- Open Access
A note on ‘A best proximity point theorem for Geraghty-contractions’
© Zhang et al.; licensee Springer 2013
- Received: 3 January 2013
- Accepted: 26 March 2013
- Published: 12 April 2013
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.
- fixed point
- best proximity point
- weak P-property
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 ()
Theorem 1.2 ()
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 ()
Now we need the following notations and basic facts.
In , the authors give sufficient conditions for when the sets and are nonempty. In , 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 ()
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 , the authors give a generalization of Theorem 1.2 by considering a nonself map and they get the following theorem.
Theorem 1.5 ()
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 , 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.
Before giving our main results, we first introduce the notion of weak P-property.
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 .
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 .
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 . In fact, our proof process is less than one page. However, the proof process in  is three pages.
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 .
We can see that the pair satisfies the weak P-property but not the P-property.
This project is supported by the National Natural Science Foundation of China under grant (11071279).
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