On unification of the strong convergence theorems for a finite family of total asymptotically nonexpansive mappings in Banach spaces

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.Comment: 22 pages, Journal of Applied Mathematics (in press

A New Iteration Process for Approximation of Common Fixed Points for Finite Families of Total Asymptotically Nonexpansive Mappings

LetEbe a real Banach space, andKa closed convex nonempty subset ofE. LetT1,T2,…,Tm:K→Kbemtotal asymptotically nonexpansive mappings. A simple iterative sequence{xn}n≥1is constructed inEand necessary and sufficient conditions for this sequence to converge to a common fixed point of{Ti}i=1mare given. Furthermore, in the case thatEis a uniformly convex real Banach space, strong convergence of the sequence{xn}n=1∞to a common fixed point of the family{Ti}i=1mis proved. Our recursion formula is much simpler and much more applicable than those recently announced by several authors for the same problem

Approximating Fixed Points of The General Asymptotic Set Valued Mappings

  The purpose of this paper is to introduce a new generalization of asymptotically non-expansive set-valued mapping  and to discuss its demi-closeness principle. Then, under certain conditions, we prove that the sequence defined by  yn+1 = tn z+ (1-tn )un ,  un in Gn( yn ) converges strongly to some fixed point in reflexive Banach spaces.  As an application, existence theorem for an iterative differential equation as well as convergence theorems for a fixed point iterative method designed to approximate this solution is prove

Convergece Theorems for Finite Families of Asymptotically Quasi-Nonexpansive Mappings

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

Some Weak Convergence Theorems for a Family of Asymptotically Nonexpansive Nonself Mappings

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

Effective results on compositions of nonexpansive mappings

This paper provides uniform bounds on the asymptotic regularity for iterations associated to a finite family of nonexpansive mappings. We obtain our quantitative results in the setting of $(r,\delta)$-convex spaces, a class of geodesic spaces which generalizes metric spaces with a convex geodesic bicombing