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# Best proximity point theorems for generalized contractions in partially ordered metric spaces

- Jingling Zhang
^{1}, - Yongfu Su
^{1}Email author and - Qingqing Cheng
^{1}

**2013**:83

https://doi.org/10.1186/1687-1812-2013-83

© Zhang et al.; licensee Springer 2013

**Received:**17 January 2013**Accepted:**14 March 2013**Published:**4 April 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.

## Keywords

- generalized contraction
- fixed point
- best proximity point
- weak
*P*-monotone property

## 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 well-known 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*.

*Γ*the functions $\beta :[0,\mathrm{\infty})\to [0,1)$ satisfying the following condition:

In 1973, Geraghty introduced the Geraghty-contraction and obtained Theorem 1.2 as follows.

**Definition 1.1** [1]

*Geraghty-contraction*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]

- (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 Geraghty-contraction as follows.

**Definition 1.8** [6]

*A*,

*B*be two nonempty subsets of a metric space $(X,d)$. A mapping $T:A\to B$ is said to be a

*Geraghty-contraction*if there exists $\beta \in \Gamma $ such that for any $x,y\in A$,

*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 *P*-property.

**Definition 1.9** [9]

*P*-

*property*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 non-self 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 Geraghty*-*contraction satisfying* $T({A}_{0})\subseteq {B}_{0}$. *Suppose that the pair* $(A,B)$ *has the* *P*-*property*. *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 *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 $(A,B)$ 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*

*P*-

*monotone 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* *P*-*monotone 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

*P*-monotone 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, $f({\overline{A}}_{0})\subseteq {B}_{0}$.

*P*-monotone property and

*f*is increasing, we have

*f*is continuous, then we have

*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 *P*-monotone property, we have ${P}_{{A}_{0}}f{x}_{0}\ge {x}_{0}$.

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* *P*-*monotone 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

*P*-monotone property, by the definition of

*f*, we have

By the weak *P*-monotone property, we have ${P}_{{A}_{0}}f{x}_{0}\ge {x}_{0}$.

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* *P*-*monotone 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

*P*-monotone property and

*ψ*is nondecreasing, by the definition of

*T*, we have

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 *P*-monotone property, we have ${x}_{0}\le {P}_{{A}_{0}}T{x}_{0}$.

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* *P*-*monotone 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

*P*-monotone property and

*ψ*is nondecreasing, by the definition of

*T*, we have

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 *P*-monotone property, we have ${x}_{0}\le {P}_{{A}_{0}}T{x}_{0}$.

Therefore, ${x}^{\ast}$ is the one in ${A}_{0}$ such that $d({x}^{\ast},T{x}^{\ast})=d(A,B)$. □

## Declarations

### Acknowledgements

This project is supported by the National Natural Science Foundation of China under grant (11071279).

## Authors’ Affiliations

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