A best proximity point theorem for Geraghty-contractions

*Correspondence: ksadaran@dma.ulpgc.es Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, Las Palmas de Gran Canaria, 35017, Spain Abstract 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).


Introduction
Let A and B be nonempty subsets of a metric space (X, d).
An operator T : A → B is said to be a k-contraction if there exists k ∈ [, ) such that d(Tx, Ty) ≤ kd(x, y) for any x, y ∈ A. 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 β : [, ∞) → [, ) satisfying the following condition: Theorem . ([]) Let (X, d) be a complete metric space and T : X → X be an operator. Suppose that there exists β ∈ F such that for any x, y ∈ X, d(Tx, Ty) ≤ β d(x, y) · d(x, y). (
The aim of this paper is to give a generalization of Theorem . 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 (X, d) and T : A → X be a mapping. The solutions of the equation Tx = x are fixed points of T. Consequently, T(A) ∩ A = ∅ is a necessary condition for the existence of a fixed point for the operator T. If this necessary condition does not hold, then d(x, Tx) >  for any x ∈ A and the mapping T : A → X does not have any fixed point. In this setting, our aim is to find an element x ∈ A such that d(x, Tx) 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 T : A → B.
A natural question is whether one can find an element for any x ∈ A, the optimal solution to this problem will be the one for which the value d(A, B) is attained by the real valued function Some results about best proximity points can be found in [-].

Notations and basic facts
Let A and B be two nonempty subsets of a metric space (X, d).
We denote by A  and B  the following sets: Therefore, every Geraghty-contraction is a contractive mapping.
In [], the author introduces the following definition.
It is easily seen that for any nonempty subset A of (X, d), the pair (A, A) has the P-property.
In [], the author proves that any pair (A, B) of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property.

Main results
We start this section presenting our main result.

Theorem . Let (A, B) be a pair of nonempty closed subsets of a complete metric space
Since (A, B) has the P-property, we have that d(x n , x n+ ) = d(Tx n- , Tx n ) for any n ∈ N.
Taking into account that T is a Geraghty-contraction, for any n ∈ N, we have that In this case, and consequently, Tx n  - = Tx n  . Therefore, and this is the desired result.
In the contrary case, suppose that d(x n , x n+ ) >  for any n ∈ N. http://www.fixedpointtheoryandapplications.com/content/2012/1/231 By (), (d(x n , x n+ )) is a decreasing sequence of nonnegative real numbers, and hence there exists r ≥  such that In the sequel, we prove that r = .
Assume r > , then from () we have The last inequality implies that lim n→∞ β(d(x n- , x n )) =  and since β ∈ F , we obtain r =  and this contradicts our assumption. Therefore, In what follows, we prove that (x n ) is a Cauchy sequence.
In the contrary case, we have that lim sup m,n→∞ d(x n , x m ) > .
By using the triangular inequality, By () and since d(x n+ , x m+ ) = d(Tx n , Tx m ), by the above mentioned comment, we have which gives us Since lim sup m,n→∞ d(x n , x m ) >  and by (), lim sup n→∞ d(x n , x n+ ) = , from the last inequality it follows that lim sup m,n→∞ Therefore, lim sup m,n→∞ β(d(x n , x m )) = . Taking into account that β ∈ F , we get lim sup m,n→∞ d(x n , x m ) =  and this contradicts our assumption. http://www.fixedpointtheoryandapplications.com/content/2012/1/231 Therefore, (x n ) is a Cauchy sequence. Since (x n ) ⊂ A and A is a closed subset of the complete metric space (X, d), we can find x * ∈ A such that x n → x * .
Since any Geraghty-contraction is a contractive mapping and hence continuous, we have Tx n → Tx * .
This implies that d(x n+ , Tx n ) → d(x * , Tx * ). Taking into account that the sequence (d(x n+ , Tx n )) is a constant sequence with value d (A, B), we deduce d x * , Tx * = d (A, B).
This means that x * is a best proximity point of T.
This proves the part of existence of our theorem. For the uniqueness, suppose that x  and x  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. Therefore, x  = x  . This finishes the proof.

Examples
In order to illustrate our results, we present some examples. In the sequel, we check that T is a Geraghty-contraction. http://www.fixedpointtheoryandapplications.com/content/2012/1/231 In fact, for (, x), (, y) ∈ A with x = y, we have Now, we prove that Suppose that x > y (the same reasoning works for y > x).
This proves (). Taking into account () and (), we have where φ(t) = ln( + t) for t ≥ , and β(t) = φ(t) t for t >  and β() = . Obviously, when x = y, the inequality () is satisfied. It is easily seen that β(t) = ln(+t) t ∈ F by using elemental calculus. Therefore, T is a Geraghty-contraction. Notice that the pair (A, B) satisfies the P-property. d(A, B) = , then x  = y  and x  = y  and consequently, By Theorem ., T has a unique best proximity point.
Obviously, this point is (, ) ∈ A. http://www.fixedpointtheoryandapplications.com/content/2012/1/231 The condition A and B are nonempty closed subsets of the metric space (X, d) is not a necessary condition for the existence of a unique best proximity point for a Geraghtycontraction T : A → B as it is proved with the following example.
Example . Consider X = R  with the usual metric and the subsets of X given by Obviously, d(A, B) =  and B is not a closed subset of X.
Note that A  =  × [, π  ) and B  = B. We consider the mapping T : A → B defined as Now, we check that T is a Geraghty-contraction.
In fact, for (, x), (, y) ∈ A with x = y, we have In what follows, we need to prove that In fact, suppose that x > y (the same argument works for y > x).
Put arctan x = α and arctan y = β (notice that α > β since the function φ(t) = arctan t for t ≥  is strictly increasing).
Taking into account that tan(αβ) = tan αtan β  + tan α · tan β and since α, β ∈ [, π  ), we have that tan α, tan β ∈ [, ∞), and consequently, from the last inequality it follows that Applying φ (notice that φ(t) = arctan t) to the last inequality and taking into account the increasing character of φ, we have αβ ≤ arctan(tan αtan β), or equivalently, arctan xarctan y = αβ ≤ arctan(xy), and this proves (). http://www.fixedpointtheoryandapplications.com/content/2012/1/231 By () and (), we get where β(t) = arctan t t for t >  and β() = . Obviously, the inequality () is satisfied for (, x), (, y) ∈ A with x = y. Now, we prove that β ∈ F . In fact, if β(t n ) = arctan t n t n → , then the sequence (t n ) is a bounded sequence since in the contrary case, t n → ∞ and thus β(t n ) → . Suppose that t n . This means that there exists >  such that, for each n ∈ N, there exists p n ≥ n with t p n ≥ . The bounded character of (t n ) gives us the existence of a subsequence (t k n ) of (t p n ) with (t k n ) convergent. Suppose that t k n → a. From β(t n ) → , we obtain arctan t kn t kn → arctan a a =  and, as the unique solution of arctan x = x is x  = , we obtain a = .
Thus, t k n →  and this contradicts the fact that t k n ≥ for any n ∈ N. Therefore, t n →  and this proves that β ∈ F . A similar argument to the one used in Example . proves that the pair (A, B) has the P-property.
On the other hand, the point (, ) ∈ A is a best proximity point for T since Taking into account that the unique solution of this equation is x = , 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 (A, A) satisfies the P-property, we have the following corollary.
Corollary . Let (X, d) be a complete metric space and A be a nonempty closed subset of X. Let T : A → A be a Geraghty-contraction. Then T has a unique fixed point.
Proof Using Theorem . when A = B, the desired result follows.