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Ishikawa iteration process for asymptotic pointwise nonexpansive mappings in metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 98 (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
Throughout this paper, denotes a metric space. Let us assume that there exists a family ℱ of metric segments such that any two points x, y in M are endpoints of a unique metric segment ( is an isometric image of the real line interval ). For any , there exists a unique such that
and we will write
These metric spaces are usually called convex metric spaces . A hyperbolic metric space is a convex metric space, which satisfies the following condition:
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 
Let be a hyperbolic metric space. We say that M is uniformly convex if for any , for every , and for each
Throughout this work, M is a hyperbolic metric space.
The following theorem is a metric version of the parallelogram identity.
Theorem 2.1 
Let be uniformly convex hyperbolic metric space. Fix . For each and for each , denote
where the infimum is taken over all such that , , and . Then for any and for each . Moreover, for a fixed , we have
is a nondecreasing function of ε;
if , then .
Definition 2.2 We say that is 2-uniformly convex if
From the definition of , we obtain the following inequality:
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 ).
A geodesic metric space is a space if every geodesic triangle satisfies the following inequality:
for all and all comparison points . If x, , are points of a space and is the midpoint of the segment , then the inequality implies:
which is the (CN) inequality of Bruhat and Tits . As for the Hilbert space, the (CN) inequality implies that spaces are uniformly convex with
One may also find the modulus of uniform convexity via similar triangles. The (CN) inequality also implies that
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 
Assume that is 2-uniformly convex. Then for any , there exists such that
for any .
The following technical result is a generalization of Lemma 2.3 of .
Lemma 2.1 Let be 2-uniformly convex. Assume that be bounded away from 0 and 1, i.e., there exists two real numbers α, β such that and there exists such that
Proof By replacing α by , x by and y by in Theorem 2.2, we get
for any . Let be a nontrivial ultrafilter over ℕ. Then , . Therefore,
But , . Since t is bounded away from 0 and 1, then
Since was an arbitrary nontrivial ultrafilter over ℕ, we get
Recall that is called a type if there exists in M such that
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.
In fact, if M is 2-uniformly convex, and τ is a type defined on a nonempty closed, bounded and convex subset C of M, then there exists a unique such that
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 
Let C be a nonempty subset of . A self mapping T on C is said to be asymptotic pointwise nonexpansive if for any , there exists a sequence such that
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 .
Let C be a nonempty, closed, convex and bounded subset of a complete hyperbolic 2-uniformly convex metric space . Let T be an asymptotic pointwise nonexpansive self mapping on C. Let be bounded away from 0 and 1 and . The modified Ishikawa iteration process is defined by
In order to prove our main result, the following lemmas are needed.
Lemma 3.1 Let C be a nonempty, closed, convex and bounded subset of a complete hyperbolic 2-uniformly convex metric space . Let T be an asymptotic pointwise nonexpansive self mapping on C. Assume that , for any , where is as in Definition 2.3. Let be bounded away from 0 and 1 and in the modified Ishikawa iteration process
where and is a fixed arbitrary point. Then for any , exists.
Now let be the diameter of C. Hence,
for any . If we let , we get
for any . Note that is convergent. Indeed since , then is bounded. Moreover,
Since is convergent, then is also convergent.
Next, we let and get
Since C is bounded, we conclude that , which implies the desired conclusion. □
Lemma 3.2 Let , C and T be as in Lemma 3.1. Assume that is bounded away from 0 and 1, and is bounded away from 1. Define
for any . Then
Proof Using Theorem 2.4, T has a fixed point . Lemma 3.1 implies that exists. Set . Without loss of generality, we may assume . Let be a nontrivial ultrafilter over ℕ. Then and , for any and . Moreover, we have
Since was an arbitrary nontrivial ultrafilter over ℕ, we get
for any , and . Using Lemma 2.1, with and , we obtain
Lemma 3.3 Let , C, T, and be as in Lemma 3.2. Then
provided that , i.e., T is uniformly Lipschitzian mapping on C.
Proof Using Theorem 2.4, T has a fixed point . We have from Lemma 3.1
Moreover, we have
Let us prove that . Indeed we have
Using Lemma 3.1 and Lemma 3.2, we get
Now use (3.1) and (3.5) to get
Since M is 2-uniformly convex, (2.2) implies
Let be a nontrivial ultrafilter over ℕ. Then
Using Lemma 3.1, relations (3.3) and (3.6), we get
where depends only on M. We have
Now we distinguish two cases for s.
Case 1. If , then .
Case 2. If , we have
Since C is bounded, we get
if used with (3.7), we will get
On the other hand, we have
which implies . Since was an arbitrary nontrivial ultrafilter over ℕ, we get
Lemma 3.4 Let , C, T, and be as in Lemma 3.1. Assume , i.e., T is uniformly Lipschitzian mapping on C. Then
Proof Note that
for any . Since
for any . Moreover, we have
Also, we have
Substituting (3.10), (3.11) and (3.12) into (3.9), we get
Let be an arbitrary nontrivial ultrafilter over ℕ, then
Using Lemma 3.2 and (3.7), we get
which implies , for any nontrivial ultrafilter over ℕ. Therefore, we have
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 .
Proof For any , we have
If we let , we get
Using the definition of the type, we get
for any . Since
for any . Since ω is the minimum point of τ, we get
for any . Therefore, we have
for any . This implies that . Since
for any , we conclude that , i.e., . □
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The author declares that he has no competing interests.
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Bin Dehaish, B.A. Ishikawa iteration process for asymptotic pointwise nonexpansive mappings in metric spaces. Fixed Point Theory Appl 2013, 98 (2013). https://doi.org/10.1186/1687-1812-2013-98
- asymptotically nonexpansive mapping
- asymptotic pointwise nonexpansive mapping
- fixed point
- inequality of Bruhat and Tits
- Ishikawa iteration process
- uniformly convex metric space
- uniformly Lipschitzian mapping