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Best proximity point theorems for generalized contractions in partially ordered metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 83 (2013)
Abstract
The purpose of this paper is to obtain four best proximity point theorems for generalized contractions in partially ordered metric spaces. Further, our P-operator technique, which changes a non-self mapping to a self-mapping, plays an important role. Some recent results in this area have been improved.
MSC:47H05, 47H09, 47H10.
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 is as follows: Let be a complete metric space, and let be a contraction of X into itself. Then T has a unique fixed point in X.
In the sequel, we denote by Γ the functions satisfying the following condition:
In 1973, Geraghty introduced the Geraghty-contraction and obtained Theorem 1.2 as follows.
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 ,
Theorem 1.2 [1]
Let be a complete metric space, and let 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 the Banach contraction principle. Recently, the generalized contraction principle has been studied by many authors in metric spaces or more generalized metric spaces. Some results have been got in partially ordered metric spaces as follows.
Theorem 1.3 [2]
Let be a partially ordered set, and suppose that there exists a metric d such that is a complete metric space. Let be an increasing mapping such that there exists an element with . Suppose that there exists such that
Assume that either f is continuous or X is such that if an increasing sequence , then , ∀n. Besides, if for each , there exists which is comparable to x and y, then f has a unique fixed point.
Theorem 1.4 [3]
Let be a partially ordered set, and suppose that there exists a metric such that is a complete metric space. Let be a continuous and nondecreasing mapping such that
where is a continuous and nondecreasing function such that ψ is positive in , and . If there exists with , then f has a fixed point.
Definition 1.5 [4]
An altering distance function is a function which satisfies
-
(i)
ψ is continuous and nondecreasing.
-
(ii)
if and only if .
Theorem 1.6 [4]
Let X be a partially ordered set, and suppose that there exists a metric d in X such that is a complete metric space. Let be a continuous and nondecreasing mapping such that
where ψ is an altering distance function and is a continuous function with the condition for all . If there exists such that , then T has a fixed point.
Theorem 1.7 [5]
Let be a partially ordered set, and suppose that there exists a metric such that is a complete metric space. Let be a continuous and nondecreasing mapping such that
where ψ and ϕ are altering distance functions. If there exists with , then f has a fixed point.
In 2012, Caballero et al. introduced a generalized Geraghty-contraction as follows.
Definition 1.8 [6]
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 ,
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 [7], the authors give sufficient conditions for when the sets and are nonempty. In [8], the authors prove that any pair of nonempty, closed convex subsets of a uniformly convex Banach space satisfies the P-property.
Definition 1.9 [9]
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 [6], the authors give a generalization of Theorem 1.2 by considering a non-self mapping, and they get the following theorem.
Theorem 1.10 [6]
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 .
Inspired by [6], the purpose of this paper is to obtain four best proximity point theorems for generalized contractions in partially ordered metric spaces. Further, a series of best proximity point problems can be solved by our P-operator technique, which changes a non-self mapping to a self-mapping. Some recent results in this area have been improved.
2 Main results
Before giving our main results, we first introduce the weak P-monotone property.
Weak P-monotone property Let be a pair of nonempty subsets of a partially ordered metric space with . Then the pair is said to have the weak P-monotone property if for any and ,
furthermore, implies .
Now we are in a position to give our main results.
Theorem 2.1 Let be a partially ordered set, and let be a complete metric space. Let be a pair of nonempty closed subsets of X such that . Let be an increasing mapping with , and let there exist such that
Assume that either f is continuous or that is such that if an increasing sequence , then , ∀n. Suppose that the pair has the weak P-monotone property. And for some , there exists such that and . Besides, if for each , there exists which is comparable to x and y, then there exists an in A such that .
Proof We first prove that is closed. Let be a sequence such that . It follows from the weak P-monotone property that
as , where and , . Then is a Cauchy sequence so that converges strongly to a point . By the continuity of a 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 f and the closeness of , we have . That is, .
Define an operator , by . Since the pair has the weak P-monotone property and f is increasing, we have
for any . Obviously, is increasing. Let , , , when f is continuous, then we have
Then is continuous. When is such that if an increasing sequence , then (∀n), we need not prove the continuity of . For some , there exists such that and . That is,
By the weak P-monotone property, we have .
This shows that is a contraction satisfying all the conditions in Theorem 1.3. Using Theorem 1.3, we can get has a unique fixed point . That is, , which implies that
Therefore, is the unique one in such that . □
Theorem 2.2 Let be a partially ordered set, and let be a complete metric space. Let be a pair of nonempty closed subsets of X such that . Let be a continuous and nondecreasing mapping with , and let f satisfy
where is a continuous and nondecreasing function such that ψ is positive in , and . Suppose that the pair has the weak P-monotone property. If for some , there exists such that and , then there exists an in A such that .
Proof In Theorem 2.1, we have proved that is closed and . Now we define an operator by . Since the pair has the weak P-monotone property, by the definition of f, we have
for any . Obviously, is continuous and nondecreasing. For some , there exists such that and . That is,
By the weak P-monotone property, we have .
This shows that is a contraction satisfying all the conditions in Theorem 1.4. Using Theorem 1.4, we can get has a fixed point . That is, , which implies that
Therefore, is the one in such that . □
Theorem 2.3 Let X be a partially ordered set, and let be a complete metric space. Let be a pair of nonempty closed subsets of X such that . Let be a continuous and nondecreasing mapping with , and let T satisfy
where ψ is an altering distance function and is a continuous function with the condition for all . Suppose that the pair has the weak P-monotone property. If for some , there exists such that and , then there exists an in A such that .
Proof In Theorem 2.1, we have proved that is closed and . Define an operator by . Since the pair has the weak P-monotone property and ψ is nondecreasing, by the definition of T, we have
for any . Since

this shows that is continuous and nondecreasing. Because there exist and such that and , by the weak P-monotone property, we have .
This shows that is a contraction from a complete metric subspace into itself and satisfies all the conditions in Theorem 1.6. Using Theorem 1.6, we can get has a fixed point . That is, , which implies that
Therefore, is the one in such that . □
Theorem 2.4 Let be a partially ordered set, and let be a complete metric space. Let be a pair of nonempty closed subsets of X such that . Let be a continuous and nondecreasing mapping with , and let T satisfy
where ψ and ϕ are altering distance functions. Suppose that the pair has the weak P-monotone property. If for some , there exists such that and , then there exists an in A such that .
Proof In Theorem 2.1, we have proved that is closed and . Define an operator by . Since the pair has the weak P-monotone property and ψ is nondecreasing, by the definition of T, we have
for any . Since

This shows that is continuous and nondecreasing. Because there exist and such that and , by the weak P-monotone property, we have .
This shows that is a contraction from a complete metric subspace into itself and satisfies all the conditions in Theorem 1.7. Using Theorem 1.7, we can get has a fixed point . That is, , which implies that
Therefore, is the one in such that . □
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Acknowledgements
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. Best proximity point theorems for generalized contractions in partially ordered metric spaces. Fixed Point Theory Appl 2013, 83 (2013). https://doi.org/10.1186/1687-1812-2013-83
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DOI: https://doi.org/10.1186/1687-1812-2013-83