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Convergence to Common Fixed Point for Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces

Yali Li, Jianjun Liu, Lei Deng
2008 International Journal of Mathematics and Mathematical Sciences  
LetKbe a nonempty closed convex subset of a reflexive and strictly convex Banach spaceEwith a uniformlyGâteauxdifferentiable norm,ℱ={T(h):h≥0}a generalized asymptotically nonexpansive self-mapping semigroup  ...  ofK, andf:K→Ka fixed contractive mapping with contractive coefficientβ∈(0,1).  ...  In addition, for all n ∈ N, if k Yali Li et al. 3 A one-parameter generalized asymptotically nonexpansive semigroup is a family F {T h : h ≥ 0} of self-mapping of K such that i T 0 x x for x ∈ K; ii  ... 
doi:10.1155/2008/687815 fatcat:gs22hwnv2jcnzpyjzdc7rfuwgm

Strong Convergence Theorems for Semigroups of Asymptotically Nonexpansive Mappings in Banach Spaces

D. R. Sahu, Ngai-Ching Wong, Jen-Chih Yao
2013 Abstract and Applied Analysis  
We will show that the implicit iteration scheme: = +(1− ) ( ) , for all ∈ N, converges strongly to a common fixed point of the semigroup F for some suitably chosen parameters { } and { }.  ...  Let be a real reflexive Banach space with a weakly continuous duality mapping . Let be a nonempty weakly closed star-shaped (with respect to ) subset of .  ...  The authors would like to thank the referees for their careful reading and helpful comments.  ... 
doi:10.1155/2013/202095 fatcat:fmaab24kezfutlgi3zwlvugk2m

Invariant Means and Reversible Semigroup of Relatively Nonexpansive Mappings in Banach Spaces

Kyung Soo Kim
2014 Abstract and Applied Analysis  
The purpose of this paper is to study modified Halpern type and Ishikawa type iteration for a semigroup of relatively nonexpansive mappingsI={T(s):s∈S}on a nonempty closed convex subsetCof a Banach space  ...  We prove that, given some mild conditions, we can generate iterative sequences which converge strongly to a common element of the set of fixed pointsF(I), whereF(I)=⋂{F(T(s)):s∈S}.  ...  Acknowledgments The author would like to thank Professor Anthony To-Ming Lau and Professor Jong Kyu Kim for their helpful suggestions.  ... 
doi:10.1155/2014/694783 fatcat:wq7nmshrvnhcpodthniunjhdfm

The split common fixed point problem for asymptotically nonexpansive semigroups in Banach spaces

Lin Wang, Zhaoli Ma
2016 Fixed Point Theory and Applications  
In this paper, we propose an iteration method for finding a split common fixed point of asymptotically nonexpansive semigroups in the setting of two Banach spaces, and we obtain some weak and strong convergence  ...  The results presented in the paper are new and improve and extend some recent corresponding results. MSC: 47H09; 47J25  ...  Acknowledgements The authors would like to express their thanks to the reviewers and editors for their helpful suggestions and advices.  ... 
doi:10.1186/s13663-016-0588-2 fatcat:iwswafdnk5edvpqj3ifjvtcypq

Convergence theorems of modified Mann iterations

Jinzuo Chen, Dingping Wu
2013 Fixed Point Theory and Applications  
In this paper, we introduce the modified iterations of Mann's type for nonexpansive mapping and asymptotically nonexpansive mapping to have the strong and weak convergence in a uniformly convex Banach  ...  We also proved strong convergence theorems of our modified Mann's iteration processes for nonexpansive semigroups and asymptotically nonexpansive semigroups.  ...  If Fix(T) = ∅, then {x n } ∞ n= converges weakly to a fixed point of T. 4 Strong convergence to a common fixed point of asymptotically nonexpansive semigroups and nonexpansive semigroups 4.1 Strong  ... 
doi:10.1186/1687-1812-2013-282 fatcat:dm3uxv3fabfq7j3b6r4yjath3a

Page 7937 of Mathematical Reviews Vol. , Issue 2000k [page]

2000 Mathematical Reviews  
Then {u(t): t € G} is weakly convergent (to a common fixed point of ) if and only if it is weakly asymptotically regular (that is, {u(At) — u(t)} converges to 0 weakly for every h € G).” 47H Nonlinear  ...  [“Weak convergence theorems for asymptotically nonexpansive mappings and semigroups”, Nonlinear Anal., to appear] to ob- tain weakly convergent iteration processes for nonexpansive map- pings and nonexpansive  ... 

Asymptotic behaviour for an almost-orbit of nonexpansive semigroups in Banach spaces

Jong Kyu Kim, Gang Li
2000 Bulletin of the Australian Mathematical Society  
Then {u(t) : t £ G} is weakly convergent (to a common fixed point of S) if and only if it is weakly asymptotically regular (that is, {u(ht) -u(t)} converges to 0 weakly for every h £ G).  ...  In this paper, by using the technique of product nets, we are able to prove a weak convergence theorem for an almost-orbit of right reversible semigroups of nonexpansine mappings in a general Banach space  ...  Then for each x £ C, {T n x} converges weakly to a fixed point of T if and only if T is weakly asymptotically regular, that is, T n+1 x -T n x converges weakly to 0.  ... 
doi:10.1017/s0004972700022358 fatcat:v2pcsxgo3jbbriwwnatxbweo4a

Page 6866 of Mathematical Reviews Vol. , Issue 99j [page]

1999 Mathematical Reviews  
We prove that if S is an amenable semigroup and .Y = {T,: t € S} is a nonexpansive semigroup on a closed, convex subset C in a uniformly convex Banach space EF such that the set F(.Y) of common fixed points  ...  Since the topology in such spaces is generated by a modular which does not need to be very regular (e.g. nonexpansive), the translation of the Banach fixed-point theorem to this setting requires a number  ... 

Strong convergence for total quasi-ϕ-asymptotically nonexpansive semigroups in Banach spaces

Jing Quan, Shih-sen Chang, Xiongrui Wang
2012 Fixed Point Theory and Applications  
The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-φ-asymptotically nonexpansive semigroups to prove strong convergence in Banach spaces.  ...  The main results presented in this paper extend and improve the corresponding results of many authors. MSC: 47H05; 47H09; 49M05  ...  Chang [] established some new approximation theorems of common fixed points for a countable family of total asymptotically nonexpansive mappings in Banach spaces.  ... 
doi:10.1186/1687-1812-2012-142 fatcat:a4wwvubgzjczvb6c7qnqjyvuay

Applications of uniform asymptotic regularity to fixed point theorems

Sławomir Borzdyński, Andrzej Wiśnicki
2016 Journal of Fixed Point Theory and Applications  
In particular, we prove that a jointly continuous left amenable or left reversible semigroup generated by firmly nonexpansive mappings on a bounded τ-compact subset of a Banach space has a common fixed  ...  point, and give a qualitative complement to the Markov-Kakutani theorem.  ...  Let S be a subsurjective semigroup on a bounded subset C of a Banach space whose generators consist of firmly nonexpansive mappings. Then S has a common fixed point.  ... 
doi:10.1007/s11784-016-0300-5 fatcat:qins7q7lhjdyrj3djubjrecu7i

Fixed point theorems for a semigroup of total asymptotically nonexpansive mappings in uniformly convex Banach spaces

Suthep Suantai, Withun Phuengrattana
2014 Opuscula Mathematica  
In this paper, we provide existence and convergence theorems of common fixed points for left (or right) reversible semitopological semigroups of total asymptotically nonexpansive mappings in uniformly  ...  convex Banach spaces.  ...  Acknowledgments The authors would like to thank the referees for valuable suggestions on the paper. The first author would like to thank Chiang Mai University, Thailand for financial support.  ... 
doi:10.7494/opmath.2014.34.1.183 fatcat:5rgrgqaravh6vjd6gi3kmltv7i

The structure of fixed-point sets of Lipschitzian type semigroups

DR Sahu, RP Agarwal, Donal O'Regan
2012 Fixed Point Theory and Applications  
The purpose of this paper is to establish some results on the structure of fixed point sets for one-parameter semigroups of nonlinear mappings which are not necessarily Lipschitzian in Banach spaces.  ...  Our results improve several known existence and convergence fixed point theorems for semigroups which are not necessarily Lipschitzian. MSC: Primary 47H09; 47H10; secondary 47B20; 54C15  ...  A general formula for WCS(X) in an arbitrary Banach space is not known.  ... 
doi:10.1186/1687-1812-2012-163 fatcat:5uyeuf2hrvbqraira3hdgr5kwy

On the split common fixed point problem for nonexpansive semigroups in Banach spaces

Jian-Qiang Zhang, Gang Wang
2016 International Mathematical Forum  
In this paper, an iteration method is proposed for finding a split common fixed point of two nonexpansive semigroups in Banach spaces.  ...  The iterative scheme proposed is proved to converge weakly and strongly to a split common fixed point of two nonexpansive semigroups in Banach spaces under some suitable conditions.  ...  Iterative approximation of fixed point for nonexpansive mappings, asymptotically nonexpansive mappings, nonexpansive semi-groups, and asymptotically nonexpansive semigroupps in Hilbert or Banach spaces  ... 
doi:10.12988/imf.2016.6331 fatcat:44hlmyqa2ranpggbn3t2hsdh2u

Implicit Ishikawa Approximation Methods for Nonexpansive Semigroups in CAT(0) Spaces

Zhi-bin Liu, Yi-shen Chen, Xue-song Li, Yi-bin Xiao
2013 Abstract and Applied Analysis  
This paper is devoted to the convergence of the implicit Ishikawa iteration processes for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces.  ...  Under certain and different conditions, we also get the strong convergence theorems of implicit Ishikawa iteration sequences for nonexpansive semigroups in the CAT(0) spaces.  ...  Under various and appropriate conditions, we obtain that { } generated by (11) converges strongly to a common fixed point of Γ.  ... 
doi:10.1155/2013/503198 fatcat:yp7xrfisczbkrdg5pu3d3xrqsm

Fixed Point Theorems for Certain Classes of Semigroups of Mappings

Mo Tak Kiang
1974 Transactions of the American Mathematical Society  
This is followed by a fixed point theorem for commutative semigroups of continuous asymptotically-nonexpansive self-mappings on a weakly compact, convex subset of a strictly convex Banach space.  ...  Fixed point theorems for commuting semigroups of self-mappings are considered in this paper. A generalization of the classical Markov-Kakutani theorem is first given.  ...  We conclude this paper with a method suggested by the above counterexample to locate common fixed points for commuting semigroups of continuous asymptotically-nonexpansive self-mappings when the Banach  ... 
doi:10.2307/1996847 fatcat:ihgb37gs4fdwrnsddwuc335kt4
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