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])
Letbe a complete metric space andbe an operator. Suppose that there existssatisfying the condition
IfTsatisfies the following inequality:
thenThas a unique fixed point.
It is very natural that some mappings, especially nonselfmappings 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 [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 be a metric space and A and B be nonempty subsets of a metric space . A mapping is called a kcontraction if there exists such that for any . It is clear that a kcontraction coincides with the celebrated Banach fixed point theorem if one takes , where A is a complete subset of X.
Let A and B be two nonempty subsets of a metric space . We denote by and the following sets:
In [13], the authors presented sufficient conditions which determine when the sets and are nonempty. In [19], the author introduced the following definition.
Definition 1.1 Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the Pproperty if and only if for any and ,
It can be easily seen that for any nonempty subset A of , the pair has the Pproperty. In [19], the author proved that any pair of nonempty closed convex subsets of a real Hilbert space H satisfies the Pproperty. Now, we introduce the class F of those functions satisfying the following condition:
Definition 1.2 (See [6])
Let A, B be two nonempty subsets of a metric space . A mapping is said to be a Geraghtycontraction if there exists such that
Theorem 1.2 (See [6])
Letbe a pair of nonempty closed subsets of a complete metric spacesuch thatis nonempty. Letbe a continuous Geraghtycontraction satisfying. Suppose that the pairhas thePproperty. Then there exists a uniqueinAsuch 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 ψGeraghtycontractions.
2 Main results
We start this section with the definition of the following auxiliary function. Let Ψ denote the class of functions which satisfy the following conditions:
(a) ψ is nondecreasing;
(b) ψ is subadditive, that is, ;
(c) ψ is continuous;
We introduce the following contraction.
Definition 2.1 Let A, B be two nonempty subsets of a metric space . A mapping is said to be a ψGeraghty contraction if there exists such that
Remark 2.1 Notice that since , we have
We are now ready to state and prove our main theorem.
Theorem 2.1Letbe a pair of nonempty closed subsets of a complete metric spacesuch thatis nonempty. Letbe aψGeraghty contraction satisfying. Suppose that the pairhas thePproperty. Then there exists a uniqueinAsuch that.
Proof Regarding that is nonempty, we take . Since , we can find such that . Analogously, regarding the assumption , we determine such that . Recursively, we obtain a sequence in satisfying
Since has the Pproperty, we derive that
If there exists such that , then the proof is completed. Indeed,
and consequently, . On the other hand, due to (8) we have
Therefore, we conclude that
For the rest of the proof, we suppose that for any . Since T is a ψGeraghty contraction, for any ℕ, we have that
Consequently, is a nonincreasing sequence and bounded below, and so exists. Let . Assume that . Then, from (6), we have
On the other hand, since , we conclude , that is,
Notice that since for any , for fixed , we have , and since satisfies the Pproperty, . In what follows, we prove that is a Cauchy sequence. On the contrary, assume that we have
By using the triangular inequality,
By (12) and since , by the comment mentioned above, regarding the discussion on the Pproperty above together with (12), (15) and the property of the function ψ, we derive that
By a simple manipulation, (16) yields that
By taking the properties of the function ψ into account, together with (13) and and , the last inequality yields
Therefore . By taking the fact into account, we get
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 .
We claim that . Suppose, on the contrary, that . This means that we can find such that for each , there exists with
Due to the properties of ψ, we get
Using the fact that T is a ψGeraghty contraction, we have
for any . Since and , we can find such that for
Regarding the fact that the sequence is a constant sequence with value , we derive
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.
We shall show the uniqueness of the best proximity point of T. Suppose that and are two distinct best proximity points of T, that is, . This implies 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 satisfies the Pproperty for any nonempty subset A of X. Consequently, we have the following corollary.
Corollary 2.1Letbe a complete metric space andAbe a nonempty closed subset of X. Letbe aψGeraghtycontraction. ThenThas 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 2.2Letbe a complete metric space andAbe a nonempty closed subset of X. Letbe a Geraghtycontraction. ThenThas a unique fixed point.
Proof Apply Theorem 2.1 with and . □
In order to illustrate our results, we present the following example.
Example 2.1 Suppose that with the metric
and consider the closed subsets
Set to be the mapping defined by
Since , the pair has the Pproperty.
Without loss of generality, we assume that . Moreover,
and
We have
Therefore, T is a ψGeraghtycontraction. Notice that the pair satisfies the Pproperty. Indeed, if
Therefore, since the assumptions of Theorem 2.1 are satisfied, by Theorem 2.1 there exists a unique such that
More precisely, the point is the best proximity point of T.
Example 2.2 Let and be a metric on X. Suppose and are two closed subsets of ℝ. Define by . Define by and by . Clearly, . Now we have
Also, . Further, clearly, the pair has the Pproperty. Let . Note that, if , then condition (6) holds. Hence, we assume that . We shall show that (6) holds. Suppose, on the contrary, there exist such that
and so
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.
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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