Approximating common fixed points in hyperbolic spaces
© Fukhar-ud-din and Khamsi; licensee Springer. 2014
Received: 14 February 2014
Accepted: 16 April 2014
Published: 8 May 2014
We establish strong convergence and Δ-convergence theorems of an iteration scheme associated to a pair of nonexpansive mappings on a nonlinear domain. In particular we prove that such a scheme converges to a common fixed point of both mappings. Our results are a generalization of well-known similar results in the linear setting. In particular, we avoid assumptions such as smoothness of the norm, necessary in the linear case.
MSC:47H09, 46B20, 47H10, 47E10.
for two quasi-nonexpansive mappings S, T on a nonempty closed and convex subset of a strictly convex Banach space. Takahashi and Tamura  proved weak convergence of (1.1) to a common fixed point of two nonexpansive mappings in a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable and strong convergence in a strictly convex Banach space (see also [3, 4]). Mann and Ishikawa iterative procedures are well-defined in a vector space through its built-in convexity. In the literature, several mathematicians have introduced the notion of convexity in metric spaces; for example [5–8]. In this work, we follow the original metric convexity introduced by Menger  and used by many authors like Kirk [5, 6] and Takahashi . Note that Mann iterative procedures were also investigated in hyperbolic metric spaces [10, 11].
In this paper we investigate the results published in  and generalize them to uniformly convex hyperbolic spaces. A particular example of such metric spaces is the class of -spaces (in the sense of Gromov) and ℝ-trees (in the sense of Tits). Heavy use of the linear structure of Banach spaces in  presents some difficulties when extending these results to metric spaces. For example, a key assumption in many of their results is the smoothness of the norm which is hard to extend to metric spaces.
2 Menger convexity in metric spaces
for any and any .
An example of hyperbolic spaces is the family of Banach vector spaces or any normed vector spaces. Hadamard manifolds , the Hilbert open unit ball equipped with the hyperbolic metric , and the spaces [6, 16–20] (see Example 2.1) are examples of nonlinear structures which play a major role in recent research in metric fixed point theory. A subset C of a hyperbolic space X is said to be convex if , whenever (see also ).
for any . X is said to be uniformly convex whenever , for any and .
We have . Moreover, is an increasing function of ε.
- (ii)For , we have
Next we give a very important example of uniformly convex hyperbolic metric space.
Example 2.1 
Let be a metric space. A geodesic from x to y in X is a mapping c from a closed interval to X such that , , and for all . In particular, c is an isometry and . The image α of c is called a geodesic (or metric) segment joining x and y. The space is said to be a geodesic space if every two points of X are joined by a geodesic and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each , which will be denoted by , and called the segment joining x to y.
A geodesic triangle in a geodesic metric space X consists of three points , , in X (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for geodesic triangle in is a triangle in such that for . Such a triangle always exists (see ).
A geodesic metric space is said to be a space if all geodesic triangles of appropriate size satisfy the following comparison axiom.
ρ will denote the asymptotic radius of with respect to X. A point is said to be an asymptotic center of with respect to C if . We denote with , the set of asymptotic centers of with respect to C. When , we call ξ an asymptotic center of and we use the notation instead of . In general, the set of asymptotic centers of a bounded sequence may be empty or may even contain infinitely many points. Note that in the study of the geometry of Banach spaces, the function is also known as a type. For more on types in metric spaces, we refer to .
Definition 2.2 A bounded sequence in X is said to Δ-converge to if x is the unique asymptotic center of every subsequence of . We write ( Δ-converges to x).
In this paper, we study the iteration schemes (2.1)-(2.2) for nonexpansive mappings. We study strong convergence of these iterates in strictly convex hyperbolic spaces and prove Δ-convergence results in uniformly convex hyperbolic spaces without requiring any condition similar to norm Fréchet differentiability.
In the sequel, the following results will be needed.
Let X be a hyperbolic metric spaces. Assume that X is uniformly convex. Let C be a nonempty, closed and convex subset of X. Then every bounded sequence has a unique asymptotic center with respect to C.
Let X be a hyperbolic metric spaces. Assume that X is uniformly convex. Let C be a nonempty, closed and convex subset of X. Let C be a nonempty closed and convex subset of X, and be a bounded sequence in C such that and . If is a sequence in C such that , then .
for all , such that , , and .
for and any , then we must have .
The following result is very useful.
Lemma 2.4 
But since we use convex combinations other than the middle point, we will need the following generalization obtained by using Lemma 2.3.
A subset C of a metric space X is Chebyshev if for every , there exists such that for all , . In other words, for each point of the space, there is a well-defined nearest point of C. We can then define the nearest point projection by sending x to . We have the following result.
Lemma 2.6 
i.e., C is Chebyshev.
3 Convergence in strictly convex hyperbolic space
The first result of this work discusses the convergence behavior of the sequence generated by (2.1).
if and , with , then implies ;
if and , with , then implies ;
if , with , then implies . In this case, we have .
- (1)If and , then
The strict convexity of X will imply .
- (2)If and , then
- (3)If and , then
If , then and . Hence .
Let us finish the proof of Theorem 3.1. Note that (i) implies and . If , then the conclusion (2) above implies . Otherwise the conclusion (4) will imply . This proves (i).
For (ii), notice that and . Hence the conclusion (3) will imply which proves (ii).
we get , which completes the proof of (iii). □
If we assume compactness, Theorem 3.1 will imply the following result.
Theorem 3.2 Let X be a strictly convex hyperbolic space. Let C be a nonempty bounded, closed and convex subset of X. Let be two nonexpansive mappings. Assume that . Fix . Assume that is a compact subset of C. Define as in (2.1) where , with , and is the initial element of the sequence. Then converges strongly to a common fixed point of S and T.
Proof We have . Since is compact, has a convergent subsequence , i.e., . By Theorem 3.1, we have and . □
we have . In particular, we have . Continuity of T implies . Since X is strictly convex, then is a nonempty convex subset of X. Since T and S commute, we have . Moreover, since the closure of is compact, we see that is compact. The above proof shows that S has a fixed point in , i.e., .
The case gives the following result.
Theorem 3.3 Let C be a nonempty closed and convex subset of a complete strictly convex hyperbolic space X. Let be a nonexpansive mapping such that is a compact subset of C, where . Define by (2.2), where , and or and , with . Then converges strongly to a fixed point of T.
Proof We saw that in this case, we have . Since . Then there exists a subsequence of such that . By Theorem 3.1, we have and . □
4 Convergence in uniformly convex hyperbolic spaces
The following result is similar to the well-known demi-closedness principle discovered by Göhde in uniformly convex Banach spaces .
Lemma 4.1 Let C be a nonempty, closed and convex subset of a complete uniformly convex hyperbolic space X. Let be a nonexpansive mapping. Let be an approximate fixed point sequence of T, i.e., . If is the asymptotic center of with respect to C, then x is a fixed point of T, i.e., . In particular, if is an approximate fixed point sequence of T, such that , then .
By the uniqueness of the asymptotic center, we get . □
The following theorem is necessary to discuss the behavior of the iterates in (2.1).
- (i)If , where , then
- (ii)If and , with , then
- (iii)If , with , then
Using Lemma 2.5, we get .
Using Lemma 2.5, we get .
we conclude that . □
The conclusion of Theorem 4.1(iii) is amazing because the sequence generated by (2.1) gives an approximate fixed point sequence for both S and T without assuming that these mappings commute.
As a direct consequence to Theorem 4.1 and Remark 4.1, we get the following result which discusses the Δ-convergence of the iterative sequence defined by (2.1).
if and , with , then and ;
if and , with , then and ;
if , with , then and .
Since y is the unique asymptotic center of , we get . This proves that Δ-converges to y.
Following the same proof as given above for (i), we get Δ-converges to its unique asymptotic center which is a fixed point of T.
The conclusion (iii) follows easily from (i) and (ii). □
As a corollary to Theorem 4.2, we get the following result when .
Corollary 4.1 Let C be a nonempty, closed and convex subset of a complete uniformly convex hyperbolic space X. Let be a nonexpansive mapping with a fixed point. Suppose that is given by (2.2), where and or and , with . Then , with .
Using the concept of near point projection, we establish the following amazing convergence result.
Theorem 4.3 Let C be a nonempty, closed and convex subset of a complete uniformly convex hyperbolic space X. Let be nonexpansive mappings such that . Let P be the nearest point projection of C onto F. For an initial value , define as given in (2.1), where , with . Then converges strongly to the asymptotic center of .
Since , we conclude that , which implies which is our desired contradiction. Therefore converges strongly to y. □
The authors are grateful to King Fahd University of Petroleum and Minerals for supporting research project IN121055.
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