Abstract
In Caballero et al. (Fixed Point Theory Appl. (2012). doi:10.1186/1687-1812-2012-231), the authors prove a best proximity point theorem for Geraghty nonself contraction. In this note, not only P-property has been weakened, but also an improved best proximity point theorem will be presented by a short and simple proof. An example which satisfies weak P-property but not P-property has been presented to demonstrate our results.
MSC:47H05, 47H09, 47H10.
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1 Introduction and preliminaries
Let A and B be nonempty subsets of a metric space . An operator is said to be contractive if there exists such that for any . The well-known Banach contraction principle says: Let be a complete metric space, and be a contraction of X into itself. Then T has a unique fixed point in X.
In 1973, Geraghty introduced the Geraghty-contraction and obtained Theorem 1.2.
Definition 1.1 ([1])
Let be a metric space. A mapping is said to be a Geraghty-contraction if there exists such that for any
where the class Γ denotes those functions satisfying the following condition:
Theorem 1.2 ([1])
Let be a complete metric space and be an operator. Suppose that there exists such that for any ,
Then T has a unique fixed point.
Obviously, Theorem 1.2 is an extensive version of Banach contraction principle. In 2012, Caballero et al. introduced generalized Geraghty-contraction as follows.
Definition 1.3 ([2])
Let A, B be two nonempty subsets of a metric space . A mapping is said to be a Geraghty-contraction if there exists such that for any
where the class denotes those functions satisfying the following condition:
Now we need the following notations and basic facts.
Let A and B be two nonempty subsets of a metric space . We denote by and the following sets:
where .
In [3], the authors give sufficient conditions for when the sets and are nonempty. In [4], the author presents the following definition and proves that any pair of nonempty, closed and convex subsets of a real Hilbert space H satisfies the P-property.
Definition 1.4 ([2])
Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the P-property if and only if for any and ,
Let A, B be two nonempty subsets of a complete metric space and consider a mapping . The best proximity point problem is whether we can find an element such that . Since for any , in fact, the optimal solution to this problem is the one for which the value is attained.
In [2], the authors give a generalization of Theorem 1.2 by considering a nonself map and they get the following theorem.
Theorem 1.5 ([2])
Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty. Let be a Geraghty-contraction satisfying . Suppose that the pair has the P-property. Then there exists a unique in A such that .
Remark In [2], the proof of Theorem 1.5 is unnecessarily complex. In this note, not only P-property has been weakened, but also an improved best proximity point theorem will be presented by a short and simple proof. An example which satisfies weak P-property but not P-property has been presented to demonstrate our results.
2 Main results
Before giving our main results, we first introduce the notion of weak P-property.
Weak P-property Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the weak P-property if and only if for any and ,
Now we are in a position to give our main results.
Theorem 2.1 Let be a pair of nonempty closed subsets of a complete metric space such that . Let be a Geraghty-contraction satisfying . Suppose that the pair has the weak P-property. Then there exists a unique in A such that .
Proof We first prove that is closed. Let be a sequence such that . It follows from the weak P-property that
as , where and , . Then is a Cauchy sequence so that converges strongly to a point . By the continuity of metric d we have , that is, , and hence is closed.
Let be the closure of , we claim that . In fact, if , then there exists a sequence such that . By the continuity of T and the closeness of , we have . That is .
Define an operator , by . Since the pair has weak P-property and T is a Geraghty-contraction, we have
for any . This shows that is a Geraghty-contraction from complete metric subspace into itself. Using Theorem 1.2, we can get has a unique fixed point . That is . It implies that
Therefore, is the unique one in such that . It is easy to see that is also the unique one in A such that . □
Remark In Theorem 2.1, P-property is weakened to weak P-property. Therefore, Theorem 2.1 is an improved result of Theorem 1.5. In addition, our proof is shorter and simpler than that in [2]. In fact, our proof process is less than one page. However, the proof process in [2] is three pages.
3 Example
Now we present an example which satisfies weak P-property but not P-property.
Consider , where d is the Euclidean distance and the subsets and .
Obviously, , and . Furthermore,
however,
We can see that the pair satisfies the weak P-property but not the P-property.
References
Geraghty M: On contractive mappings. Proc. Am. Math. Soc. 1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5
Caballero J, et al.: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. 2012. doi:10.1186/1687–1812–2012–231
Kirk WA, Reich S, Veeramani P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 2003, 24: 851–862. 10.1081/NFA-120026380
Sankar Raj, V: Banach contraction principle for non-self mappings. Preprint
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This project is supported by the National Natural Science Foundation of China under grant (11071279).
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Zhang, J., Su, Y. & Cheng, Q. A note on ‘A best proximity point theorem for Geraghty-contractions’. Fixed Point Theory Appl 2013, 99 (2013). https://doi.org/10.1186/1687-1812-2013-99
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DOI: https://doi.org/10.1186/1687-1812-2013-99