A one-step implicit iterative method for two finite families of asymptotically nonexpansive mappings in a hyperbolic space

We introduce a one-step implicit iterative method for two finite families of asymptotically nonexpansive mappings in a hyperbolic space and use it to approximate common fixed points of these families. The results presented in this paper are new in the setting of hyperbolic spaces. On top, these are generalizations of several results in literature from Banach spaces to hyperbolic spaces. At the end of the paper, we give an example to validate our results.

APPROXIMATING COMMON FIXED POINTS OF NEARLY ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

We use an iteration scheme to approximate common fixed points of nearly asymptotically nonexpansive mappings. We generalize corresponding theorems of [1] to the case of two nearly asymptotically nonexpansive mappings and those of [9] not only to a larger class of mappings but also with better rate of convergence.

A Weak Convergence Theorem for A Finite Family of Asymptotically Nonexpansive Mappings

The purpose of this paper is to prove a new weak convergence theorem for a finite family of asymptotically nonexpansive mappings in uniformly convex Banach space.

Strong Convergence Theorems for Mixed Type Asymptotically Nonexpansive Mappings

The purpose of this paper is to study a new two-step iterative scheme with mean errors of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and prove strong convergence theorems for the new two-step iterative scheme in uniformly convex Banach spaces.

Approximation of the Nearest Common Fixed Point of Asymptotically Nonexpansive Mappings in Banach Spaces

In this paper, the iteration xn＋l =αny ＋ （1 -αn）Ti（n）k（n）xn for a family of asymptotically nonexpansive mappings T1, T2, ..., TN is originally introduced in an uniformly convex Banach space. Motivated by recent papers, we prove that under suitable conditions the iteration scheme converges strongly to the nearest common fixed point of the family of asymptotically nonexpansive mappings. The results presented in this paper expand and improve correponding ones from Hilbert spaces to uniformly convex Banach spaces, or from nonexpansive mappings to asymptotically nonexpansive mappings.

Strongly Convergent Approximations to Fixed Points of Total Asymptotically Nonexpansive Mappings

In this work we prove a new strong convergence result of the regularized successive approximation method given by yn＋1 = qnz0 ＋ （1 - qn）T^nyn, n = 1, 2,…,where lim n→∞ qn = 0 and ∞∑n=1 qn=∞ for T a total asymptotically nonexpansive mapping, i.e., T is such that ││T^n x - T^n y││ ≤ x - y ││ ＋ kn^（1）φ（││x - y││） ＋ kn^（2）,where kn^1 and kn^2 are real null convergent sequences and φ：R^＋→R^＋ is continuous such that φ（0）=0 and limt→∞φ（t）/t≤ C for a certain constant C 〉 0. Among other features, our results essentially generalize existing results on strong convergence for T nonexpansive and asymptotically nonexpansive. The convergence and stability analysis is given for both self- and nonself-mappings.

COMMON FIXED POINT THEOREM FOR NONCOMMUTING MAPPINGS SATISFYING A GENERALIZED ASYMPTOTICALLY NONEXPANSIVE CONDITION

We present a common fixed point theorem for generalized asymptotically non- expansive and noncommuting mappings in normed linear spaces.

ITERATIVE APPROXIMATION OF FIXED POINTS OF (ASYMPTOTICALLY) NONEXPANSIVE MAPPINGS

Let E be a uniformly convex Banach space which satisfies Opial's condition or has a Frechet differentiable norm,and C be a bounded closed convex subset of E. If T∶C→C is (asymptotically)nonexpansive,then the modified Ishikawa iteration process defined byx n+1 =t nT ns nT nx n+1-s nx n+(1-t n)x n,converges weakly to a fixed point of T ,where {t n} and {s n} are sequences in [0,1] with some restrictions.

ASYMPTOTIC BEHAVIOR FOR COMMUTATIVE SEMIGROUPS OF ALMOST ASYMPTOTICALLY NONEXPANSIVE TYPE MAPPINGS

This article introduces the concept of commutative semigroups of almost asymptotically nonexpansive-type mappings in a Banach space X which has the Opial property and whose norm is UKK, and establishes the weak convergence theorems for almostorbits of this class of commutative semigroups. The author improves, extends and develops some recent and earlier results.

INTERATIVE APPROXIMATION OF FIXED POINTS FOR ALMOST ASYMPTOTICALLY NONEXPANSIVE TYPE MAPPINGS IN BANACH SPACES

A new class of almost asymptotically nonexpansive type mappings in Banach spaces is introduced,which includes a number of known classes of nonlinear Lipschitzian mappings and non_Lipschitzian mappings in Banach spaces as special cases; for example,the known classes of nonexpansive mappings,asymptotically nonexpansive mappings and asymptotically nonexpansive type mappings.The convergence problem of modified Ishikawa iterative sequences with errors for approximating fixed points of almost asymptotically nonexpansive type mappings is considered.Not only S.S.Chang's inequality but also H.K.Xu's one for the norms of Banach spaces are applied to make the error estimate between the exact fixed point and the approximate one.Moreover,Zhang Shi_sheng's method(Applied Mathematics and Mechanics(English Edition),2001,22(1):25-34) for making the convergence analysis of modified Ishikawa iterative sequences with errors is extended to the case of almost asymptotically nonexpansive type mappings. The new convergence criteria of modified Ishikawa iterative sequences with errors for finding fixed points of almost asymptotically nonexpansive type mappings in uniformly convex Banach spaces are presented. Also,the new convergence criteria of modified Mann iterative sequences with errors for this class of mappings are immediately obtained from these criteria.The above results unify,improve and generalize Zhang Shi_sheng's ones on approximating fixed points of asymptotically nonexpansive type mappings by the modified Ishikawa and Mann iterative sequences with errors.

ON THE ITERATIVE APPROXIMATION PROBLEM OF FIXED OINTS FOR ASYMPTOTICALLY NON-EXPANSIVE TYPE MAPPINGS IN BANACH SPACES

Some iterative approximation theorems of fixed points for asymptotically nonexpansive type mappings in Banach spaces are obtained.

CONVERGENT PROBLEM OF ITERATIVE SEQUENCES FOR NONLINEAR MAPPINGS WITH ERROR MEMBERS IN BANACH SPACES

In this paper,the approximation problems of Ishikawa iteration with errors of fixed points for asymptotically nonexpansive mappings and asymptotically pseudocontractive mappings in arbitrary real Banach spaces are investigated.Some necessary condition and sufficient condition for the convergence of iterative sequences are given respectively.The results thus extend and improve some recent corresponding results.

Asymptotic Behavior of Asymptotically Nonexpansive Type Mappings in Banach Space

Let X be a uniformly convex Banach space X such that its dual X^＊ has the KK property. Let C be a nonempty bounded closed convex subset of X and G be a directed system. Let ={Tt ： t ∈ G} be a family of asymptotically nonexpansive type mappings on C. In this paper, we investigate the asymptotic behavior of {Ttx0 ： t∈ G} and give its weak convergence theorem.

ON THE EXISTENCE OF FIXED POINTS FOR MAPPINGS OF ASYMPTOTICALLY NONEXPANSIVE TYPE

Let C be a nonempty weakly compact convex subset of a Banach space X, and T : C →C a mapping of asymptotically nonexpansive type. Then there hold the following conclusions: (i) if X has uniform normal structure and limsup |||TjN||| < N(X)~1/(N(X)) , where|||TjN||| is the exact Lipschitz constant of TjN , N is some positive integer, and N(X) is the normal structure coefficient of X, then T has a fixed point; (ii) if X is uniformly convex in every direction and has weak uniform normal structure, then T has a fixed point.

Viscosity Approximation Method for Infinitely Many Asymptotically Nonexpansive Maps in Banach Spaces

In the framework of reflexive Banach spaces satisfying a weakly continuous duality map, the author uses the viscosity approximation method to obtain the strong convergence theorem for iterations with infinitely many asymptotically nonexpansive mappings. The main results obtained in this paper improve and extend some recent results.

ON REICH'S OPEN QUESTION

Under more general form and more general conditions an affirmative answer to Reich's open question is given. The results presented also extend and improve some recent results of Reich, Shioji, Takahashi and Wittmann.

UNIFORM NORMAL STRUCTURE AND SOLUTIONS OF REICH’S OPEN QUESTION

The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive mapping, an affirmative answer to Reich＇ s open question is given. The results presented extend and improve Zhang Shisheng＇ s recent ones in the following aspects ： （i） Zhang＇ s stronger condition that the sequence of iterative parameters converges to zero is removed; （ii） Zhang＇ s stronger assumption that the asymptotically nonexpansive mapping has a fixed point is removed; （iii） Zhang＇ s stronger condition that the sequence generated by the Banach Contraction Principle is strongly convergent is also removed. Moreover, these also extend and improve the corresponding ones obtained previously by several authors including Reich, Shioji, Takahashi,Ueda and Wittmann.

Strong Convergence and Certain Control Conditions for Modified Ishikawa Type Iteration

In this paper, we establish the strong convergent theorems of an iterative algorithm for asymptotically nonexpansive mappings in Banach spaces and nonexpansive mappings in uniformly smooth Banach spaces, respectively. The results presented in this paper not only give an affirmative partial answer to Reich's open question, but also generalize and improve the corresponding results of Chang, Lee and Chan [7] and Kim and Xu [10] .

COMMON FIXED POINTS WITH APPLICATIONS TO BEST SIMULTANEOUS APPROXIMATIONS

For a subset K of a metric space （X,d） and x ∈ X,Px（x）={y ∈ K ： d（x,y） = d（x,K）≡ inf{d（x,k） ： k ∈ K}}is called the set of best K-approximant to x. An element go E K is said to be a best simulta- neous approximation of the pair y1,y2 E ∈ if max{d（y1,go）,d（y2,go）}=inf g∈K max {d（y1,g）,d（y2,g）}.In this paper, some results on the existence of common fixed points for Banach operator pairs in the framework of convex metric spaces have been proved. For self mappings T and S on K, results are proved on both T- and S- invariant points for a set of best simultaneous approximation. Some results on best K-approximant are also deduced. The results proved generalize and extend some results of I. Beg and M. Abbas^[1], S. Chandok and T.D. Narang^[2], T.D. Narang and S. Chandok^[11], S.A. Sahab, M.S. Khan and S. Sessa^[14], P. Vijayaraju^[20] and P. Vijayaraju and M. Marudai^[21].

Asymptotic Behavior of Non-Lipschitzian Mappings in Banach Spaces *

Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space with a Fréchet differentiable norm, and G be a directed system , let T= {T t:t∈G} be asymptotically nonexpansive type mappings on C . We give the weak convergence theorem of {T t:t∈G} in this paper.