Best proximity points of generalized almost ψ-Geraghty contractive non-self-mappings
© Aydi et al.; licensee Springer. 2014
Received: 11 November 2013
Accepted: 24 January 2014
Published: 11 February 2014
In this paper, we introduce the new notion of almost ψ-Geraghty contractive mappings and investigate the existence of a best proximity point for such mappings in complete metric spaces via the weak P-property. We provide an example to validate our best proximity point theorem. The obtained results extend, generalize, and complement some known fixed and best proximity point results from the literature.
MSC:47H10, 54H25, 46J10, 46J15.
1 Introduction and preliminaries
Non-self-mappings are among the intriguing research directions in fixed point theory. This is evident from the increase of the number of publications related with such maps. A great deal of articles on the subject investigate the non-self-contraction mappings on metric spaces. Let be a metric space and A and B be nonempty subsets of X. A mapping is said to be 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 (Banach contraction principle)  if one takes where the induced metric space is complete.
In nonlinear analysis, the theory of fixed points is an essential instrument to solve the equation for a self-mapping T defined on a subset of an abstract space such as a metric space, a normed linear space or a topological vector space. Following the Banach contraction principle, most of the fixed point results have been proved for a self-mapping defined on an abstract space. It is quite natural to investigate the existence and uniqueness of a non-self-mapping which does not possess a fixed point. If a non-self-mapping has no fixed point, then the answer of the following question makes sense: Is there a point such that the distance between x and Tx is closest in some sense? Roughly speaking, best proximity theory investigates the existence and uniqueness of such a closest point x. We refer the reader to [2–9] and [10–32] for further discussion of best proximity.
It is clear that the notion of a fixed point coincided with the notion of a best proximity point when the underlying mapping is a self-mapping.
In , the authors presented sufficient conditions for the sets and to be nonempty.
The author defined contraction mappings via functions from this class and proved the following result.
Theorem 1.1 (Geraghty )
where , then T has a unique fixed point.
Recently, Caballero et al.  introduced the following contraction.
Definition 1.2 ()
Based on Definition 1.2, the authors  obtained the following result.
Theorem 1.2 (See )
Let be a pair of nonempty closet 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 .
The P-property mentioned in the theorem above has been introduced in .
It is easily seen that for any nonempty subset A of , the pair has the P-property. In , the author proved that any pair of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property.
Recently, Zhang et al.  defined the following notion, which is weaker than the P-property.
ψ is nondecreasing;
ψ is subadditive, that is, ;
ψ is continuous;
The notion of ψ-Geraghty contraction has been introduced very recently in , as an extension of Definition 1.2.
In , the author also proved the following best proximity point theorem.
Theorem 1.3 (See )
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 .
2 Main results
Our main results are based on the following definition which is a generalization of Definition 1.5.
Now, we state and prove our main theorem about existence and uniqueness of a best proximity point for a non-self-mapping satisfying a generalized almost ψ-Geraghty contraction.
Theorem 2.1 Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a generalized almost ψ-Geraghty contraction satisfying . Assume that the pair has the weak P-property. Then T has a unique best proximity point in A.
that is, . Therefore, . This implies that , which is a contradiction. Therefore, is a Cauchy sequence.
and so , which is a contradiction. Therefore, , that is, . In other words, is a best proximity point of T. This completes the proof of the existence of a best proximity point.
which is not possible, since ψ is nondecreasing. Therefore, we must have . This completes the proof. □
To illustrate our result given in Theorem 2.1, we present the following example, which shows that Theorem 2.1 is a proper generalization of Theorem 1.2.
Obviously, . Let be defined by . Notice that , and . Also, it is clear that the pair has the weak P-property.
We shall show that T is a generalized almost ψ-Geraghty contraction. Without loss of generality, consider the case where . Then we have and .
Thus, all hypotheses of Theorem 2.1 are satisfied, and is the unique best proximity point of the map T.
Then Theorem 1.2 (the main result of Caballero et al. ) is not applicable.
Similarly, we cannot apply Theorem 1.3 because T is not a ψ-Geraghty contraction. Let , and with . Then T does not satisfy (8).
If in Theorem 2.1 we take for all , then we deduce the following corollary.
Assume that the pair has the weak P-property. Then T has a unique best proximity point in A.
If further in the above corollary we take where , then we deduce another particular result.
Assume that the pair has the weak P-property. Then T has a unique best proximity point in A.
3 Application to fixed point theory
The case in Theorem 2.1 corresponds to a self-mapping and results in an existence and uniqueness theorem for a fixed point of the map T. We state this case in the next theorem.
Then T has a unique fixed point.
Finally, taking in Theorem 3.1, we get another fixed point result.
Then T has a unique fixed point.
Remark 3.1 The best proximity theorem given in this work, more precisely Theorem 2.1, is a quite general result. It is a generalization of Theorem 2.1 in , Theorem 8 in , and also Theorem 1.2 given in Section 1. In addition, Corollary 3.1 improves Theorem 1.1.
Remark 3.2 Very recently, Karapınar and Samet  proved that the function on the set X, where is also a metric on X. Therefore, some of the fixed theorems regarding contraction mappings defined via auxiliary functions from the set Ψ can be in fact deduced from the existing ones in the literature. However, our main result given in Theorem 2.1 is not a consequence of any existing theorems due to the fact that the contraction condition contains the term .
On the other hand, the definition of can be used to show that Theorem 3.1 follows from Corollary 3.1. Nevertheless, Corollary 3.1 and hence Theorem 3.1 are still new results.
The authors thank to the referees for their careful reading and valuable comments and remarks which contributed to the improvement of the article.
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