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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.
Common Fixed Point Theorems for Totally Quasi-G-Asymptotically Nonexpansive Semigroups with the Generalized f-Projection  [PDF]
Chunjie Wang, Yuanheng Wang
Applied Mathematics (AM) , 2014, DOI: 10.4236/am.2014.51004
Abstract:

In this paper, we introduce some new classes of the totally quasi-G-asymptotically nonexpansive mappings and the totally quasi-G-asymptotically nonexpansive semigroups. Then, with the generalized f-projection operator, we prove some strong convergence theorems of a new modified Halpern type hybrid iterative algorithm for the totally quasi-G-asymptotically nonexpansive semigroups in Banach space. The results presented in this paper extend and improve some corresponding ones by many 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.
Convergece Theorems for Finite Families of Asymptotically Quasi-Nonexpansive Mappings
Chidume CE,Ali Bashir
Journal of Inequalities and Applications , 2007,
Abstract: Let be a real Banach space, a closed convex nonempty subset of , and asymptotically quasi-nonexpansive mappings with sequences (resp.) satisfying as , and . Let be a sequence in . Define a sequence by , , , , , . Let . Necessary and sufficient conditions for a strong convergence of the sequence to a common fixed point of the family are proved. Under some appropriate conditions, strong and weak convergence theorems are also proved.
On Unification of the Strong Convergence Theorems for a Finite Family of Total Asymptotically Nonexpansive Mappings in Banach Spaces
Farrukh Mukhamedov,Mansoor Saburov
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/281383
Abstract: We unify all known iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically I-nonexpansive mappings. Note that such a scheme contains a particular case of the method introduced by (C. E. Chidume and E. U. Ofoedu, 2009). We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically I-nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed-convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.
On unification of the strong convergence theorems for a finite family of total asymptotically nonexpansive mappings in Banach spaces  [PDF]
Farrukh Mukhamedov,Mansoor Saburov
Mathematics , 2010,
Abstract: In this paper, we unify all know iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically $I$-nonexpansive mappings. Note that such a scheme contains as a particular case of the method introduced in [C.E. Chidume, E.U. Ofoedu, \textit{Inter. J. Math. & Math. Sci.} \textbf{2009}(2009) Article ID 615107, 17p]. We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically $I-$nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.
Convergence theorems for asymptotically nonexpansive mappings in Banach spaces
Yongfu Su, Xiaolong Qin and Meijuan Shang
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE , 2008,
Abstract: . Let E be a uniformly convex Banach space, and let K be a nonempty convex closed subset which is also a nonexpansive retract of E. Let T K E be an asymptotically nonexpansive mapping with {kn} ì [1, ¥) such that ( from n=1 to ¥) (kn - 1) ¥ and let F(T) be nonempty, where F(T) denotes the fixed points set of T. Let {an}, {bn}, {gn}, {a¢n}, {b¢n}, {g¢n}, {a¢¢n}, {b¢¢n} and {g¢¢n} be real sequences in [0, 1] such that an + bn + gn = a¢n + b¢n + g¢n = a¢¢n + b¢¢n + g¢¢n = 1 and e £ an, a¢n, a¢¢n £ 1 - e for all n N and some e > 0, starting with arbitrary x1 K, define the sequence { xn} by setting
Convergence Theorems for Asymptotically Nonexpansive Mappings in Banach Spaces
Yongfu Su,Xiaolong Qin,Meijuan Shang
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE , 2008,
Abstract: Let E be a uniformly convex Banach space, and let K be a nonempty convex closed subset which is also a nonexpansive retract of E. Let T: K E be an asymptotically nonexpansive mapping with {kn} ì [1, ¥) such that ( from n=1 to ¥)(kn - 1) < ¥ and let F(T) be nonempty, where F(T) denotes the fixed points set of T. Let{an}, {bn},{gn}, {a¢n}, {b¢n}, {g¢n}, {a¢¢n}, {b¢¢n}and {g¢¢n}be real sequences in [0, 1] such that an +bn +gn =a¢n +b¢n +g¢n =a¢¢n +b¢¢n +g¢¢n = 1 and e £ an,a¢n, a¢¢n £ 1 - e for all n N and some e > 0, starting with arbitrary x1 K,define the sequence { xn} by setting zn = P(a¢¢nT(PT)n-1xn+ b¢¢nxn+ g¢¢nwn), yn =P(a¢nT(PT)n-1zn+ b¢nxn+ g¢nvn), xn+1 =P(anT(PT)n-1yn+ bnxn+ gnun), with the restrictions ( from n=1 to ¥) (gn) < ¥,( from n=1 to ¥) (g¢n) < ¥ and ( from n=1 to ¥) (g¢¢n) < ¥ where { wn} , { vn} and { un} are bounded sequences in K. (i) If E is realuniformly convex Banach space satisfying Opial's condition, then weak convergence of { xn} to some p F(T) is obtained; (ii) If T satisfies condition (A), then { xn} convergence strongly to some p F(T).
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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