Abstract
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.
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1 Introduction
Let C be a nonempty subset of a metric space and be a mapping. Denote the set of fixed points of T by . Then T is (i) nonexpansive if for (ii) quasi-nonexpansive if and for and . For an initial value , Das and Debata [1] studied the strong convergence of Ishikawa iterates defined by
for two quasi-nonexpansive mappings S, T on a nonempty closed and convex subset of a strictly convex Banach space. Takahashi and Tamura [2] 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 [9] and used by many authors like Kirk [5, 6] and Takahashi [8]. Note that Mann iterative procedures were also investigated in hyperbolic metric spaces [10, 11].
In this paper we investigate the results published in [2] 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 [2] 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
Let be a metric space. Assume that for any x and y in X, there exists a unique metric segment , which is an isometric copy of the real line interval . Note by ℱ the family of the metric segments in X. For any , there exists a unique point such that
Throughout this paper we will denote such point by . Such metric spaces are usually called convex metric spaces [9]. Moreover, if we have
for all p, q, x, y in X and , then X is said to be a hyperbolic metric space (see [11–13]). For , the hyperbolic inequality reduces to the convex structure inequality [8]. Throughout this paper, we will assume
for any and any .
An example of hyperbolic spaces is the family of Banach vector spaces or any normed vector spaces. Hadamard manifolds [14], the Hilbert open unit ball equipped with the hyperbolic metric [15], 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 [21]).
Let be a hyperbolic metric space. For any and , set
for any . X is said to be uniformly convex whenever , for any and .
Throughout this paper we assume that if X is a uniformly convex hyperbolic space, then for every and , there exists such that
Remark 2.1
-
(i)
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 [16]
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 [18]).
A geodesic metric space is said to be a space if all geodesic triangles of appropriate size satisfy the following comparison axiom.
Let Δ be a geodesic triangle in X and let be a comparison triangle for Δ. Then Δ is said to satisfy the inequality if for all and all comparison points ,
Complete spaces are often called Hadamard spaces (see [16]). If x, , are points of a space, then the inequality implies
The above inequality is known as the (CN) inequality of Bruhat and Tits [24]. The (CN) inequality implies that spaces are uniformly convex with
In a hyperbolic space X, (1.1) is written as
where . If in (2.1), it reduces to the following Ishikawa iteration process of one mapping:
where . Let be a bounded sequence in a metric space X and C be a nonempty subset. Define , by
The asymptotic radius of with respect to C is given by
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 [25].
The Δ-convergence, introduced independently several years ago by Kuczumow [26] and Lim [27], is shown in spaces to behave similarly as the weak convergence in Banach spaces.
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 .
The following lemma [29] will be useful in studying the sequence generated by (2.1) in uniformly convex metric spaces. Here we give a proof based on the ideas developed in [25].
Lemma 2.3 Let X be a uniformly convex hyperbolic space. Then for arbitrary positive numbers and , and , we have
for all , such that , , and .
Proof Without loss of generality, we may assume . In this case, we have . Let be fixed and . Set . Since
the uniform convexity of X will imply . Using the uniform convexity of X, we get
Hence
□
Remark 2.2 If is uniformly convex, then is strictly convex, i.e., whenever
for and any , then we must have .
The following result is very useful.
Lemma 2.4 [25]
Let be a uniformly convex hyperbolic space. Let be such that
Then we have
But since we use convex combinations other than the middle point, we will need the following generalization obtained by using Lemma 2.3.
Lemma 2.5 Let be a uniformly convex hyperbolic space. Let be such that , , and
where , with . Then we have
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 [25]
Let be a complete uniformly convex hyperbolic space. Let C be nonempty, convex and closed subset of X. Let be such that . Then there exists a unique best approximant of x in C, i.e., there exists a unique such that
i.e., C is Chebyshev.
3 Convergence in strictly convex hyperbolic space
Let be a hyperbolic metric space. Let C be a nonempty closed convex subset of X. Let be two nonexpansive mappings. Throughout the paper, assume that . Let and (assuming F is not empty). Set . Then
is nonempty and invariant by both S and T. Therefore one may always assume that C is bounded provided that S and T have a common fixed point. Moreover, if is the sequence generated by (2.1), then we have
where . This proves that is decreasing, which implies that exists. Using the above inequalities, we get
The first result of this work discusses the convergence behavior of the sequence generated by (2.1).
Theorem 3.1 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 . Let and be given by (2.1). Then the following hold:
-
(i)
if and , with , then implies ;
-
(ii)
if and , with , then implies ;
-
(iii)
if , with , then implies . In this case, we have .
Proof Assume that . Let . Without loss of generality, we may assume and . Since is decreasing, we get
The above inequalities imply
Set . Without loss of generality we may assume otherwise most of the conclusions in the theorem are trivial. Assume that . Then . Hence
which implies . If we assume that , then , which implies .
-
(1)
If and , then
The strict convexity of X will imply .
-
(2)
If and , then
The strict convexity of X will imply .
-
(3)
If and , then
The strict convexity of X will imply .
-
(4)
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).
For (iii), notice that . Hence the conclusion (4) will imply . Since
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 . □
The existence of a common fixed point T and S is crucial. If one assumes that T and S commute, i.e., , then a common fixed point exists under the assumptions of Theorem 3.2. Indeed, fix and define
for and . Then
That is, is a contraction. By the Banach contraction principle (BCP), has a unique fixed point in C. Since the closure of is compact, there exists a subsequence of such that . Since is bounded 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 [30].
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 .
Proof Let be an approximate fixed point sequence of T. Let be the unique asymptotic center of with respect to C. Since
we get
By the uniqueness of the asymptotic center, we get . □
The following theorem is necessary to discuss the behavior of the iterates in (2.1).
Theorem 4.1 Let C be a nonempty, closed and convex subset of a complete uniformly convex hyperbolic space X. Let be nonexpansive mappings such that . Fix and generate by (2.1). Set
for any .
-
(i)
If , where , then
-
(ii)
If and , with , then
-
(iii)
If , with , then
Proof Let . Then the sequence is decreasing. Set . If , then all the conclusions are trivial. Therefore we will assume that . Note that we have
and
for any . In order to prove (i), assume that , where . From the inequalities (4.1) and (4.2), we get
which implies . Indeed, let be an ultrafilter over ℕ. Then we have and . Hence
Since , we get . Since was an arbitrary ultrafilter, we get as claimed. Therefore we have
Using Lemma 2.5, we get .
Next we prove (ii). Assume and , with . First note that from (4.1) and (4.2), we get
which implies . Since , we conclude that . Since , we get in a similar fashion . Therefore we have
Using Lemma 2.5, we get .
Finally let us prove (iii). Assume that , with . Then from (i) and (ii), we know that
Since
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.
Remark 4.1 If we assume that and , then
Indeed, if we assume this not to be so, then there exists a subsequence and such that
for any . In particular, it is clear, since is bounded, that . Without loss of generality, we may assume that , for . The proof of (ii) will imply
which is a contradiction since is a bounded sequence. Therefore we must have
In particular, if we assume , then we get
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).
Theorem 4.2 Let C be a nonempty, closed and convex subset of a complete uniformly convex hyperbolic space X. Let be two nonexpansive mappings such that . Fix and generate by (2.1). Then
-
(i)
if and , with , then and ;
-
(ii)
if and , with , then and ;
-
(iii)
if , with , then and .
Proof Let us prove (i). Assume and , with . Theorem 4.1 and Remark 4.1 imply that is an approximate fixed point sequence of S, i.e.,
Let y be the unique asymptotic center of . Then Lemma 4.1 implies that . Let us prove that in fact Δ-converges to y. Let be a subsequence of . Let z be the unique asymptotic center of . Again since is an approximate fixed point sequence of S, we get . Hence
Since , we get
Since y is the unique asymptotic center of , we get . This proves that Δ-converges to y.
Next we prove (ii). Assume and , with . Then Theorem 4.1 implies that is an approximate fixed point sequence of T, i.e.,
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 .
Proof First, we claim that
We prove (4.3) by induction on . For , we have
That is,
for . Assume that (4.3) is true for . That is,
for . Hence
This completes the proof of (4.3). Let us complete the proof of Theorem 4.3. We know from Theorem 4.2(iii) that Δ-converges to its unique asymptotic center y, which is in F. Let us prove that converges strongly to y. Assume not, i.e., there exist and a subsequence such that , for any . It is clear that we must have , otherwise is a constant sequence. Since
we get
for any . Using the properties of the modulus of uniform convexity, there exists such that
for any . Hence
for any . Using the definition of the nearest point projection P, we get
for any . Using the inequality (4.3) above, we get
for any and . Since , we know that is decreasing (in n and fixed ). Hence
for any . Since y is the asymptotic center of , we get
for any . Finally since , if we let , we get
Since , we conclude that , which implies which is our desired contradiction. Therefore converges strongly to y. □
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The authors are grateful to King Fahd University of Petroleum and Minerals for supporting research project IN121055.
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Fukhar-ud-din, H., Khamsi, M.A. Approximating common fixed points in hyperbolic spaces. Fixed Point Theory Appl 2014, 113 (2014). https://doi.org/10.1186/1687-1812-2014-113
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DOI: https://doi.org/10.1186/1687-1812-2014-113