The purpose of this paper is to provide sufficient conditions for the existence of a unique best proximity point for Geraghty-contractions.
Our paper provides an extension of a result due to Geraghty (Proc. Am. Math. Soc. 40:604-608, 1973).
Keywords:fixed point; Geraghty-contraction; P-property; best proximity point
An operator is said to be a k-contraction if there exists such that for any . Banach’s contraction principle states that when A is a complete subset of X and T is a k-contraction which maps A into itself, then T has a unique fixed point in A.
A huge number of generalizations of this principle appear in the literature. Particularly, the following generalization of Banach’s contraction principle is due to Geraghty .
Theorem 1.1 ()
ThenThas a unique fixed point.
Remark 1.1 Since the functions belonging to ℱ are strictly smaller than one, condition (1) implies that
The aim of this paper is to give a generalization of Theorem 1.1 by considering a non-self map T.
First, we present a brief discussion about a best proximity point.
Let A be a nonempty subset of a metric space and be a mapping. The solutions of the equation are fixed points of T. Consequently, is a necessary condition for the existence of a fixed point for the operator T. If this necessary condition does not hold, then for any and the mapping does not have any fixed point. In this setting, our aim is to find an element such that is minimum in some sense. The best approximation theory and best proximity point analysis have been developed in this direction.
A natural question is whether one can find an element such that . Since for any , the optimal solution to this problem will be the one for which the value is attained by the real valued function given by .
2 Notations and basic facts
In , the authors present sufficient conditions which determine when the sets and are nonempty.
Now, we present the following definition.
Therefore, every Geraghty-contraction is a contractive mapping.
In , the author introduces the following definition.
Definition 2.2 ()
In , the author proves that any pair of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property.
3 Main results
We start this section presenting our main result.
Theorem 3.1Letbe a pair of nonempty closed subsets of a complete metric spacesuch thatis nonempty. Letbe a Geraghty-contraction satisfying. Suppose that the pairhas theP-property. Then there exists a uniqueinAsuch that.
In this case,
and this is the desired result.
In the contrary case, we have that
By using the triangular inequality,
which gives us
This proves the part of existence of our theorem.
This means that
Using the P-property, we have
Using the fact that T is a Geraghty-contraction, we have
which is a contradiction.
This finishes the proof. □
In order to illustrate our results, we present some examples.
Let A and B be the subsets of X defined by
In the sequel, we check that T is a Geraghty-contraction.
Now, we prove that
This proves (5).
Taking into account (4) and (5), we have
Therefore, T is a Geraghty-contraction.
By Theorem 3.1, T has a unique best proximity point.
The condition A and B are nonempty closed subsets of the metric space is not a necessary condition for the existence of a unique best proximity point for a Geraghty-contraction as it is proved with the following example.
Now, we check that T is a Geraghty-contraction.
In what follows, we need to prove that
Taking into account that
and this proves (8).
By (7) and (8), we get
In fact, if , then the sequence is a bounded sequence since in the contrary case, and thus . Suppose that . This means that there exists such that, for each , there exists with . The bounded character of gives us the existence of a subsequence of with convergent. Suppose that . From , we obtain and, as the unique solution of is , we obtain .
and this gives us
Notice that in this case B is not closed.
Notice that when , Corollary 4.1 is Theorem 1.1 due to Gerahty .
The authors declare that they have no competing interests.
The three authors have contributed equally in this paper. They read and approval the final manuscript.
This research was partially supported by ‘Universidad de Las Palmas de Gran Canaria’, Project ULPGC 2010-006.
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