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
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 .
First, we introduce the class ℱ of those functions satisfying the following condition:
Theorem 1.1 ()
Let be a complete metric space and be an operator. Suppose that there exists such that for any ,
ThenThas a unique fixed point.
Since the constant functions , where , belong to ℱ, Theorem 1.1 extends Banach’s contraction principle.
Remark 1.1 Since the functions belonging to ℱ are strictly smaller than one, condition (1) implies that
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 .
We denote by and the following sets:
In , the authors present sufficient conditions which determine when the sets and are nonempty.
Now, we present the following definition.
Definition 2.1 Let A, B be two nonempty subsets of a metric space . A mapping is said to be a Geraghty-contraction if there exists such that
Notice that since , we have
Therefore, every Geraghty-contraction is a contractive mapping.
In , the author introduces the following definition.
Definition 2.2 ()
Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the P-property if and only if for any and ,
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 3.1Let 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 theP-property. Then there exists a unique inAsuch that .
Proof Since is nonempty, we take .
As , we can find such that . Similarly, since , there exists such that . Repeating this process, we can get a sequence in satisfying
Since has the P-property, we have that
Taking into account that T is a Geraghty-contraction, for any , we have that
Suppose that there exists such that .
In this case,
and consequently, .
and this is the desired result.
In the contrary case, suppose that for any .
By (2), is a decreasing sequence of nonnegative real numbers, and hence there exists such that
In the sequel, we prove that .
Assume , then from (2) we have
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.
In the contrary case, we have that
By using the triangular inequality,
By (2) and since , by the above mentioned comment, we have
which gives us
Since and by (3), , from the last inequality it follows that
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 .
Taking into account that the sequence is a constant sequence with value , we deduce
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 .
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.
Example 4.1 Consider with the usual metric.
Let A and B be the subsets of X defined by
Obviously, and A, B are nonempty closed subsets of X.
Moreover, it is easily seen that and .
Let be the mapping defined as
In the sequel, we check that T is a Geraghty-contraction.
In fact, for with , we have
Now, we prove that
Suppose that (the same reasoning works for ).
Then, since is strictly increasing in , we have
This proves (5).
Taking into account (4) and (5), we have
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.
then and and consequently,
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.
Example 4.2 Consider with the usual metric and the subsets of X given by
Obviously, and B is not a closed subset of X.
Note that and .
We consider the mapping defined as
Now, we check that T is a Geraghty-contraction.
In fact, for with , we have
In what follows, we need to prove that
In fact, suppose that (the same argument works for ).
Put and (notice that since the function for is strictly increasing).
Taking into account that
and since , we have that , and consequently, from the last inequality it follows that
Applying ϕ (notice that ) to the last inequality and taking into account the increasing character of ϕ, we have
and this proves (8).
By (7) and (8), we get
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.
On the other hand, the point is a best proximity point for T since
Moreover, is the unique best proximity point for T.
Indeed, if is a best proximity point for T, then
and this gives us
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 4.1Let be a complete metric space andAbe a nonempty closed subset of X. Let be a Geraghty-contraction. ThenThas 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 .
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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