On best proximity point of ψ-Geraghty contractions
© Karapınar; licensee Springer 2013
Received: 12 March 2013
Accepted: 17 May 2013
Published: 24 July 2013
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 P-property. In this paper, we introduce the notion of ψ-Geraghty 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.
1 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 well-known examples in which fixed point theory is used.
The most remarkable paper in this field was reported by Banach  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 .
Theorem 1.1 (Geraghty )
then T has a unique fixed point.
It is very natural that some mappings, especially non-self-mappings defined on a complete metric space , do not necessarily possess a fixed point, that is, for all . In such situations, it is reasonable to search for the existence (and uniqueness) of a point such that is an approximation of an such that . In other words, one speculates to determine an approximate solution that is optimal in the sense that the distance between and is minimum. Here, the point is called a best proximity point.
This research subject has attracted attention of a number of authors; for example, see [2–23]. In this paper we generalize and improve certain results of Caballero et al. in . Notice also that in the best proximity point theory, we usually consider a non-self-mapping. In fixed point theory, almost all maps are self-mappings. For the sake of completeness, we recall some basic definitions and fundamental results on the best proximity theory.
Let be a metric space and A and B be nonempty subsets of a metric space . A mapping is called a k-contraction if there exists such that for any . It is clear that a k-contraction coincides with the celebrated Banach fixed point theorem if one takes , where A is a complete subset of X.
Definition 1.2 (See )
Theorem 1.2 (See )
Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a continuous Geraghty-contraction satisfying . Suppose that the pair has the P-property. Then there exists a unique in A such that .
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 ψ-Geraghty-contractions.
2 Main results
ψ is nondecreasing;
ψ is subadditive, that is, ;
ψ is continuous;
We introduce the following contraction.
We are now ready to state and prove our main theorem.
Theorem 2.1 Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a ψ-Geraghty contraction satisfying . Suppose that the pair has the P-property. Then there exists a unique in A such that .
Regarding the properties of the function ψ, the limit above contradicts the assumption (14). Therefore, is a Cauchy sequence.
Since and A is a closed subset of the complete metric space , we can find such that .
a contradiction. Therefore, .
which is equivalent to saying that is the best proximity point of T. This completes the proof of the existence of a best proximity point.
a contradiction. This completes the proof. □
Notice that the pair satisfies the P-property for any nonempty subset A of X. Consequently, we have the following corollary.
Corollary 2.1 Let be a complete metric space and A be a nonempty closed subset of X. Let be a ψ-Geraghty-contraction. Then T has a unique fixed point.
Proof Apply Theorem 2.1 with . □
If we take we obtain Theorem 1.2 as a corollary of Theorem 2.1.
Corollary 2.2 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 Apply Theorem 2.1 with and . □
In order to illustrate our results, we present the following example.
and and .
Since , the pair has the P-property.
Notice that and and .
where is defined as .
More precisely, the point is the best proximity point of T.
which yields that , a contradiction. Therefore condition (6) holds for all . Hence, the conditions of Theorem 2.1 hold and T has a unique best proximity point. Here, is the best proximity point of T.
The author expresses his gratitude to the anonymous referees for constructive and useful remarks, comments and suggestions.
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