Abstract
Very recently, Caballero, Harjani and Sadarangani (Fixed Point Theory Appl. 2012:231, 2012) observed some best proximity point results for Geraghty contractions by using the Pproperty. In this paper, we introduce the notion of ψGeraghty contractions and show the existence and uniqueness of the best proximity point of such contractions in the setting of a metric space. We state examples to illustrate our result.
MSC: 41A65, 90C30, 47H10.
Keywords:
best proximity point; nonself mapping; partial order; metric space; fixed point1 Introduction and preliminaries
In nonlinear functional analysis, fixed point theory and best proximity point theory play a crucial role in the establishment of the existence of certain differential and integral equations. As a consequence, fixed point theory is very useful for various quantitative sciences that involve such equations. To list a few, certain branches of computer sciences, engineering and economics are wellknown examples in which fixed point theory is used.
The most remarkable paper in this field was reported by Banach [1] in 1922. In this paper, Banach proved that every contraction in a complete metric space has a unique fixed point. Following this outstanding paper, many authors have extended, generalized and improved this remarkable fixed point theorem of Banach by changing either the conditions of the mappings or the construction of the space. In particular, one of the notable generalizations of Banach fixed point theorem was reported by Geraghty [2].
Theorem 1.1 (Geraghty [2])
Let
IfTsatisfies the following inequality:
thenThas a unique fixed point.
It is very natural that some mappings, especially nonselfmappings defined on a complete
metric space
This research subject has attracted attention of a number of authors; for example, see [223]. In this paper we generalize and improve certain results of Caballero et al. in [6]. Notice also that in the best proximity point theory, we usually consider a nonselfmapping. In fixed point theory, almost all maps are selfmappings. For the sake of completeness, we recall some basic definitions and fundamental results on the best proximity theory.
Let
Let A and B be two nonempty subsets of a metric space
where
In [13], the authors presented sufficient conditions which determine when the sets
Definition 1.1 Let
It can be easily seen that for any nonempty subset A of
Definition 1.2 (See [6])
Let A, B be two nonempty subsets of a metric space
Theorem 1.2 (See [6])
Let
In the following section, we improve the theorem above by using a distance function ψ in Definition 1.2. In particular, we introduce Definition 2.1 and broaden the scope of Theorem 1.2 to ψGeraghtycontractions.
2 Main results
We start this section with the definition of the following auxiliary function. Let
Ψ denote the class of functions
(a) ψ is nondecreasing;
(b) ψ is subadditive, that is,
(c) ψ is continuous;
(d)
We introduce the following contraction.
Definition 2.1 Let A, B be two nonempty subsets of a metric space
Remark 2.1 Notice that since
We are now ready to state and prove our main theorem.
Theorem 2.1Let
Proof Regarding that
Since
If there exists
and consequently,
Therefore, we conclude that
For the rest of the proof, we suppose that
Consequently,
for each
On the other hand, since
Notice that since
By using the triangular inequality,
By (12) and since
By a simple manipulation, (16) yields that
By taking the properties of the function ψ into account, together with (13) and
Therefore
Regarding the properties of the function ψ, the limit above contradicts the assumption (14). Therefore,
Since
We claim that
Due to the properties of ψ, we get
Using the fact that T is a ψGeraghty contraction, we have
for any
Consequently, for
a contradiction. Therefore,
Regarding the fact that the sequence
which is equivalent to saying that
We shall show the uniqueness of the best proximity point of T. Suppose that
Using the Pproperty, we have
Using the fact that T is a ψGeraghty contraction, we have
a contradiction. This completes the proof. □
Notice that the pair
Corollary 2.1Let
Proof Apply Theorem 2.1 with
If we take
Corollary 2.2Let
Proof Apply Theorem 2.1 with
In order to illustrate our results, we present the following example.
Example 2.1 Suppose that
and consider the closed subsets
and
Set
Since
Notice that
Without loss of generality, we assume that
and
We have
where
Therefore, T is a ψGeraghtycontraction. Notice that the pair
then
Therefore, since the assumptions of Theorem 2.1 are satisfied, by Theorem 2.1 there
exists a unique
More precisely, the point
Example 2.2 Let
Also,
and so
which yields that
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author expresses his gratitude to the anonymous referees for constructive and useful remarks, comments and suggestions.
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