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.

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.

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.

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.

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.

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.

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

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).

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.

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.