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Weak Convergence of the Projection Type Ishikawa Iteration Scheme for Two Asymptotically Nonexpansive Nonself-Mappings
Tanakit Thianwan
Abstract and Applied Analysis , 2011, DOI: 10.1155/2011/745451
Abstract: We study weak convergence of the projection type Ishikawa iteration scheme for two asymptotically nonexpansive nonself-mappings in a real uniformly convex Banach space which has a Fréchet differentiable norm or its dual ? has the Kadec-Klee property. Moreover, weak convergence of projection type Ishikawa iterates of two asymptotically nonexpansive nonself-mappings without any condition on the rate of convergence associated with the two maps in a uniformly convex Banach space is established. Weak convergence theorem without making use of any of the Opial's condition, Kadec-Klee property, or Fréchet differentiable norm is proved. Some results have been obtained which generalize and unify many important known results in recent literature.
Approximating fixed points of nonexpansive type mappings
Hemant K. Pathak,Mohammad S. Khan
International Journal of Mathematics and Mathematical Sciences , 2001, DOI: 10.1155/s0161171201003003
Abstract: In a uniformly convex Banach space, the convergence of Ishikawa iterates to a unique fixed point is proved for nonexpansive type mappings under certain conditions.
Approximating common fixed points of families of quasi-nonexpansive mappings  [cached]
M. K. Ghosh,Lokenath Debnath
International Journal of Mathematics and Mathematical Sciences , 1995, DOI: 10.1155/s0161171295000354
Abstract: This paper deals with a family of quasi-nonexpansive mappings in a uniformly convex Banach space, and the convergence of iterates generated by this family. A fixed point theorem for two quasi-nonexpansive mappings is then proved. This theorem is then extended for a finite family of quasinonexpansive mappings. It is shown that Ishikawa's [1] result follows as special cases of results proved in this paper.
Strong Convergence Theorems of Modified Ishikawa Iterations for Countable Hemi-Relatively Nonexpansive Mappings in a Banach Space  [cached]
Petrot Narin,Wattanawitoon Kriengsak,Kumam Poom
Fixed Point Theory and Applications , 2009,
Abstract: We prove some strong convergence theorems for fixed points of modified Ishikawa and Halpern iterative processes for a countable family of hemi-relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space by using the hybrid projection methods. Moreover, we also apply our results to a class of relatively nonexpansive mappings, and hence, we immediately obtain the results announced by Qin and Su's result (2007), Nilsrakoo and Saejung's result (2008), Su et al.'s result (2008), and some known corresponding results in the literatures.
Strong Convergence Theorems of Modified Ishikawa Iterations for Countable Hemi-Relatively Nonexpansive Mappings in a Banach Space  [cached]
Narin Petrot,Kriengsak Wattanawitoon,Poom Kumam
Fixed Point Theory and Applications , 2009, DOI: 10.1155/2009/483497
Abstract: We prove some strong convergence theorems for fixed points of modified Ishikawa and Halpern iterative processes for a countable family of hemi-relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space by using the hybrid projection methods. Moreover, we also apply our results to a class of relatively nonexpansive mappings, and hence, we immediately obtain the results announced by Qin and Su's result (2007), Nilsrakoo and Saejung's result (2008), Su et al.'s result (2008), and some known corresponding results in the literatures.
Ishikawa iteration process with errors for nonexpansive mappings
Jialin Huang
International Journal of Mathematics and Mathematical Sciences , 2001, DOI: 10.1155/s0161171201005452
Abstract: We study the construction and the convergence of the Ishikawa iterative process with errors for nonexpansive mappings in uniformly convex Banach spaces. Some recent corresponding results are generalized.
Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces  [cached]
Suzuki Tomonari
Fixed Point Theory and Applications , 2005,
Abstract: In 1979, Ishikawa proved a strong convergence theorem for finite families of nonexpansive mappings in general Banach spaces. Motivated by Ishikawa's result, we prove strong convergence theorems for infinite families of nonexpansive mappings.
Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces  [cached]
Tomonari Suzuki
Fixed Point Theory and Applications , 2005, DOI: 10.1155/fpta.2005.103
Abstract: In 1979, Ishikawa proved a strong convergence theorem for finite families of nonexpansive mappings in general Banach spaces. Motivated by Ishikawa's result, we prove strong convergence theorems for infinite families of nonexpansive mappings.
Convergence Theorems of an Implicit Iterates with Errors for Non-Lipschitzian Asymptotically Quasi-Nonexpansive Type Mappings
G. S. Saluja
International Journal of Analysis and Applications , 2013,
Abstract: The aim of this paper is to study an implicit iterative process with errors for two finite families of non-Lipschitzian asymptotically quasi-nonexpansive type mappings in the framework of real Banach spaces. In this paper, we have obtained a necessary and sufficient condition to converge to common fixed points for proposed scheme and mappings and also obtained strong convergence theorems by using semi-compactness and Condition (B’).
On Common Random Fixed Points of a New Iteration with Errors for Nonself Asymptotically Quasi-Nonexpansive Type Random Mappings  [PDF]
R. A. Rashwan,P. K. Jhade,Dhekra Mohammed Al-Baqeri
International Journal of Analysis , 2013, DOI: 10.1155/2013/312685
Abstract: We prove some strong convergence of a new random iterative scheme with errors to common random fixed points for three and then nonself asymptotically quasi-nonexpansive-type random mappings in a real separable Banach space. Our results extend and improve the recent results in Kiziltunc, 2011, Thianwan, 2008, Deng et al., 2012, and Zhou and Wang, 2007 as well as many others. 1. Introduction and Preliminaries The theory of random operators is an important branch of probabilistic analysis which plays a key role in many applied areas. The study of random fixed points forms a central topic in this area. Research of this direction was initiated by Prague School of Probabilistic in connection with random operator theory [1–3]. Random fixed point theory has attracted much attention in recent times since the publication of the survey article by Bharucha-Reid [4] in 1976, in which the stochastic versions of some well-known fixed point theorems were proved. A lot of efforts have been devoted to random fixed point theory and applications (e.g. see [5–10] and many others). Let be a measurable space, a nonempty subset of a separable Banach space . A mapping is called measurable if for every Borel subset of . A mapping is said to be random mapping if for each fixed , the mapping is measurable. A measurable mapping is called a random fixed point of the random mapping if for each . Throughout this paper, we denote the set of all random fixed points of random mapping by and by for the th iterate of . The class of asymptotically nonexpansive mappings is a natural generalization of the important class of nonexpansive mappings. Goebel and Kirk [11] proved that if is a nonempty closed and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point. Iterative techniques for asymptotically nonexpansive self-mappings in Banach spaces including Mann type and Ishikawa type iteration processes have been studied extensively by various authors (e.g. see [12–15]). The strong and weak convergences of the sequence of Mann iterates to a fixed point of quasi-nonexpansive mappings were studied by Petryshyn and Williamson [16]. Subsequently, the convergence of Ishikawa iterates of quasi-nonexpansive mappings in Banach spaces were discussed by Ghosh and Debnath [17]. The previous results and some obtained necessary and sufficient conditions for Ishikawa iterative sequence to converge a fixed point for asymptotically quasi-nonexpansive mappings were extended by Liu [18, 19]. In 2000, Noor [20] introduced a three-step iterative
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