Ishikawa iteration process for asymptotic pointwise nonexpansive mappings in metric spaces
© Bin Dehaish; licensee Springer 2013
Received: 19 September 2012
Accepted: 28 March 2013
Published: 12 April 2013
Let be a complete 2-uniformly convex metric space, C be a nonempty, bounded, closed and convex subset of M, and T be an asymptotic pointwise nonexpansive self mapping on C. In this paper, we define the modified Ishikawa iteration process in M, i.e.,
and we investigate when the Ishikawa iteration process converges weakly to a fixed point of T.
MSC:06F30, 46B20, 47E10.
The class of asymptotic nonexpansive mapping have been extensively studied in fixed point theory since the publication of the fundamental paper . Kirk and Xu  studied the asymptotic nonexpansive mapping in uniformly convex Banach spaces. Their result has been generalized by Hussain and Khamsi  to metric spaces. Khamsi and Kozlowski  extended their result to modular function spaces. In almost all papers, authors do not describe any algorithm for constructing a fixed point for the asymptotic nonexpansive mapping. Ishikawa  and Mann  iterations are two of the most popular methods to check that these two iterations were originally developed to provide ways of computing fixed points for which repeated function iteration failed to converge. Espinola et al.  examined the convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces. Kozlowski  proved convergence to a fixed point of some iterative algorithms applied to asymptotic pointwise mappings in Banach spaces. In , the authors discussed the convergence of these iterations in modular function spaces. In a recent paper , the authors investigate the existence of a fixed point of asymptotic pointwise nonexpansive mappings and study the convergence of the modified Mann iteration in hyperbolic metric spaces. It is well known that the iteration processes for generalized nonexpansive mappings have been successfully used to develop efficient and powerful numerical methods for solving various nonlinear equations and variational problems. The purpose of this paper is to discuss the behavior of the modified Ishikawa iteration process associated with asymptotic pointwise mappings, defined in hyperbolic metric spaces.
For more on metric fixed point theory, the reader may consult the book of Khamsi and Kirk .
2 Basic definitions and results
for all p, q, x, y in M, and (see ).
Obviously, the class of hyperbolic spaces includes convex subsets of normed linear spaces. As nonlinear examples, one can consider the spaces [14–16] (see Example 2.1) as well as the Hilbert open unit ball equipped with the hyperbolic metric .
Definition 2.1 
Throughout this work, M is a hyperbolic metric space.
The following theorem is a metric version of the parallelogram identity.
Theorem 2.1 
is a nondecreasing function of ε;
if , then .
for any and .
Example 2.1 Let be a metric space. A geodesic space is a metric space such that every can be joined by a geodesic map were , , and for all . Moreover, c is an isometry and . X is said to be uniquely geodesic if for every there is exactly one geodesic joining them, which will be denoted by , and called the segment joining x to y.
A geodesic triangle in a geodesic metric space consists of three points , , in X (the vertices of Δ) and three geodesic segments between each pair of vertices (the edges of Δ). A comparison triangle for in is a triangle in such that for . Such a triangle always exists (see ).
This clearly implies that any space is 2-uniformly convex with .
The following inequality plays an important role in the study of the convergence of Ishikawa iterations.
Theorem 2.2 
for any .
The following technical result is a generalization of Lemma 2.3 of .
Theorem 2.3 
Assume that is complete and uniformly convex. Let C be any nonempty closed, bounded and convex subset of M. Let τ be a type defined on C. Then any minimizing sequence of τ is convergent. Its limit is independent of the minimizing sequence.
for any . In this inequality, one may find an analogy with the Opial property used in the study of the fixed point property in Banach and metric spaces.
Definition 2.3 
for any , and . A point is called a fixed point of T if . The set of all fixed points of T is denoted by .
Let . Set . Then we have , , and for all y in C, and . In other words, in the above definition, we will always assume for all , and .
In , the authors gave the following existence fixed point theorem for asymptotic pointwise nonexpansive mappings in hyperbolic metric spaces.
Theorem 2.4 Let be a complete hyperbolic metric space which is 2-uniformly convex. Let C be a nonempty, closed, convex and bounded subset of M. Let T be an asymptotic pointwise nonexpansive self mapping on C. Then T has a fixed point in C. Moreover, the fixed point set is convex.
3 Ishikawa iteration process
In this section, we define and prove the weak convergence theorem of the modified Ishikawa iteration process for asymptotic pointwise nonexpansive mappings in a complete hyperbolic 2-uniformly convex metric space .
In order to prove our main result, the following lemmas are needed.
where and is a fixed arbitrary point. Then for any , exists.
Since is convergent, then is also convergent.
Since C is bounded, we conclude that , which implies the desired conclusion. □
provided that , i.e., T is uniformly Lipschitzian mapping on C.
Now we distinguish two cases for s.
Case 1. If , then .
We conclude this paper by a result connecting the sequence and .
Theorem 3.1 Let , C, T, and be as in Lemma 3.1. Define the type on C. If ω is the minimum point of τ, i.e., , then .
for any , we conclude that , i.e., . □
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