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Some new results on cyclic relatively nonexpansive mappings in convex metric spaces
Journal of Inequalities and Applications volume 2014, Article number: 350 (2014)
Abstract
In this article, we prove a best proximity point theorem for generalized cyclic contractions in convex metric spaces. Then we investigate the structure of minimal sets of cyclic relatively nonexpansive mappings in the setting of convex metric spaces. In this way, we obtain an extension of the Goebel-Karlovitz lemma, which is a key lemma in fixed point theory.
MSC:47H10, 47H09, 46B20.
1 Introduction
Let be a metric space, and let A, B be subsets of X. A mapping is said to be cyclic provided that and . We begin by recalling the following extension of the Banach contraction principle.
Theorem 1.1 ([1])
Let A and B be nonempty closed subsets of a complete metric space . Suppose that T is a cyclic mapping such that
for some and for all , . Then T has a unique fixed point in .
In [2] Eldred and Veeramani introduced the class of cyclic contractions. Before stating the definition we recall that
denotes the distance between the subsets A and B of X.
Definition 1.2 Let A and B be nonempty subsets of a metric space X. A mapping is said to be a cyclic contraction if T is cyclic and
for some and for all , .
Let T be a cyclic mapping. A point is said to be a best proximity point for T provided that .
For a uniformly convex Banach space X, Eldred and Veeramani proved the following theorem.
Theorem 1.3 ([2])
Let A and B be nonempty, closed, and convex subsets of a uniformly convex Banach space X and let be a cyclic contraction map. For , define for each . Then there exists a unique such that and .
An interesting extension of Theorem 1.3 can be found in [3, 4].
Recently, Suzuki et al. in [5] introduced the notion of the property UC, which is a kind of geometric property for subsets of a metric space X.
Definition 1.4 ([5])
Let A and B be nonempty subsets of a metric space . Then is said to satisfy property UC if the following holds:
If and are sequences in A and is a sequence in B such that and , then we have .
We mention that if A and B are nonempty subsets of a uniformly convex Banach space X such that A is convex, then satisfies the property UC. Other examples of pairs having the property UC can be found in [5].
The next theorem guarantees the existence, uniqueness, and convergence of a best proximity point for cyclic contractions in metric spaces by using the notion of the property UC.
Theorem 1.5 ([5])
Let be a metric space and let A and B be nonempty subsets of X such that satisfies the property UC. Assume that A is complete. Let be a generalized cyclic contraction, that is, there exists such that
for all and . Then T has a unique best proximity point z in A, and for every the sequence converges to z.
We mention that in [6] the authors proved Theorem 1.5 without using property UC and obtained the existence and not convergence of best proximity points for generalized cyclic contractions in Banach spaces (for more information one can refer to [7]).
We also recall that the weaker notion of the property UC was introduced in [8], called the WUC property, in order to study of the existence, uniqueness, and convergence of a best proximity point for cyclic contraction mappings.
Let be a nonempty pair of subsets of a metric space . A mapping is said to be a cyclic relatively nonexpansive if T is cyclic and for all . It is clear that every nonexpansive mapping is relatively nonexpansive.
Eldred et al. [9] established the existence of best proximity points for cyclic relatively nonexpansive mappings by using a geometric notion of proximal normal structure in the setting of Banach spaces. For related results, we refer the reader to [10–18].
In this article, motivated by Theorem 1.5, we establish a best proximity point theorem for generalized cyclic contraction mappings in convex metric spaces. We also study the structure of minimal sets for cyclic relatively nonexpansive mappings. In this way, we obtain an extension of the Goebel-Karlovitz lemma which plays an important role in fixed point theory.
2 Preliminaries
The notion of convexity in metric spaces was introduced by Takahashi as follows.
Definition 2.1 ([19])
Let be a metric space and . A mapping is said to be a convex structure on X provided that, for each and ,
A metric space together with a convex structure is called a convex metric space, which is denoted by . A Banach space and each of its convex subsets are convex metric spaces. But a Fréchet space is not necessarily a convex metric space. Other examples of convex metric spaces which are not embedded in any Banach space can be found in [19].
Here, we recall some notations and definitions of [6, 19].
Definition 2.2 ([19])
A subset K of a convex metric space is said to be a convex set provided that for all and .
Proposition 2.3 ([19])
Let be a convex metric space and let denote the closed ball centered at with radius . Then is a convex subset of X.
Proposition 2.4 ([19])
Let be a family of convex subsets of X, then is also a convex subset of X.
Definition 2.5 ([19])
A convex metric space is said to have property (C) if every bounded decreasing net of nonempty, closed, and convex subsets of X has a nonempty intersection.
For example every weakly compact convex subset of a Banach space has property (C). The next example ensures that condition (C) is natural as well in the metrical setting.
Example 2.1 ([20])
Let ℋ be a Hilbert space and let X be a nonempty closed subset of such that if and with , then and , where . Let for all , where is the inner product of ℋ. If we define the convex structure with , then is a complete convex metric space which has the property (C) (for more information see Example 2 of [20]).
Let A and B be two nonempty subsets of a convex metric space . We shall say that a pair in a convex metric space satisfies a property if both A and B satisfy that property. For instance, is convex if and only if both A and B are convex; , and . We shall also adopt the following notations:
The closed and convex hull of a set A will be denoted by and is defined by
The pair is said to be proximal in if . Moreover, we set
Note that if is a nonempty, weakly compact, and convex pair of subsets of a Banach space X, then so is the pair , and it is easy to see that .
Definition 2.6 A pair of sets is said to be proximal if and .
The following result follows from the proof of Theorem 2.1 in [9].
Lemma 2.7 Let be a nonempty weakly compact convex pair of a Banach space X and a cyclic relatively nonexpansive mapping. Then there exists which is minimal with respect to being nonempty, closed, convex, and a T-invariant pair of subsets of such that
Moreover, the pair is proximal.
Definition 2.8 ([21])
Let be a nonempty pair of subsets of a metric space . We say that the pair is proximal compactness provided that every net of satisfying the condition that , has a convergent subnet in . Also, we say that A is semi-compactness if is proximal compactness.
It is clear that if is a compact pair in a metric space then is proximal compactness.
Definition 2.9 Let be a nonempty pair of sets in a Banach space X. A point p in A (q in B) is said to be a diametral point with respect to B (w.r.t. A) if (). A pair in is diametral if both points p and q are diametral.
3 Main results
In this section, we study the structure of minimal sets of cyclic relatively nonexpansive mappings in the setting of convex metric spaces.
3.1 Generalized cyclic contractions in convex metric spaces
We begin our main results of this paper with the following existence theorem.
Theorem 3.1 Let be a nonempty, bounded, closed, and convex pair in a convex metric space . Suppose that is a generalized cyclic contraction. If X has the property (C) then T has a best proximity pair.
Proof Let Σ denote the set of all nonempty, bounded, closed, and convex pairs which are subsets of such that T is cyclic on . Note that . Also, Σ is partially ordered by reverse inclusion, that is, . By the fact that X has the property (C), every increasing chain in Σ is bounded above. So, by using Zorn’s lemma we obtain a maximal element, say . We note that is a nonempty, bounded, closed, and convex pair in X and . Furthermore,
and also
that is, T is cyclic on . It now follows from the maximality of that
Let , then . Now, if we have
Therefore, for all we have
and then
Thus,
from which one concludes that
So,
Similarly, if we obtain
Put
Note that and and we have
Further, if then by (3), , i.e. and also, by (4), . This proves that T is cyclic on . Maximality of implies that and . We now conclude that
So,
Now, for each pair we must have
which completes the proof of the theorem. □
Remark 3.1 Note that Theorem 3.1 holds once the maximal sets and have been fixed and the cyclic mapping satisfies the condition that there exists such that
for all .
The next corollary, obtained from Theorem 3.1, immediately follows.
Corollary 3.2 Let be a nonempty, bounded, closed, and convex pair in a reflexive Banach space X. Suppose that is a generalized cyclic contraction. Then T has a best proximity point.
Let us illustrate Theorem 3.1 with the following example.
Example 3.1 Let and define a metric d on X by
Define with
for each and (see [22]). Then is a convex stricture on X. In this order, let and . We may assume that . Then for each we have
This implies that is a convex metric space. Now, let E be a nonempty convex subset of X. Then for each and . If , then we conclude that . Therefore, the convex metric space must have the property (C). Suppose that and . Then is a bounded, closed, and convex pair of subsets of X. Let be a mapping defined by
Clearly, T is cyclic on . On the other hand, T is generalized cyclic contraction for each . It now follows from Theorem 3.1 that T has a best proximity point which is a fixed point in this case.
3.2 Extension of Goebel-Karlovitz lemma
The following result is another version of Lemma 2.7 in the setting of convex metric spaces.
Lemma 3.3 Let be a nonempty, bounded, closed, and convex pair of a convex metric space such that is nonempty and is proximal compactness. Assume that is a cyclic relatively nonexpansive mapping. If X has the property (C) then there exists a pair which is minimal with respect to being nonempty, closed, convex, and a T-invariant pair of subsets of such that
Proof Let Σ denote the set of all nonempty, closed, and convex pairs which are subsets of such that T is cyclic on and for some . Note that by the fact that is nonempty. Also, Σ is partially ordered by reverse inclusion. Assume that is a increasing chain in Σ. Set and . Since X has the property (C), we conclude that is a nonempty pair. Also, by Proposition 2.4, is a closed and convex pair. Moreover,
Similarly we can see that , that is, T is cyclic on . Now, let be such that . Since is proximal compactness, has a convergent subsequence, say , such that and . Thus,
Therefore, there exists an element such that . Hence, every increasing chain in Σ is bounded above with respect to a reverse inclusion relation. Then by using Zorn’s lemma we can get a maximal element, say , which is minimal with respect to set inclusion and so, is minimal with respect to being nonempty, closed, convex, and a T-invariant pair of subsets of such that
□
Lemma 3.4 Let be a nonempty, bounded, closed, and convex pair of a convex metric space such that is nonempty, X has the property (C) and is proximal compactness. Let be a cyclic relatively nonexpansive mapping. Suppose that is a minimal, closed, convex pair which is T-invariant such that . Then each pair with contains a diametral point (with respect to ).
Proof By the fact that T is cyclic, a similar argument to Theorem 3.1 implies that T is also cyclic on . Let be such that . The relatively nonexpansiveness of T implies that
So, . Now, by the minimality of , we must have
Assume that such that and suppose there is no diametral point in , that is,
Put and . Let and define
Note that is a nonempty, closed, and convex pair in X by Propositions 2.3 and 2.4, and since ,
It is not difficult to see that, for ,
We now prove that T is cyclic on . Let . We must verify that , that is, . By the relatively nonexpansiveness of T, for we have
then , which implies that . Therefore, and hence, . Thus, . Similarly, we can see that . Therefore, T is cyclic on . Now, the minimality of implies that and . Hence, and so, for each , . Therefore, we obtain
which is a contradiction. Thus, our assumption was wrong and either p or q must be a diametral point for . □
Definition 3.5 Let be a nonempty pair of subsets of a metric space . Suppose that is a cyclic mapping. Then a sequence in is said to be an approximate best proximity point sequence for T if
Note that if , then the sequence is said to be an approximate fixed point sequence for T.
The following lemma guarantees the existence of approximate best proximity sequences for cyclic relatively nonexpansive mappings.
Lemma 3.6 Let be a nonempty, bounded, closed, and convex pair of a convex metric space such that is nonempty, X has the property (C) and is proximal compactness. Let be a cyclic relatively nonexpansive mapping. Then there exists an approximate best proximity point sequence for T in A.
Proof It follows from Lemma 3.3 that there exists a pair which is minimal with respect to being nonempty, closed, convex, and a T-invariant pair of subsets of and there exists such that
For any put . Then . Define the mapping as follows:
Since T is cyclic and is a convex pair in convex metric space , we conclude that is cyclic on . Now, for each we have
Hence, for each we have
By using Remark 3.1, we conclude that the cyclic mapping has a best proximity point, say , for each . Thus,
If in the above relation, we obtain
That is, there exists a sequence in A such that , which completes the proof. □
Here, we state the main result of this paper.
Theorem 3.7 Let be a nonempty, bounded, closed, and convex pair of a convex metric space such that X has the property (C). Assume that is nonempty and is proximal compactness and satisfies the property UC. Let be a cyclic relatively nonexpansive mapping. Let be a minimal closed and convex pair which is T-invariant such that , and a sequence in such that . Then for all with we have
Moreover, the set of best proximity points of T is nonempty.
Proof We see from Lemma 3.4 that each point with contains a diametral point, that is,
Let be the sequence given by the statement, which exists because of Lemma 3.6, and assume that there exists and such that and
Note that
as . Since has the property UC,
Set
and
Note that is a nonempty, bounded, and closed pair in X. We show that is also convex. Let and . We have
Thus, is convex. Similarly, we can see that is convex. Besides, by the fact that we conclude that . We now verify that T is cyclic on . Suppose that . Then and . It now follows from (6) that
that is, . Now, let . Then and
that is, . Hence, T is cyclic on . From the minimality of one deduces that and . Since and is proximal compactness, we may assume that and . Therefore, and we have
for all . Therefore and , which is a contradiction by the fact that the pair contains a diametral point. Hence,
On the other hand, and
We now conclude that
and so, for each pair we have
This completes the proof of theorem. □
The next corollary is an extension of the classical Goebel-Karlovitz lemma [23, 24] in convex metric spaces.
Corollary 3.8 Let A be a nonempty, bounded, closed, and convex subset of a convex metric space such that X has the property (C). Assume that A is semi-compactness. Let be a nonexpansive mapping. Suppose that is a minimal closed and convex subset which is T-invariant and is an approximate fixed point sequence in . Then for each we have
Moreover, the set of fixed points of T is nonempty.
Proof If we consider and in Theorem 3.7, then has the property UC and the result follows by observing that . □
The following theorem is another version of Theorem 3.7, in the setting of reflexive Banach spaces.
Theorem 3.9 ([7])
Let be a nonempty, bounded, closed, and convex pair of a Banach space X such that satisfies property UC. Suppose that is a cyclic relatively nonexpansive mapping. Let be a minimal closed and convex pair which is T-invariant such that , and a sequence in such that . Then for all with we have
Proof At first, we note that every reflexive Banach space is a convex metric space which has the property (C). It is sufficient to consider a convex structure with . Moreover, is nonempty. Indeed, if is a sequence in such that , as X is reflexive and is a bounded and closed pair in X, the sequence has a subsequence such that and , where ‘⇀’ denotes the weak convergence. It now follows from the weak lower semicontinuity of the norm that
that is, is nonempty. By a similar argument to Theorem 3.7, the result follows. □
Remark 3.2 Note that in Theorem 3.9, we cannot deduce the existence of a best proximity point unless we add another condition. For instance, if the sequence , considered in Theorem 3.9, converges to a point in A then the best proximity point set of T will be nonempty.
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Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second author acknowledges with thanks DSR for financial support.
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Gabeleh, M., Shahzad, N. Some new results on cyclic relatively nonexpansive mappings in convex metric spaces. J Inequal Appl 2014, 350 (2014). https://doi.org/10.1186/1029-242X-2014-350
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DOI: https://doi.org/10.1186/1029-242X-2014-350