for an asymptotically nonexpansive mapping in -uniformly smooth and strictly convex Banach spaces. We also proved the strong convergence theorems of implicit and explicit schemes. The results obtained in this paper extend and improve many recent ones announced by many others.
Iterative scheme for a nonexpansive mapping, an η-strictly pseudo-contractive mapping and variational inequality problems in a uniformly convex and 2-uniformly smooth Banach space
Atid Kangtunyakarn
Fixed Point Theory and Applications , 2013, DOI: 10.1186/1687-1812-2013-23
Abstract: In this paper, we introduce an iterative scheme by the modification of Mann’s iteration process for finding a common element of the set of solutions of a finite family of variational inequality problems and the set of fixed points of an η-strictly pseudo-contractive mapping and a nonexpansive mapping. Moreover, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of -strictly pseudo-contractive mappings for every in uniformly convex and 2-uniformly smooth Banach spaces.
Compact asymptotic center and common fixed point in strictly convex Banach spaces  [PDF]
Ali Abkar,Mohammad Eslamian
Mathematics , 2011,
Abstract: In this paper, we present some common fixed point theorems for a commuting pair of mappings, including a generalized nonexpansive single valued mapping and a generalized nonexpansive multivalued mapping in strictly convex Banach spaces. The results obtained in this paper extend and improve some recent known results.
A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces  [PDF]
Pongsakorn Sunthrayuth,Poom Kumam
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/560671
Abstract: We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others. 1. Introduction Throughout this paper we denoted by and the set of all positive integers and all positive real numbers, respectively. Let be a real Banach space, and let be a nonempty closed convex subset of . A mapping of into itself is said to be nonexpansive if for each . We denote by the set of fixed points of . We know that is nonempty if is bounded; for more detail see [1]. A one-parameter family from of into itself is said to be a nonexpansive semigroup on if it satisfies the following conditions: (i) ;(ii) for all ;(iii)for each the mapping is continuous;(iv) for all and . We denote by the set of all common fixed points of , that is, . We know that is nonempty if is bounded; see [2]. Recall that a self-mapping is a contraction if there exists a constant such that for each . As in [3], we use the notation to denote the collection of all contractions on , that is, . Note that each has a unique fixed point in . In the last ten years, the iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [3–5]. Let be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let be a strongly positive bounded linear operator on : that is, there is a constant with property A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space : where is the fixed point set of a nonexpansive mapping on and is a given point in . In 2003, Xu [3] proved that the sequence generated by converges strongly to the unique solution of the minimization problem (1.2) provided that the sequence satisfies certain conditions. Using the viscosity approximation method, Moudafi [6] introduced the iterative process for nonexpansive mappings (see [3, 7] for further developments in both Hilbert and Banach spaces) and proved that if is a real Hilbert space, the sequence generated by the following algorithm: where is a contraction mapping with constant and satisfies certain conditions, converges strongly to a fixed point of in which is unique solution of the variational inequality: In 2006, Marino and Xu [8] combined the iterative method (1.3)
Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces  [PDF]
Rattanaporn Wangkeeree
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/643740
Abstract: Let be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from to . Let be a nonexpansive semigroup on such that , and is a contraction on with coefficient . Let be -strongly accretive and -strictly pseudocontractive with and a positive real number such that . When the sequences of real numbers and satisfy some appropriate conditions, the three iterative processes given as follows: , , , , and , converge strongly to , where is the unique solution in of the variational inequality , . Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others. 1. Introduction Let be a real Banach space. A mapping of into itself is said to be nonexpansive if for each . We denote by the set of fixed points of . A mapping is called -contraction if there exists a constant such that for all . A family of mappings of into itself is called a nonexpansive semigroup on if it satisfies the following conditions: (i) for all ; (ii) for all ; (iii) for all and ; (iv) for all , the mapping is continuous. We denote by the set of all common fixed points of , that is, In [1], Shioji and Takahashi introduced the following implicit iteration in a Hilbert space where is a sequence in and is a sequence of positive real numbers which diverges to . Under certain restrictions on the sequence , Shioji and Takahashi [1] proved strong convergence of the sequence to a member of . In [2], Shimizu and Takahashi studied the strong convergence of the sequence defined by in a real Hilbert space where is a strongly continuous semigroup of nonexpansive mappings on a closed convex subset of a Banach space and . Using viscosity method, Chen and Song [3] studied the strong convergence of the following iterative method for a nonexpansive semigroup with in a Banach space: where is a contraction. Note however that their iterate at step is constructed through the average of the semigroup over the interval . Suzuki [4] was the first to introduce again in a Hilbert space the following implicit iteration process: for the nonexpansive semigroup case. In 2002, Benavides et al. [5], in a uniformly smooth Banach space, showed that if satisfies an asymptotic regularity condition and fulfills the control conditions , , and , then both the implicit iteration process (1.5) and the explicit iteration process (1.6), converge to a same point of . In 2005, Xu [6] studied the strong convergence of the implicit iteration process (1.2) and (1.5) in a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping.
Convergence of Ishikawa iterates of generalized nonexpansive mappings  [cached]
M. K. Ghosh,L. Debnath
International Journal of Mathematics and Mathematical Sciences , 1997, DOI: 10.1155/s0161171297000707
Abstract: This paper is concerned with the convergence of Ishikawa iterates of generalized nonexpansive mappings in both uniformly convex and strictly convex Banach spaces. Several fixed point theorems are discussed.
Strong Convergence Theorems for Nonexpansive Semigroups and Variational Inequalities in Banach Spaces
Haiqing Wang,Yongfu Su,Hong Zhang
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/641479
Abstract: Let be a uniformly convex Banach space and ={()∶0≤<∞} be a nonexpansive semigroup such that ?()=>0(())≠?. Consider the iterative method that generates the sequence {} by the algorithm
Weak and strong convergence theorems for nonexpansive semigroups in Banach spaces  [cached]
Atsushiba Sachiko,Takahashi Wataru
Fixed Point Theory and Applications , 2005,
Abstract: We introduce an implicit iterative process for a nonexpansive semigroup and then we prove a weak convergence theorem for the nonexpansive semigroup in a uniformly convex Banach space which satisfies Opial's condition. Further, we discuss the strong convergence of the implicit iterative process.
Weak and strong convergence theorems for nonexpansive semigroups in Banach spaces
Sachiko Atsushiba,Wataru Takahashi
Fixed Point Theory and Applications , 2005, DOI: 10.1155/fpta.2005.343
Abstract: We introduce an implicit iterative process for a nonexpansive semigroup and then we prove a weak convergence theorem for the nonexpansive semigroup in a uniformly convex Banach space which satisfies Opial's condition. Further, we discuss the strong convergence of the implicit iterative process.
Common Solutions of Generalized Mixed Equilibrium Problems, Variational Inclusions, and Common Fixed Points for Nonexpansive Semigroups and Strictly Pseudocontractive Mappings  [PDF]
Poom Kumam,Usa Hamphries,Phayap Katchang
Journal of Applied Mathematics , 2011, DOI: 10.1155/2011/953903
Abstract: We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. Our results improve and extend the corresponding results of Martinez-Yanes and Xu (2006), Shehu (2011), Zhang et al. (2008), and many authors. 1. Introduction Throughout this paper, we assume that is a real Hilbert space with inner product and norm are denoted by and , respectively. Let denote the family of all subsets of , and let be a closed-convex subset of . Recall that a mapping is said to be a -strict pseudocontraction [1] if there exists such that where denotes the identity operator on . When , is said to be nonexpansive [2] if And when , is said to be pseudocontraction if Clearly, the class of -strict pseudocontraction falls into the one between classes of nonexpansive mappings and pseudocontraction mapping. We denote the set of fixed points of by . A family of mappings of into itself is called a nonexpansive semigroup on if it satisfies the following conditions: (i) for all ,(ii) for all ,(iii) for all and ,(iv)for all is continuous. We denote by the set of all common fixed points of , that is, . It is known that is closed and convex. Let be a single-valued nonlinear mapping, and let be a set-valued mapping. We consider the following variational inclusion problem, which is to find a point such that where is the zero vector in H. The set of solutions of problem (1.4) is denoted by . Let the set-valued mapping be a maximal monotone. We define the resolvent operator associate with and as follows: where is a positive number. It is worth mentioning that the resolvent operator is single-valued, nonexpansive, and 1-inverse-strongly monotone ([3, 4]). Let be a bifunction of into , where is the set of real numbers, let be a mapping, and let be a real-valued function. The generalized mixed equilibrium problem is for finding such that The set of solutions of (1.6) is denoted by , that is, If , then the problem (1.6) is reduced into the
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