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Weak and strong convergence theorems of a multistep iteration to a common fixed point of a family of nonself asymptotically nonexpansive mappings in banach spaces
Banerjee,Shrabani; Choudhury,Binayak S;
Cubo (Temuco) , 2012, DOI: 10.4067/S0719-06462012000300010
Abstract: in this paper we have defined a multistep iterative scheme with errors involving a family of asymptotically nonexpansive nonself mappings in banach spaces. a retraction has been used in the construction of theiteration. we prove here weak and strong convergences of the iteration to common fixed points of the family of asymptotically nonexpansive nonself mappings. we have used several concepts of banach space geometry. our results improve and extend some recent results.
Weak and strong convergence theorems of a multistep iteration to a common fixed point of a family of nonself asymptotically nonexpansive mappings in banach spaces  [cached]
Shrabani Banerjee,Binayak S Choudhury
Cubo : A Mathematical Journal , 2012,
Abstract: In this paper we have defined a multistep iterative scheme with errors involving a family of asymptotically nonexpansive nonself mappings in Banach spaces. A retraction has been used in the construction of theiteration. We prove here weak and strong convergences of the iteration to common fixed points of the family of asymptotically nonexpansive nonself mappings. We have used several concepts of Banach space geometry. Our results improve and extend some recent results. En este artículo definimos un esquema de multi paso iterativo con errores que involucran una familia de aplicaciones no expansivas y no auto asintóticamente en espacios deBanach. Una retracción se ha usado en la construcción de la iteración. Probamos convergencias débiles y fuertes de las iteraciones a puntos fijos clásicos de la familia de aplicaciones no expansivas no auto asintóticamente. Hemos usado varios conceptos de geometría en espacios de Banach. Nuestro resultado mejora y extiende algunos resultados recientes.
Fixed-point iteration processes for non-Lipschitzian mappings of asymptotically quasi-nonexpansive type
Daya Ram Sahu,Jong Soo Jung
International Journal of Mathematics and Mathematical Sciences , 2003, DOI: 10.1155/s0161171203302182
Abstract: We study convergences of Mann and Ishikawa iteration processes for mappings of asymptotically quasi-nonexpansive type in Banach spaces.
Convergence Theorems on a New Iteration Process for Two Asymptotically Nonexpansive Nonself-Mappings with Errors in Banach Spaces
Murat Ozdemir,Sezgin Akbulut,Hukmi Kiziltunc
Discrete Dynamics in Nature and Society , 2010, DOI: 10.1155/2010/307245
Abstract: We introduce a new two-step iterative scheme for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space. Weak and strong convergence theorems are established for this iterative scheme in a uniformly convex Banach space. The results presented extend and improve the corresponding results of Chidume et al. (2003), Wang (2006), Shahzad (2005), and Thianwan (2008).
Implicit predictor-corrector iteration process for finitely many asymptotically (quasi-)nonexpansive mappings
Ceng LC,Wong NC,Yao JC
Journal of Inequalities and Applications , 2006,
Abstract: We study an implicit predictor-corrector iteration process for finitely many asymptotically quasi-nonexpansive self-mappings on a nonempty closed convex subset of a Banach space . We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. In the case is a uniformly convex Banach space and the mappings are asymptotically nonexpansive, we verify the weak (resp., strong) convergence of this iteration process to a common fixed point of these mappings if Opial's condition is satisfied (resp., one of these mappings is semicompact). Our results improve and extend earlier and recent ones in the literature.
Strong Convergence of a Hybrid Iteration Scheme for Equilibrium Problems, Variational Inequality Problems and Common Fixed Point Problems, of Quasi-?-Asymptotically Nonexpansive Mappings in Banach Spaces
Jing Zhao
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/516897
Abstract: We introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed quasi-ϕ-asymptotically nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality problem for a γ-inverse strongly monotone mapping in Banach spaces. Then we study the strong convergence of the algorithm. Our results improve and extend the corresponding results announced by many others.
Approximating Common Fixed Points for a Finite Family of Asymptotically Nonexpansive Mappings Using Iteration Process with Errors Terms  [PDF]
Seyit Temir,Adem Kili?man
Abstract and Applied Analysis , 2013, DOI: 10.1155/2013/974317
Abstract: Let be a real Banach space and a nonempty closed convex subset of . Let be asymptotically nonexpansive mappings with sequence , , and , where is the set of fixed points of . Suppose that ,?? ,?? are appropriate sequences in and ,?? are bounded sequences in such that for . We give defined by The purpose of this paper is to study the above iteration scheme for approximating common fixed points of a finite family of asymptotically nonexpansive mappings and to prove weak and some strong convergence theorems for such mappings in real Banach spaces. The results obtained in this paper extend and improve some results in the existing literature. 1. Introduction Let be a nonempty subset of a real Banach space and let be a mapping. Let be the set of fixed points of . A mapping is called nonexpansive if for all . Similarly, is called asymptotically nonexpansive if there exists a sequence with such that for all and . The mapping is called uniformly L-Lipschitzian if there exists a positive constant such that for all and . It is easy to see that if is asymptotically nonexpansive, then it is uniformly -Lipschitzian with the uniform Lipschitz constant . The class of asymptotically nonexpansive mappings which is an important generalization of the class nonexpansive maps was introduced by Goebel and Kirk [1]. They proved that every asymptotically nonexpansive self-mapping of a nonempty closed convex bounded subset of a uniformly convex Banach space has a fixed point. The main tool for approximation of fixed points of generalizations of nonexpansive mappings remains iterative technique. Iterative techniques for nonexpansive self-mappings in Banach spaces including Mann type (one-step), Ishikawa type (two-step), and three-step iteration processes have been studied extensively by various authors; see, for example, ([2–8]). Recently, Chidume and Ali [9] defined (4) and constructed the sequence for the approximation of common fixed points of finite families of asymptotically nonexpansive mappings. Y?ld?r?m and ?zdemir [10] introduced an iteration scheme for approximating common fixed points of a finite family of asymptotically quasi-nonexpansive self-mappings and proved some strong and weak convergence theorems for such mappings in uniformly convex Banach spaces. Quan et al. [11] studied sufficient and necessary conditions for finite step iterative schemes with mean errors for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces to converge to a common fixed point. Peng [12] proved the convergence of finite step iterative schemes with mean errors for
Convergence of Viscosity Iteration Process for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mappings  [PDF]
Zhiming Cheng
Journal of Applied Mathematics , 2014, DOI: 10.1155/2014/642167
Abstract: We introduce a general iteration method for a finite family of generalized asymptotically quasi-nonexpansive mappings. The results presented in the paper extend and improve some recent results in the works by Shahzad and Udomene (2006); L. Qihou (2001); Khan et al. (2008). 1. Introduction and Preliminaries Let be a nonempty subset of a real Banach space and a self-mapping of . The set of fixed points of is denoted by and we assume that . The mapping is said to be(i)contractive mapping if there exists a constant in such that , for all ;(ii)asymptotically nonexpansive mapping if there exists a sequence in with such that , for all and ;(iii)asymptotically quasi-nonexpansive if there exists a sequence in with such that , for all , and ;(iv)generalized asymptotically quasi-nonexpansive [1] if there exist two sequences , in with and such that where ;(v)uniformly -Lipschitzian if there exists a constant such that , for all and ;(vi) uniform -Lipschitz if there are constants and such that , for all and ;(vii)semicompact if for a sequence in with , there exists a subsequence of such that . In (1), if for all , then becomes an asymptotically quasi-nonexpansive mapping; if and for all , then becomes a quasi-nonexpansive mapping. It is known that an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive and a uniformly -Lipschitzian mapping is uniform -Lipschitz. The mapping is said to be demiclosed at if for each sequence converging weakly to and converging strongly to , we have . A Banach space is said to satisfy Opial’s property if for each and each sequence weakly convergent to , the following condition holds for all : Let be a nonempty closed convex subset of a real Banach space and a finite family of asymptotically nonexpansive mappings of into itself. Suppose that , , and . Then we consider the following mapping of into itself: where (identity mapping). Such a mapping is called the modified -mapping generated by and (see [2, 3]). In the sequel, we assume that . In 2008, Khan et al. [4] introduced the following iteration process for a family of asymptotically quasi-nonexpansive mappings, for an arbitrary : where , , , and proved that the iterative sequence defined by (4) converges strongly to a common fixed point of the family of mappings if and only if , where . With the help of (3), we write (4) as Recently, Chang et al. [5] introduced the following iteration process of asymptotically nonexpansive mappings in Banach space: where and is a fixed contractive mapping, and necessary and sufficient conditions are given for the iterative
Some Weak Convergence Theorems for a Family of Asymptotically Nonexpansive Nonself Mappings  [cached]
Yan Hao,Sun Young Cho,Xiaolong Qin
Fixed Point Theory and Applications , 2010, DOI: 10.1155/2010/218573
Abstract: A one-step iteration with errors is considered for a family of asymptotically nonexpansive nonself mappings. Weak convergence of the purposed iteration is obtained in a Banach space.
Some Weak Convergence Theorems for a Family of Asymptotically Nonexpansive Nonself Mappings  [cached]
Hao Yan,Cho SunYoung,Qin Xiaolong
Fixed Point Theory and Applications , 2010,
Abstract: A one-step iteration with errors is considered for a family of asymptotically nonexpansive nonself mappings. Weak convergence of the purposed iteration is obtained in a Banach space.
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