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

R E S E A R C H

Open Access

Best proximity point theorems for

generalized contractions in partially ordered

metric spaces

Jingling Zhang, Yongfu Su

_{and Qingqing Cheng}

*_{Correspondence:}

suyongfu@gmail.com; suyongfu@tjpu.edu.cn

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, P.R. China

Abstract

The purpose of this paper is to obtain four best proximity point theorems for generalized contractions in partially ordered metric spaces. Further, ourP-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

LetAandBbe nonempty subsets of a metric space (X,d). An operatorT:A→Bis said to be contractive if there existsk∈[, ) such thatd(Tx,Ty)≤kd(x,y) for anyx,y∈A. The well-known Banach contraction principle is as follows: Let (X,d) be a complete metric space, and letT:X→Xbe a contraction ofXinto itself. ThenThas a unique fixed point inX.

In the sequel, we denote byΓ the functionsβ: [,∞)→[, ) satisfying the following condition:

β(tn)→ ⇒ tn→.

In , Geraghty introduced the Geraghty-contraction and obtained Theorem . as fol-lows.

Definition .[] Let (X,d) be a metric space. A mappingT :X→X is said to be a

Geraghty-contractionif there existsβ∈Γ such that for anyx,y∈X,

d(Tx,Ty)≤βd(x,y)d(x,y).

Theorem .[] Let(X,d)be a complete metric space,and let T:X→X be an operator.

Suppose that there existsβ∈Γ such that for any x,y∈X,

d(Tx,Ty)≤βd(x,y)d(x,y).

Then T has a unique fixed point.

Obviously, Theorem . is an extensive version of the Banach contraction principle. Re-cently, 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 .[] Let(X,≤)be a partially ordered set,and suppose that there exists a

met-ric d such that(X,d)is a complete metric space.Let f :X→X be an increasing mapping such that there exists an element x∈X with x≤f(x).Suppose that there existsβ∈Γ such that

df(x),f(y)≤βd(x,y)d(x,y), ∀y≥x.

Assume that either f is continuous or X is such that if an increasing sequence xn→x∈X, then xn≤x,∀n.Besides,if for each x,y∈X,there exists z∈X which is comparable to x and y,then f has a unique fixed point.

Theorem .[] Let(X,≤)be a partially ordered set,and suppose that there exists a met-ric d∈X such that(X,d)is a complete metric space.Let f :X→X be a continuous and nondecreasing mapping such that

df(x),f(y)≤d(x,y) –ψd(x,y) for y≥x,

whereψ: [,∞)→[,∞)is a continuous and nondecreasing function such thatψis pos-itive in(,∞),ψ() = andlimt→∞ψ(t) =∞.If there exists x∈X with x≤f(x),then f has a fixed point.

Definition .[] An altering distance function is a functionψ: [,∞)→[,∞) which

satisfies

(i) ψis continuous and nondecreasing. (ii) ψ(t) = if and only ift= .

Theorem .[] 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→X be a continuous and nondecreasing mapping such that

ψd(Tx,Ty)≤φd(x,y), ∀y≥x,

whereψis an altering distance function andφ: [,∞)→[,∞)is a continuous function with the conditionψ(t) >φ(t)for all t> .If there exists x∈X such that x≤Tx,then T has a fixed point.

Theorem .[] Let(X,≤)be a partially ordered set,and suppose that there exists a met-ric d∈X such that(X,d)is a complete metric space.Let f :X→X be a continuous and nondecreasing mapping such that

whereψandφare altering distance functions.If there exists x∈X with x≤f(x),then f has a fixed point.

In , Caballeroet al.introduced a generalized Geraghty-contraction as follows.

Definition .[] LetA,Bbe two nonempty subsets of a metric space (X,d). A mapping

T :A→Bis said to be aGeraghty-contractionif there exists β∈Γ such that for any

x,y∈A,

d(Tx,Ty)≤βd(x,y)d(x,y).

Now we need the following notations and basic facts. LetAandBbe two nonempty subsets of a metric space (X,d). We denote byAandBthe following sets:

A=x∈A:d(x,y) =d(A,B) for somey∈B,

B=y∈B:d(x,y) =d(A,B) for somex∈A,

whered(A,B) =inf{d(x,y) :x∈Aandy∈B}.

In [], the authors give sufficient conditions for when the setsAandBare nonempty. In [], the authors prove that any pair (A,B) of nonempty, closed convex subsets of a uni-formly convex Banach space satisfies theP-property.

Definition .[] Let (A,B) be a pair of nonempty subsets of a metric space (X,d) with

A=∅. Then the pair (A,B) is said to have theP-propertyif and only if for anyx,x∈A

andy,y∈B,

⎧ ⎨ ⎩

d(x,y) =d(A,B)

d(x,y) =d(A,B) ⇒ d(x,x) =d(y,y).

LetA,Bbe two nonempty subsets of a complete metric space, and consider a mapping

T :A→B. The best proximity point problem is whether we can find an elementx∈A

such thatd(x,Tx) =min{d(x,Tx) :x∈A}. Sinced(x,Tx)≥d(A,B) for anyx∈A, in fact, the optimal solution to this problem is the one for which the valued(A,B) is attained.

In [], the authors give a generalization of Theorem . by considering a non-self map-ping, and they get the following theorem.

Theorem .[] Let(A,B)be a pair of nonempty closed subsets of a complete metric

space(X,d)such that Ais nonempty.Let T:A→B be a Geraghty-contraction satisfying T(A)⊆B.Suppose that the pair(A,B)has the P-property.Then there exists a unique x* in A such that d(x*,Tx*) =d(A,B).

2 Main results

Before giving our main results, we first introduce the weakP-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=∅. Then the pair (A,B) is said to have theweak P-monotone propertyif for anyx,x∈Aandy,y∈B,

⎧ ⎨ ⎩

d(x,y) =d(A,B)

d(x,y) =d(A,B)

⇒ d(x,x)≤d(y,y);

furthermore,y≥yimpliesx≥x.

Now we are in a position to give our main results.

Theorem . 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=∅.Let f :A→B be an increasing mapping with f(A)⊆B,and let there existβ∈Γ such that

df(x),f(y)≤βd(x,y)d(x,y), ∀y≥x.

Assume that either f is continuous or that Ais such that if an increasing sequence xn→x∈ A,then xn≤x,∀n.Suppose that the pair(A,B)has the weak P-monotone property.And for some x∈A,there existsxˆ ∈Bsuch that d(x,xˆ ) =d(A,B)andxˆ ≤f(x).Besides,

if for each x,y∈A,there exists z∈Awhich is comparable to x and y,then there exists an x*_{in A such that d(x}*_,fx*_{) =d(A,B).}

Proof We first prove thatBis closed. Let{yn} ⊆Bbe a sequence such thatyn→q∈B.

It follows from the weakP-monotone property that

d(yn,ym)→ ⇒ d(xn,xm)→,

asn,m→ ∞, wherexn,xm∈Aandd(xn,yn) =d(A,B),d(xm,ym) =d(A,B). Then{xn}is a

Cauchy sequence so that{xn}converges strongly to a pointp∈A. By the continuity of a

metricd, we haved(p,q) =d(A,B), that is,q∈B, and henceBis closed.

LetAbe the closure ofA, we claim thatf(A)⊆B. In fact, ifx∈A\A, then there exists a sequence{xn} ⊆Asuch thatxn→x. By the continuity off and the closeness of B, we havefx=limn→∞fxn∈B. That is,f(A)⊆B.

Define an operatorPA:f(A)→A, byPAy={x∈A:d(x,y) =d(A,B)}. Since the pair (A,B) has the weakP-monotone property andf is increasing, we have

d(PAfx,PAfy)≤d(fx,fy)≤β

d(x,y)d(x,y), PAfy≥PAfx,

for anyy≥x∈A. Obviously,PAf is increasing. Letxn,x∈A,xn→x, whenf is contin-uous, then we have

ThenPAf is continuous. WhenAis such that if an increasing sequencexn→x∈A, thenxn≤x(∀n), we need not prove the continuity ofPAf. For somex∈A, there exists

ˆ

x∈Bsuch thatd(x,xˆ ) =d(A,B) andxˆ ≤f(x). That is,

d(x,xˆ ) =d(A,B), d(PAfx,fx) =d(A,B), xˆ ≤f(x).

By the weakP-monotone property, we havePAfx≥x.

This shows thatPAf :A→Ais a contraction satisfying all the conditions in The-orem .. Using TheThe-orem ., we can get PAf has a unique fixed point x

*_{. That is,}

PAfx*=x*∈A, which implies that

dx*,fx*=d(A,B).

Therefore,x*is the unique one inAsuch thatd(x*,fx*) =d(A,B).

Theorem . 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=∅.Let f :A→B be a continuous and nondecreasing mapping with f(A)⊆B,and let f satisfy

df(x),f(y)≤d(x,y) –ψd(x,y) for y≥x,

whereψ: [,∞)→[,∞)is a continuous and nondecreasing function such thatψis pos-itive in(,∞),ψ() = andlimt→∞ψ(t) =∞.Suppose that the pair(A,B)has the weak P-monotone property.If for some x∈A,there existsxˆ ∈Bsuch that d(x,xˆ ) =d(A,B)

andxˆ≤f(x),then there exists an x*in A such that d(x*,fx*) =d(A,B).

Proof In Theorem ., we have proved thatBis closed andf(A)⊆B. Now we define an operatorPA:f(A)→AbyPAy={x∈A:d(x,y) =d(A,B)}. Since the pair (A,B) has the weakP-monotone property, by the definition off, we have

d(PAfx,PAfy)≤d(fx,fy)≤d(x,y) –ψ

d(x,y), PAfy≥PAfx,

for anyy≥x∈A. Obviously,PAf is continuous and nondecreasing. For somex∈A, there existsxˆ∈Bsuch thatd(x,xˆ) =d(A,B) andxˆ≤f(x). That is,

d(x,xˆ) =d(A,B), d(PAfx,fx) =d(A,B), xˆ≤f(x).

By the weakP-monotone property, we havePAfx≥x.

This shows thatPAf :A→Ais a contraction satisfying all the conditions in Theo-rem .. Using TheoTheo-rem ., we can getPAf has a fixed pointx*. That is,PAfx*=x*∈A, which implies that

dx*,fx*=d(A,B).

Therefore,x*_{is the one inA}

Theorem . 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=∅.Let T:A→B be a continuous and nondecreasing mapping with T(A)⊆B,and let T satisfy

ψd(Tx,Ty)≤φd(x,y), ∀y≥x,

whereψ is an altering distance function andφ: [,∞)→[,∞)is a continuous func-tion with the condifunc-tionψ(t) >φ(t)for all t> .Suppose that the pair(A,B)has the weak P-monotone property.If for some x∈A,there exists y∈Bsuch that d(x,y) =d(A,B)

and y≤Tx,then there exists an x*in A such that d(x*,Tx*) =d(A,B).

Proof In Theorem ., we have proved thatBis closed andf(A)⊆B. Define an operator

PA:T(A)→AbyPAy={x∈A:d(x,y) =d(A,B)}. Since the pair (A,B) has the weak

P-monotone property andψis nondecreasing, by the definition ofT, we have

ψd(PATx,PATy)

≤ψd(Tx,Ty)≤φd(x,y), PATy≥PATx,

for anyy≥x∈A. Since

φd(x,y)→ ⇔ d(x,y)→

⇒ ψd(PATx,PATy)

→ ⇔ d(PATx,PATy)→,

this shows thatPAT is continuous and nondecreasing. Because there existx∈Aand

y∈Bsuch thatd(x,y) =d(A,B) andy≤Tx, by the weakP-monotone property, we

havex≤PATx.

This shows thatPAT:A→Ais a contraction from a complete metric subspaceA into itself and satisfies all the conditions in Theorem .. Using Theorem ., we can get

PAThas a fixed pointx*. That is,PATx*=x*∈A, which implies that

dx*,Tx*=d(A,B).

Therefore,x*_{is the one inAsuch thatd(x}*_,Tx*_{) =d(A,B).}

Theorem . 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=∅.Let T:A→B be a continuous and nondecreasing mapping with T(A)⊆B,and let T satisfy

ψdT(x),T(y)≤ψd(x,y)–φd(x,y) for y≥x,

whereψ andφare altering distance functions.Suppose that the pair(A,B)has the weak P-monotone property.If for some x∈A,there exists y∈Bsuch that d(x,y) =d(A,B)

and y≤Tx,then there exists an x*in A such that d(x*,Tx*) =d(A,B).

Proof In Theorem ., we have proved thatBis closed andf(A)⊆B. Define an operator PA:T(A)→AbyPAy={x∈A:d(x,y) =d(A,B)}. Since the pair (A,B) has the weak

P-monotone property andψis nondecreasing, by the definition ofT, we have

ψd(PATx,PATy)

for anyy≥x∈A. Since

d(x,y)→ ⇒ ψd(x,y)–φd(x,y)→

⇒ ψd(PATx,PATy)

→ ⇔ d(PATx,PATy)→.

This shows thatPATis continuous and nondecreasing. Because there existx∈Aand

y∈Bsuch thatd(x,y) =d(A,B) andy≤Tx, by the weakP-monotone property, we havex≤PATx.

This shows thatPAT:A→Ais a contraction from a complete metric subspaceA into itself and satisfies all the conditions in Theorem .. Using Theorem ., we can get

PAThas a fixed pointx*. That is,PATx*=x*∈A, which implies that

dx*,Tx*=d(A,B).

Therefore,x*is the one inAsuch thatd(x*,Tx*) =d(A,B).

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.

Acknowledgements

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

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

References

1. Geraghty, M: On contractive mappings. Proc. Am. Math. Soc.40, 604-608 (1973)

2. Amini-Harandi, A, Emami, H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal.72, 2238-2242 (2010)

3. Harjani, J, Sadarangni, K: Fixed point theorems for weakly contraction mappings in partially ordered sets. Nonlinear Anal.71, 3403-3410 (2009)

4. 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

5. Harjani, J, Sadarangni, K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal.72, 1188-1197 (2010)

6. Caballero, J, et al.: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. (2012). doi:10.1186/1687-1812-2012-231

7. Kirk, WA, Reich, S, Veeramani, P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim.24, 851-862 (2003)

8. Abkar, A, Gabeleh, M: Global optimal solutions of noncyclic mappings in metric spaces. J. Optim. Theory Appl.153, 298-305 (2012)

9. Sankar Raj, V: A best proximity point theorems for weakly contractive nonself mappings. Nonlinear Anal.74, 4804-4808 (2011)

doi:10.1186/1687-1812-2013-83