Abstract
The purpose of this paper is to provide sufficient conditions for the existence of a unique best proximity point for Geraghtycontractions.
Our paper provides an extension of a result due to Geraghty (Proc. Am. Math. Soc. 40:604608, 1973).
Keywords:
fixed point; Geraghtycontraction; Pproperty; best proximity point1 Introduction
Let A and B be nonempty subsets of a metric space
An operator
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 [1].
First, we introduce the class ℱ of those functions
Theorem 1.1 ([1])
Let
ThenThas a unique fixed point.
Since the constant functions
Remark 1.1 Since the functions belonging to ℱ are strictly smaller than one, condition (1) implies that
Therefore, any operator
The aim of this paper is to give a generalization of Theorem 1.1 by considering a nonself map T.
First, we present a brief discussion about a best proximity point.
Let A be a nonempty subset of a metric space
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
Some results about best proximity points can be found in [29].
2 Notations and basic facts
Let A and B be two nonempty subsets of a metric space
We denote by
where
In [8], the authors present sufficient conditions which determine when the sets
Now, we present the following definition.
Definition 2.1 Let A, B be two nonempty subsets of a metric space
Notice that since
Therefore, every Geraghtycontraction is a contractive mapping.
In [10], the author introduces the following definition.
Definition 2.2 ([10])
Let
It is easily seen that for any nonempty subset A of
In [10], the author proves that any pair
3 Main results
We start this section presenting our main result.
Theorem 3.1Let
Proof Since
As
Since
Taking into account that T is a Geraghtycontraction, for any
Suppose that there exists
In this case,
and consequently,
Therefore,
and this is the desired result.
In the contrary case, suppose that
By (2),
In the sequel, we prove that
Assume
The last inequality implies that
Therefore,
Notice that since
In what follows, we prove that
In the contrary case, we have that
By using the triangular inequality,
By (2) and since
which gives us
Since
Therefore,
Taking into account that
Therefore,
Since
Since any Geraghtycontraction is a contractive mapping and hence continuous, we have
This implies that
Taking into account that the sequence
This means that
This proves the part of existence of our theorem.
For the uniqueness, suppose that
This means that
Using the Pproperty, we have
Using the fact that T is a Geraghtycontraction, we have
which is a contradiction.
Therefore,
This finishes the proof. □
4 Examples
In order to illustrate our results, we present some examples.
Example 4.1 Consider
Let A and B be the subsets of X defined by
Obviously,
Moreover, it is easily seen that
Let
In the sequel, we check that T is a Geraghtycontraction.
In fact, for
Now, we prove that
Suppose that
Then, since
This proves (5).
Taking into account (4) and (5), we have
where
Obviously, when
It is easily seen that
Therefore, T is a Geraghtycontraction.
Notice that the pair
Indeed, if
then
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
Example 4.2 Consider
Obviously,
Note that
We consider the mapping
Now, we check that T is a Geraghtycontraction.
In fact, for
In what follows, we need to prove that
In fact, suppose that
Put
Taking into account that
and since
Applying ϕ (notice that
or equivalently,
and this proves (8).
By (7) and (8), we get
where
Now, we prove that
In fact, if
Thus,
Therefore,
A similar argument to the one used in Example 4.1 proves that the pair
On the other hand, the point
Moreover,
Indeed, if
and this gives us
Taking into account that the unique solution of this equation is
Notice that in this case B is not closed.
Since for any nonempty subset A of X, the pair
Corollary 4.1Let
Proof Using Theorem 3.1 when
Notice that when
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The three authors have contributed equally in this paper. They read and approval the final manuscript.
Acknowledgements
This research was partially supported by ‘Universidad de Las Palmas de Gran Canaria’, Project ULPGC 2010006.
References

Geraghty, M: On contractive mappings. Proc. Am. Math. Soc.. 40, 604–608 (1973). Publisher Full Text

Eldred, AA, Veeramani, P: Existence and convergence of best proximity points. J. Math. Anal. Appl.. 323, 1001–1006 (2006). Publisher Full Text

Anuradha, J, Veeramani, P: Proximal pointwise contraction. Topol. Appl.. 156, 2942–2948 (2009). Publisher Full Text

Markin, J, Shahzad, N: Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces. Nonlinear Anal.. 70, 2435–2441 (2009). Publisher Full Text

Sadiq Basha, S, Veeramani, P: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory. 103, 119–129 (2000). Publisher Full Text

Sankar Raj, V, Veeramani, P: Best proximity pair theorems for relatively nonexpansive mappings. Appl. Gen. Topol.. 10, 21–28 (2009)

AlThagafi, MA, Shahzad, N: Convergence and existence results for best proximity points. Nonlinear Anal.. 70, 3665–3671 (2009). Publisher Full Text

Kirk, WA, Reich, S, Veeramani, P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim.. 24, 851–862 (2003). Publisher Full Text

Sankar Raj, V: A best proximity theorem for weakly contractive nonself mappings. Nonlinear Anal.. 74, 4804–4808 (2011). Publisher Full Text

Sankar Raj, V: Banach’s contraction principle for nonself mappings. Preprint