A best proximity point theorem for Geraghty-contractions
© Caballero et al.; licensee Springer. 2012
Received: 16 May 2012
Accepted: 10 December 2012
Published: 27 December 2012
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).
Let A and B be nonempty subsets of a metric space .
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 ()
Then T has a unique fixed point.
Since the constant functions , where , belong to ℱ, Theorem 1.1 extends Banach’s contraction principle.
Therefore, any operator satisfying (1) is a continuous operator.
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.
In our context, we consider two nonempty subsets A and B of a complete metric space and a mapping .
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
Let A and B be two nonempty subsets of a metric space .
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 ()
It is easily seen that for any nonempty subset A of , the pair has the P-property.
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 3.1 Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a Geraghty-contraction satisfying . Suppose that the pair has the P-property. Then there exists a unique in A such that .
Proof Since is nonempty, we take .
Suppose that there exists such that .
and consequently, .
and this is the desired result.
In the contrary case, suppose that for any .
In the sequel, we prove that .
The last inequality implies that and since , we obtain and this contradicts our assumption.
Notice that since for any , for fixed, we have , and since satisfies the P-property, .
In what follows, we prove that is a Cauchy sequence.
Taking into account that , we get and this contradicts our assumption.
Therefore, is a Cauchy sequence.
Since and A is a closed subset of the complete metric space , we can find such that .
Since any Geraghty-contraction is a contractive mapping and hence continuous, we have .
This implies that .
This means that is a best proximity point of T.
This proves the part of existence of our theorem.
For the uniqueness, suppose that and are two best proximity points of T with .
which is a contradiction.
This finishes the proof. □
In order to illustrate our results, we present some examples.
Example 4.1 Consider with the usual metric.
Obviously, and A, B are nonempty closed subsets of X.
Moreover, it is easily seen that and .
In the sequel, we check that T is a Geraghty-contraction.
Suppose that (the same reasoning works for ).
This proves (5).
where for , and for and .
Obviously, when , the inequality (6) is satisfied.
It is easily seen that by using elemental calculus.
Therefore, T is a Geraghty-contraction.
Notice that the pair satisfies the P-property.
By Theorem 3.1, T has a unique best proximity point.
Obviously, this point is .
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.
Obviously, and B is not a closed subset of X.
Note that and .
Now, we check that T is a Geraghty-contraction.
In fact, suppose that (the same argument works for ).
Put and (notice that since the function for is strictly increasing).
and this proves (8).
where for and . Obviously, the inequality (9) is satisfied for with .
Now, we prove that .
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 .
Thus, and this contradicts the fact that for any .
Therefore, and this proves that .
A similar argument to the one used in Example 4.1 proves that the pair has the P-property.
Moreover, is the unique best proximity point for T.
Taking into account that the unique solution of this equation is , we have proved that T has a unique best proximity point which is .
Notice that in this case B is not closed.
Since for any nonempty subset A of X, the pair satisfies the P-property, we have the following corollary.
Corollary 4.1 Let be a complete metric space and A be a nonempty closed subset of X. Let be a Geraghty-contraction. Then T has a unique fixed point.
Proof Using Theorem 3.1 when , the desired result follows. □
Notice that when , Corollary 4.1 is Theorem 1.1 due to Gerahty .
This research was partially supported by ‘Universidad de Las Palmas de Gran Canaria’, Project ULPGC 2010-006.
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