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Convergence Theorem of Hybrid Iterative Algorithm for Equilibrium Problems and Fixed Point Problems of Finite Families of Uniformly Asymptotically Nonexpansive Semigroups  [PDF]
Hongbo Liu, Yi Li
Advances in Pure Mathematics (APM) , 2014, DOI: 10.4236/apm.2014.46033
Abstract:

Throughout this paper, we introduce a new hybrid iterative algorithm for finding a common element of the set of common fixed points of a finite family of uniformly asymptotically nonexpansive semigroups and the set of solutions of an equilibrium problem in the framework of Hilbert spaces. We then prove the strong convergence theorem with respect to the proposed iterative algorithm. Our results in this paper extend and improve some recent known results.

The General Iterative Methods for Asymptotically Nonexpansive Semigroups in Banach Spaces  [PDF]
Rabian Wangkeeree,Pakkapon Preechasilp
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/695183
Abstract: We introduce the general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups which is a unique solution of some variational inequalities. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. The main result extends various results existing in the current literature. 1. Introduction Let be a normed linear space. Let be a self-mapping on . Then is said to be asymptotically nonexpansive if there exists a sequence with such that for each , The class of asymptotically nonexpansive maps was introduced by Goebel and Kirk [1] as an important generalization of the class of nonexpansive maps (i.e., mappings such that , for all ). We use to denote the set of fixed points of , that is, . A self-mapping is a contraction on if there exists a constant such that We use to denote the collection of all contractions on . That is, ??is a contraction on?? . A family of mappings of into itself is called a strongly continuous semigroup of Lipschitzian mappings on if it satisfies the following conditions: (i) for all ; (ii) for all ; (iii)for each , there exists a bounded measurable function such that , for all?? ; (iv)for all , the mapping is continuous. A strongly continuous semigroup of Lipchitszian mappings is called strongly continuous semigroup of nonexpansive mappings if for all and strongly continuous semigroup of asymptotically nonexpansive if . Note that for asymptotically nonexpansive semigroup , we can always assume that the Lipchitszian constant is such that for each ,?? is nonincreasing in , and ; otherwise we replace , for each , with . We denote by the set of all common fixed points of , that is, is called uniformly asymptotically regular on [2, 3] if for all and any bounded subset of , and almost uniformly asymtotically regular on [4] if Let . Then, for each and for a nonexpansive map , there exists a unique point satisfying the following condition: since the mapping is a contraction. When is a Hilbert space and is a self-map, Browder [5] showed that converges strongly to an element of which is nearest to as . This result was extended to more various general Banach space by Morales and Jung [6], Takahashi and Ueda [7], Reich [8], and a host of other authors. Many authors (see, e.g., [9, 10]) have also shown the convergence of the path , in Banach spaces for asymptotically nonexpansive mapping self-map under some conditions on . It is an interesting problem to extend the above results to a strongly continuous semigroup of
Strong Convergence to Common Fixed Points for Countable Families of Asymptotically Nonexpansive Mappings and Semigroups  [cached]
Wattanawitoon Kriengsak,Kumam Poom
Fixed Point Theory and Applications , 2010,
Abstract: We prove strong convergence theorems for countable families of asymptotically nonexpansive mappings and semigroups in Hilbert spaces. Our results extend and improve the recent results of Nakajo and Takahashi (2003) and of Zegeye and Shahzad (2008) from the class of nonexpansive mappings to asymptotically nonexpansive mappings.
Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups  [cached]
Yongfu Su,Xiaolong Qin
Fixed Point Theory and Applications , 2006, DOI: 10.1155/fpta/2006/96215
Abstract: Strong convergence theorems are obtained from modified Halpern iterative scheme for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups, respectively. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Kim, Xu, and some others.
Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups  [cached]
Su Yongfu,Qin Xiaolong
Fixed Point Theory and Applications , 2006,
Abstract: Strong convergence theorems are obtained from modified Halpern iterative scheme for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups, respectively. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Kim, Xu, and some others.
Nonlinear ergodic theorems for asymptotically almost nonexpansive curves in a Hilbert space  [PDF]
Gang Li,Jong Kyu Kim
Abstract and Applied Analysis , 2000, DOI: 10.1155/s1085337500000312
Abstract: We introduce the notion of asymptotically almost nonexpansivecurves which include almost-orbits of commutative semigroups of asymptotically nonexpansive type mappings and study the asymptotic behavior and prove nonlinear ergodic theorems for such curves. As applications of our main theorems, we obtain the results on the asymptotic behavior and ergodicity for a commutative semigroup of non-Lipschitzian mappings with nonconvex domains in a Hilbert space.
The CQ Method for Common |Fixed Point Involving Asymptotically Nonexpansive Semigroups
渐近非扩张半群公共不动点迭代逼近的CQ方法

YANG Li,LI Jun,
杨丽
,李军

数学物理学报(A辑) , 2010,
Abstract: In this paper, by using the CQ method, the strong convergence of modified Ishikawa iterative sequence involving asymptotically nonexpansive semigroups is proved in Hilbert spaces. These results extend and improve corresponding results of others.
Hilbert空间中均衡问题与渐近非扩张半群的迭代算法的强收敛性
Strong convergence of an iterative method for equilibrium problems and asymptotically nonexpansive semigroups in Hilbert space
 [PDF]

来希雪,黄建华
福州大学学报(自然科学版) , 2015, DOI: 10.7631/issn.1000-2243.2015.04.0445
Abstract: 针对均衡问题和渐近非扩张算子半群的公共元问题,提出一个新的迭代算法,在合适的条件下,证明了由此迭代算法生成的序列的强收敛性定理.
We propose an iterative scheme for finding a common element of the solutions of an equilibrium problem and the fixed points of asymptotically nonexpansive semigroups. Under some appropriate conditions,we establish a strong convergence theorem of the sequence generated by our proposed scheme
Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces  [PDF]
Ulrich Kohlenbach,Laurentiu Leustean
Mathematics , 2007,
Abstract: This paper provides a fixed point theorem for asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces as well as new effective results on the Krasnoselski-Mann iterations of such mappings. The latter were found using methods from logic and the paper continues a case study in the general program of extracting effective data from prima-facie ineffective proofs in the fixed point theory of such mappings.
Strong Convergence Theorem for Two Commutative Asymptotically Nonexpansive Mappings in Hilbert Space  [PDF]
Jianjun Liu,Lili He,Lei Deng
International Journal of Mathematics and Mathematical Sciences , 2008, DOI: 10.1155/2008/236269
Abstract: is a bounded closed convex subset of a Hilbert space , and ∶→ are two asymptotically nonexpansive mappings such that =. We establish a strong convergence theorem for and in Hilbert space by hybrid method. The results generalize and unify many corresponding results.
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