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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.
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.
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
Browder's Convergence for Uniformly Asymptotically Regular Nonexpansive Semigroups in Hilbert Spaces  [cached]
López Acedo Genaro,Suzuki Tomonari
Fixed Point Theory and Applications , 2010,
Abstract: We give a sufficient and necessary condition concerning a Browder's convergence type theorem for uniformly asymptotically regular one-parameter nonexpansive semigroups in Hilbert spaces.
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.
Monotone Hybrid Projection Algorithms for an Infinitely Countable Family of Lipschitz Generalized Asymptotically Quasi-Nonexpansive Mappings  [PDF]
Watcharaporn Cholamjiak,Suthep Suantai
Abstract and Applied Analysis , 2009, DOI: 10.1155/2009/297565
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).
Strong convergence theorem of a hybrid projection algorithm for a family of quasi-$\phi$-asymptotically nonexpansive mappings
J.F. Tang,S.S. Chang,M. Liu,J.A. Liu
Opuscula Mathematica , 2010,
Abstract: The main purpose of this paper is by using a new hybrid projection iterative algorithm to prove some strong convergence theorems for a family of quasi-$\phi$-asymptotically nonexpansive mappings. The results presented in the paper improve and extend the corresponding results announced by some authors.
广义均衡问题、极大单调算子和全局拟-Φ-渐近非扩张半群的公共元的强收敛定理
Strong convergence theorems of an iterative method for a generalized equilibrium problem,a maximal monotone operator and total quasi-Φ-asymptotically nonexpansive semigroups
 [PDF]

吴燕林
福州大学学报(自然科学版) , 2015, DOI: 10.7631/issn.1000-2243.2015.06.0733
Abstract: 针对广义均衡问题、极大单调算子和全局拟-Φ-渐近非扩张半群的公共元,提出一个新的迭代算法,在适当的条件下,证明了由此迭代算法生成的序列的强收敛定理.
We propose an iterative scheme for finding a common element of the solutions of a generalized equilibrium problem,a maximal monotone operator and total quasi-Φ-asymptotically nonexpansive semigroups. Under some appropriate conditions,we establish some strong convergence theorems of the sequences generated by our proposed scheme
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.
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