Some new results on cyclic relatively nonexpansive mappings in convex metric spaces
© Gabeleh and Shahzad; licensee Springer 2014
Received: 19 April 2014
Accepted: 18 August 2014
Published: 11 September 2014
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.
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 ()
for some and for all , . Then T has a unique fixed point in .
denotes the distance between the subsets A and B of X.
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 ()
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 .
Recently, Suzuki et al. in  introduced the notion of the property UC, which is a kind of geometric property for subsets of a metric space X.
Definition 1.4 ()
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 .
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 ()
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  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 ).
We also recall that the weaker notion of the property UC was introduced in , 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.  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.
The notion of convexity in metric spaces was introduced by Takahashi as follows.
Definition 2.1 ()
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 .
Definition 2.2 ()
A subset K of a convex metric space is said to be a convex set provided that for all and .
Proposition 2.3 ()
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 ()
Let be a family of convex subsets of X, then is also a convex subset of X.
Definition 2.5 ()
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 ()
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 ).
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 .
Moreover, the pair is proximal.
Definition 2.8 ()
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.
which completes the proof of the theorem. □
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.
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 3.4 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 ).
which is a contradiction. Thus, our assumption was wrong and either p or q must be a diametral point for . □
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.
That is, there exists a sequence in A such that , which completes the proof. □
Here, we state the main result of this paper.
Moreover, the set of best proximity points of T is nonempty.
This completes the proof of theorem. □
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 ()
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.
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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