 Research
 Open Access
 Published:
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 Poperator technique, which changes a nonself mapping to a selfmapping, 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 (X,d). An operator T:A\to B is said to be contractive if there exists k\in [0,1) such that d(Tx,Ty)\le kd(x,y) for any x,y\in A. The wellknown Banach contraction principle is as follows: Let (X,d) be a complete metric space, and let T:X\to X 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 \beta :[0,\mathrm{\infty})\to [0,1) satisfying the following condition:
In 1973, Geraghty introduced the Geraghtycontraction and obtained Theorem 1.2 as follows.
Definition 1.1 [1]
Let (X,d) be a metric space. A mapping T:X\to X is said to be a Geraghtycontraction if there exists \beta \in \Gamma such that for any x,y\in X,
Theorem 1.2 [1]
Let (X,d) be a complete metric space, and let T:X\to X be an operator. Suppose that there exists \beta \in \Gamma such that for any x,y\in X,
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 (X,\le ) be a partially ordered set, and suppose that there exists a metric d such that (X,d) is a complete metric space. Let f:X\to X be an increasing mapping such that there exists an element {x}_{0}\in X with {x}_{0}\le f({x}_{0}). Suppose that there exists \beta \in \Gamma such that
Assume that either f is continuous or X is such that if an increasing sequence {x}_{n}\to x\in X, then {x}_{n}\le x, ∀n. Besides, if for each x,y\in X, there exists z\in X which is comparable to x and y, then f has a unique fixed point.
Theorem 1.4 [3]
Let (X,\le ) be a partially ordered set, and suppose that there exists a metric d\in X such that (X,d) is a complete metric space. Let f:X\to X be a continuous and nondecreasing mapping such that
where \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is a continuous and nondecreasing function such that ψ is positive in (0,\mathrm{\infty}), \psi (0)=0 and {lim}_{t\to \mathrm{\infty}}\psi (t)=\mathrm{\infty}. If there exists {x}_{0}\in X with {x}_{0}\le f({x}_{0}), then f has a fixed point.
Definition 1.5 [4]
An altering distance function is a function \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) which satisfies

(i)
ψ is continuous and nondecreasing.

(ii)
\psi (t)=0 if and only if t=0.
Theorem 1.6 [4]
Let X be a partially ordered set, and suppose that there exists a metric d in X such that (X,d) is a complete metric space. Let T:X\to X be a continuous and nondecreasing mapping such that
where ψ is an altering distance function and \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is a continuous function with the condition \psi (t)>\varphi (t) for all t>0. If there exists {x}_{0}\in X such that {x}_{0}\le T{x}_{0}, then T has a fixed point.
Theorem 1.7 [5]
Let (X,\le ) be a partially ordered set, and suppose that there exists a metric d\in X such that (X,d) is a complete metric space. Let f:X\to X be a continuous and nondecreasing mapping such that
where ψ and ϕ are altering distance functions. If there exists {x}_{0}\in X with {x}_{0}\le f({x}_{0}), then f has a fixed point.
In 2012, Caballero et al. introduced a generalized Geraghtycontraction as follows.
Definition 1.8 [6]
Let A, B be two nonempty subsets of a metric space (X,d). A mapping T:A\to B is said to be a Geraghtycontraction if there exists \beta \in \Gamma such that for any x,y\in A,
Now we need the following notations and basic facts. Let A and B be two nonempty subsets of a metric space (X,d). We denote by {A}_{0} and {B}_{0} the following sets:
where d(A,B)=inf\{d(x,y):x\in A\text{and}y\in B\}.
In [7], the authors give sufficient conditions for when the sets {A}_{0} and {B}_{0} are nonempty. In [8], the authors prove that any pair (A,B) of nonempty, closed convex subsets of a uniformly convex Banach space satisfies the Pproperty.
Definition 1.9 [9]
Let (A,B) be a pair of nonempty subsets of a metric space (X,d) with {A}_{0}\ne \mathrm{\varnothing}. Then the pair (A,B) is said to have the Pproperty if and only if for any {x}_{1},{x}_{2}\in {A}_{0} and {y}_{1},{y}_{2}\in {B}_{0},
Let A, B be two nonempty subsets of a complete metric space, and consider a mapping T:A\to B. The best proximity point problem is whether we can find an element {x}_{0}\in A such that d({x}_{0},T{x}_{0})=min\{d(x,Tx):x\in A\}. Since d(x,Tx)\ge d(A,B) for any x\in A, in fact, the optimal solution to this problem is the one for which the value d(A,B) is attained.
In [6], the authors give a generalization of Theorem 1.2 by considering a nonself mapping, and they get the following theorem.
Theorem 1.10 [6]
Let (A,B) be a pair of nonempty closed subsets of a complete metric space (X,d) such that {A}_{0} is nonempty. Let T:A\to B be a Geraghtycontraction satisfying T({A}_{0})\subseteq {B}_{0}. Suppose that the pair (A,B) has the Pproperty. Then there exists a unique {x}^{\ast} in A such that d({x}^{\ast},T{x}^{\ast})=d(A,B).
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 Poperator technique, which changes a nonself mapping to a selfmapping. Some recent results in this area have been improved.
2 Main results
Before giving our main results, we first introduce the weak Pmonotone property.
Weak Pmonotone property Let (A,B) be a pair of nonempty subsets of a partially ordered metric space (X,d) with {A}_{0}\ne \mathrm{\varnothing}. Then the pair (A,B) is said to have the weak Pmonotone property if for any {x}_{1},{x}_{2}\in {A}_{0} and {y}_{1},{y}_{2}\in {B}_{0},
furthermore, {y}_{1}\ge {y}_{2} implies {x}_{1}\ge {x}_{2}.
Now we are in a position to give our main results.
Theorem 2.1 Let (X,\le ) be a partially ordered set, and let (X,d) be a complete metric space. Let (A,B) be a pair of nonempty closed subsets of X such that {A}_{0}\ne \mathrm{\varnothing}. Let f:A\to B be an increasing mapping with f({A}_{0})\subseteq {B}_{0}, and let there exist \beta \in \Gamma such that
Assume that either f is continuous or that {\overline{A}}_{0} is such that if an increasing sequence {x}_{n}\to x\in {\overline{A}}_{0}, then {x}_{n}\le x, ∀n. Suppose that the pair (A,B) has the weak Pmonotone property. And for some {x}_{0}\in {A}_{0}, there exists {\stackrel{\u02c6}{x}}_{0}\in {B}_{0} such that d({x}_{0},{\stackrel{\u02c6}{x}}_{0})=d(A,B) and {\stackrel{\u02c6}{x}}_{0}\le f({x}_{0}). Besides, if for each x,y\in {\overline{A}}_{0}, there exists z\in {\overline{A}}_{0} which is comparable to x and y, then there exists an {x}^{\ast} in A such that d({x}^{\ast},f{x}^{\ast})=d(A,B).
Proof We first prove that {B}_{0} is closed. Let \{{y}_{n}\}\subseteq {B}_{0} be a sequence such that {y}_{n}\to q\in B. It follows from the weak Pmonotone property that
as n,m\to \mathrm{\infty}, where {x}_{n},{x}_{m}\in {A}_{0} and d({x}_{n},{y}_{n})=d(A,B), d({x}_{m},{y}_{m})=d(A,B). Then \{{x}_{n}\} is a Cauchy sequence so that \{{x}_{n}\} converges strongly to a point p\in A. By the continuity of a metric d, we have d(p,q)=d(A,B), that is, q\in {B}_{0}, and hence {B}_{0} is closed.
Let {\overline{A}}_{0} be the closure of {A}_{0}, we claim that f({\overline{A}}_{0})\subseteq {B}_{0}. In fact, if x\in {\overline{A}}_{0}\setminus {A}_{0}, then there exists a sequence \{{x}_{n}\}\subseteq {A}_{0} such that {x}_{n}\to x. By the continuity of f and the closeness of {B}_{0}, we have fx={lim}_{n\to \mathrm{\infty}}f{x}_{n}\in {B}_{0}. That is, f({\overline{A}}_{0})\subseteq {B}_{0}.
Define an operator {P}_{{A}_{0}}:f({\overline{A}}_{0})\to {A}_{0}, by {P}_{{A}_{0}}y=\{x\in {A}_{0}:d(x,y)=d(A,B)\}. Since the pair (A,B) has the weak Pmonotone property and f is increasing, we have
for any y\ge x\in {\overline{A}}_{0}. Obviously, {P}_{{A}_{0}}f is increasing. Let {x}_{n}, x\in {\overline{A}}_{0}, {x}_{n}\to x, when f is continuous, then we have
Then {P}_{{A}_{0}}f is continuous. When {\overline{A}}_{0} is such that if an increasing sequence {x}_{n}\to x\in {\overline{A}}_{0}, then {x}_{n}\le x (∀n), we need not prove the continuity of {P}_{{A}_{0}}f. For some {x}_{0}\in {A}_{0}, there exists {\stackrel{\u02c6}{x}}_{0}\in {B}_{0} such that d({x}_{0},{\stackrel{\u02c6}{x}}_{0})=d(A,B) and {\stackrel{\u02c6}{x}}_{0}\le f({x}_{0}). That is,
By the weak Pmonotone property, we have {P}_{{A}_{0}}f{x}_{0}\ge {x}_{0}.
This shows that {P}_{{A}_{0}}f:{\overline{A}}_{0}\to {\overline{A}}_{0} is a contraction satisfying all the conditions in Theorem 1.3. Using Theorem 1.3, we can get {P}_{{A}_{0}}f has a unique fixed point {x}^{\ast}. That is, {P}_{{A}_{0}}f{x}^{\ast}={x}^{\ast}\in {A}_{0}, which implies that
Therefore, {x}^{\ast} is the unique one in {A}_{0} such that d({x}^{\ast},f{x}^{\ast})=d(A,B). □
Theorem 2.2 Let (X,\le ) be a partially ordered set, and let (X,d) be a complete metric space. Let (A,B) be a pair of nonempty closed subsets of X such that {A}_{0}\ne \mathrm{\varnothing}. Let f:A\to B be a continuous and nondecreasing mapping with f({A}_{0})\subseteq {B}_{0}, and let f satisfy
where \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is a continuous and nondecreasing function such that ψ is positive in (0,\mathrm{\infty}), \psi (0)=0 and {lim}_{t\to \mathrm{\infty}}\psi (t)=\mathrm{\infty}. Suppose that the pair (A,B) has the weak Pmonotone property. If for some {x}_{0}\in {A}_{0}, there exists {\stackrel{\u02c6}{x}}_{0}\in {B}_{0} such that d({x}_{0},{\stackrel{\u02c6}{x}}_{0})=d(A,B) and {\stackrel{\u02c6}{x}}_{0}\le f({x}_{0}), then there exists an {x}^{\ast} in A such that d({x}^{\ast},f{x}^{\ast})=d(A,B).
Proof In Theorem 2.1, we have proved that {B}_{0} is closed and f({\overline{A}}_{0})\subseteq {B}_{0}. Now we define an operator {P}_{{A}_{0}}:f({\overline{A}}_{0})\to {A}_{0} by {P}_{{A}_{0}}y=\{x\in {A}_{0}:d(x,y)=d(A,B)\}. Since the pair (A,B) has the weak Pmonotone property, by the definition of f, we have
for any y\ge x\in {\overline{A}}_{0}. Obviously, {P}_{{A}_{0}}f is continuous and nondecreasing. For some {x}_{0}\in {A}_{0}, there exists {\stackrel{\u02c6}{x}}_{0}\in {B}_{0} such that d({x}_{0},{\stackrel{\u02c6}{x}}_{0})=d(A,B) and {\stackrel{\u02c6}{x}}_{0}\le f({x}_{0}). That is,
By the weak Pmonotone property, we have {P}_{{A}_{0}}f{x}_{0}\ge {x}_{0}.
This shows that {P}_{{A}_{0}}f:{\overline{A}}_{0}\to {\overline{A}}_{0} is a contraction satisfying all the conditions in Theorem 1.4. Using Theorem 1.4, we can get {P}_{{A}_{0}}f has a fixed point {x}^{\ast}. That is, {P}_{{A}_{0}}f{x}^{\ast}={x}^{\ast}\in {A}_{0}, which implies that
Therefore, {x}^{\ast} is the one in {A}_{0} such that d({x}^{\ast},f{x}^{\ast})=d(A,B). □
Theorem 2.3 Let X be a partially ordered set, and let (X,d) be a complete metric space. Let (A,B) be a pair of nonempty closed subsets of X such that {A}_{0}\ne \mathrm{\varnothing}. Let T:A\to B be a continuous and nondecreasing mapping with T({A}_{0})\subseteq {B}_{0}, and let T satisfy
where ψ is an altering distance function and \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is a continuous function with the condition \psi (t)>\varphi (t) for all t>0. Suppose that the pair (A,B) has the weak Pmonotone property. If for some {x}_{0}\in {A}_{0}, there exists {y}_{0}\in {B}_{0} such that d({x}_{0},{y}_{0})=d(A,B) and {y}_{0}\le T{x}_{0}, then there exists an {x}^{\ast} in A such that d({x}^{\ast},T{x}^{\ast})=d(A,B).
Proof In Theorem 2.1, we have proved that {B}_{0} is closed and f({\overline{A}}_{0})\subseteq {B}_{0}. Define an operator {P}_{{A}_{0}}:T({\overline{A}}_{0})\to {A}_{0} by {P}_{{A}_{0}}y=\{x\in {A}_{0}:d(x,y)=d(A,B)\}. Since the pair (A,B) has the weak Pmonotone property and ψ is nondecreasing, by the definition of T, we have
for any y\ge x\in {\overline{A}}_{0}. Since
this shows that {P}_{{A}_{0}}T is continuous and nondecreasing. Because there exist {x}_{0}\in {A}_{0} and {y}_{0}\in {B}_{0} such that d({x}_{0},{y}_{0})=d(A,B) and {y}_{0}\le T{x}_{0}, by the weak Pmonotone property, we have {x}_{0}\le {P}_{{A}_{0}}T{x}_{0}.
This shows that {P}_{{A}_{0}}T:{\overline{A}}_{0}\to {\overline{A}}_{0} is a contraction from a complete metric subspace {\overline{A}}_{0} into itself and satisfies all the conditions in Theorem 1.6. Using Theorem 1.6, we can get {P}_{{A}_{0}}T has a fixed point {x}^{\ast}. That is, {P}_{{A}_{0}}T{x}^{\ast}={x}^{\ast}\in {A}_{0}, which implies that
Therefore, {x}^{\ast} is the one in {A}_{0} such that d({x}^{\ast},T{x}^{\ast})=d(A,B). □
Theorem 2.4 Let (X,\le ) be a partially ordered set, and let (X,d) be a complete metric space. Let (A,B) be a pair of nonempty closed subsets of X such that {A}_{0}\ne \mathrm{\varnothing}. Let T:A\to B be a continuous and nondecreasing mapping with T({A}_{0})\subseteq {B}_{0}, and let T satisfy
where ψ and ϕ are altering distance functions. Suppose that the pair (A,B) has the weak Pmonotone property. If for some {x}_{0}\in {A}_{0}, there exists {y}_{0}\in {B}_{0} such that d({x}_{0},{y}_{0})=d(A,B) and {y}_{0}\le T{x}_{0}, then there exists an {x}^{\ast} in A such that d({x}^{\ast},T{x}^{\ast})=d(A,B).
Proof In Theorem 2.1, we have proved that {B}_{0} is closed and f({\overline{A}}_{0})\subseteq {B}_{0}. Define an operator {P}_{{A}_{0}}:T({\overline{A}}_{0})\to {A}_{0} by {P}_{{A}_{0}}y=\{x\in {A}_{0}:d(x,y)=d(A,B)\}. Since the pair (A,B) has the weak Pmonotone property and ψ is nondecreasing, by the definition of T, we have
for any y\ge x\in {\overline{A}}_{0}. Since
This shows that {P}_{{A}_{0}}T is continuous and nondecreasing. Because there exist {x}_{0}\in {A}_{0} and {y}_{0}\in {B}_{0} such that d({x}_{0},{y}_{0})=d(A,B) and {y}_{0}\le T{x}_{0}, by the weak Pmonotone property, we have {x}_{0}\le {P}_{{A}_{0}}T{x}_{0}.
This shows that {P}_{{A}_{0}}T:{\overline{A}}_{0}\to {\overline{A}}_{0} is a contraction from a complete metric subspace {\overline{A}}_{0} into itself and satisfies all the conditions in Theorem 1.7. Using Theorem 1.7, we can get {P}_{{A}_{0}}T has a fixed point {x}^{\ast}. That is, {P}_{{A}_{0}}T{x}^{\ast}={x}^{\ast}\in {A}_{0}, which implies that
Therefore, {x}^{\ast} is the one in {A}_{0} such that d({x}^{\ast},T{x}^{\ast})=d(A,B). □
References
Geraghty M: On contractive mappings. Proc. Am. Math. Soc. 1973, 40: 604–608. 10.1090/S00029939197303341765
AminiHarandi A, Emami H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 2010, 72: 2238–2242. 10.1016/j.na.2009.10.023
Harjani J, Sadarangni K: Fixed point theorems for weakly contraction mappings in partially ordered sets. Nonlinear Anal. 2009, 71: 3403–3410. 10.1016/j.na.2009.01.240
Yan F, Su Y, et al.: A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory Appl. 2012. doi:10.1186/1687–1812–2012–152
Harjani J, Sadarangni K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 2010, 72: 1188–1197. 10.1016/j.na.2009.08.003
Caballero J, et al.: A best proximity point theorem for Geraghtycontractions. 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/NFA120026380
Abkar A, Gabeleh M: Global optimal solutions of noncyclic mappings in metric spaces. J. Optim. Theory Appl. 2012, 153: 298–305. 10.1007/s1095701199664
Sankar Raj V: A best proximity point theorems for weakly contractive nonself mappings. Nonlinear Anal. 2011, 74: 4804–4808. 10.1016/j.na.2011.04.052
Acknowledgements
This project is supported by the National Natural Science Foundation of China under grant (11071279).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors contributed equally to the writing of the present article. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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/10.1186/16871812201383
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/16871812201383
Keywords
 generalized contraction
 fixed point
 best proximity point
 weak Pmonotone property