A generalized weak contraction principle with applications to coupled coincidence point problems

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R E S E A R C H

Open Access

A generalized weak contraction principle

with applications to coupled coincidence

point problems

Binayak S Choudhury

1

_{, Nikhilesh Metiya}

2

_{and Mihai Postolache}

3*

*_{Correspondence:}

[email protected]

3_{Faculty of Applied Sciences,}

University Politehnica of Bucharest, 313 Splaiul Independentei, Bucharest, 060042, Romania Full list of author information is available at the end of the article

Abstract

In this paper we establish some coincidence point results for generalized weak contractions with discontinuous control functions. The theorems are proved in metric spaces with a partial order. Our theorems extend several existing results in the current literature. We also discuss several corollaries and give illustrative examples. We apply our result to obtain some coupled coincidence point results which eﬀectively generalize a number of established results.

MSC: 54H10; 54H25

Keywords: partially ordered set; control function; coincidence point; coupled coincidence point

1 Introduction

In this paper we prove certain coincidence point results in partially ordered metric spaces for functions which satisfy a certain inequality involving three control functions. Two of the control functions are discontinuous. Fixed point theory in partially ordered metric spaces is of relatively recent origin. An early result in this direction is due to Turinici [], in which ﬁxed point problems were studied in partially ordered uniform spaces. Later, this branch of ﬁxed point theory has developed through a number of works, some of which are in [–].

Weak contraction was studied in partially ordered metric spaces by Harjaniet al.[]. In a recent result by Choudhuryet al.[], a generalization of the above result to a coincidence point theorem has been done using three control functions. Here we prove coincidence point results by assuming a weak contraction inequality with three control functions, two of which are not continuous. The results are obtained under two sets of additional con-ditions. A ﬁxed point theorem is also established. There are several corollaries and two examples. One of the examples shows that the corollaries are properly contained in their respective theorem. The corollaries are generalizations of several existing works.

We apply our result to obtain some coupled coincidence point results. Coupled ﬁxed theorems and coupled coincidence point theorems have appeared prominently in recent literature. Although the concept of coupled ﬁxed points was introduced by Guoet al.[], starting with the work of Gnana Bhaskar and Lakshmikantham [], where they established a coupled contraction principle, this line of research has developed rapidly in partially ordered metric spaces. References [–] are some examples of these works. There is a

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viewpoint from which coupled fixed and coincidence point theorems can be considered as problems in product spaces []. We adopt this approach here. Specifically, we apply our theorem to a product of two metric spaces on which a metric is defined from the metric of the original spaces. We establish a generalization of several results. We also discuss an example which shows that our result is an actual improvement over the results it general-izes.

2 Mathematical preliminaries

Let (X,) be a partially ordered set andT:X−→X. The mappingTis said to be nonde-creasing if for allx,x∈X,xximpliesTxTxand nonincreasing if for allx,x∈X, xximpliesTxTx.

Deﬁnition .([]) Let (X,) be a partially ordered set andT: X−→XandG: X−→X. The mappingTis said to beG-nondecreasingif for allx,y∈X,GxGyimpliesTxTy

andG-nonincreasingif for allx,y∈X,GxGyimpliesTxTy.

Deﬁnition . Two self-mappingsGandTof a nonempty setXare said to be commuta-tiveifGTx=TGxfor allx∈X.

Deﬁnition .([]) Two self-mappingsGandT of a metric space (X,d) are said to be

compatibleif the following relation holds:

lim

n→∞d(GTxn,TGxn) = ,

whenever{xn}is a sequence inXsuch thatlimn→∞Gxn=limn→∞Txn=xfor somex∈X is satisﬁed.

Deﬁnition .([]) Two self-mappingsGandT of a nonempty setX are said to be

weakly compatibleif they commute at their coincidence points; that is, ifGx=Txfor some

x∈X, thenGTx=TGx.

Deﬁnition .([]) Let (X,) be a partially ordered set andF:X×X−→X. The mapping

Fis said to have themixed monotone propertyifFis monotone nondecreasing in its ﬁrst argument and is monotone nonincreasing in its second argument; that is, if

x,x∈X, xx ⇒ F(x,y)F(x,y) for ally∈X;

and

y,y∈X, yy ⇒ F(x,y)F(x,y) for allx∈X.

Deﬁnition .([]) Let (X,) be a partially ordered set,F:X×X−→Xandg:X−→X. We say thatFhasthe mixed g-monotone propertyif

x,x∈X, gxgx ⇒ F(x,y)F(x,y) for ally∈X;

and

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Deﬁnition .([]) An element (x,y)∈X×Xis called acoupled ﬁxed pointof the map-pingF: X×X−→XifF(x,y) =xandF(y,x) =y.

Deﬁnition .([]) An element (x,y)∈X×Xis called acoupled coincidence pointof the mappingsF:X×X−→Xandg:X−→XifF(x,y) =gxandF(y,x) =gy.

Deﬁnition .([]) LetXbe a nonempty set. The mappingsgandF, whereg:X−→X

andF:X×X−→X, are said to becommutativeifgF(x,y) =F(gx,gy) for allx,y∈X.

Deﬁnition .([]) Let (X,d) be a metric space. The mappingsgandF, whereg:X−→ XandF:X×X−→X, are said to becompatibleif the following relations hold:

lim n→∞d

gF(xn,yn),F(gxn,gyn)

= , lim

n→∞d

gF(yn,xn),F(gyn,gxn)

= ,

whenever{xn}and{yn}are sequences inXsuch thatlimn→∞gxn=limn→∞F(xn,yn) =x andlim_n_→∞gyn=limn→∞F(yn,xn) =yfor somex,y∈Xare satisﬁed.

Deﬁnition . Let X be a nonempty set. The mappingsg andF, where g: X−→X

andF:X×X−→X, are said to beweakly compatibleif they commute at their coupled coincidence points, that is, if F(x,y) =gxandF(y,x) =gyfor some (x,y)∈X×X, then

gF(x,y) =F(gx,gy) andgF(y,x) =F(gy,gx).

Deﬁnition .([]) A functionψ: [,∞)→[,∞) is called analtering distance func-tionif the following properties are satisﬁed:

(i) ψis monotone increasing and continuous; (ii) ψ(t) = if and only ift= .

In our results in the following sections, we use the following classes of functions. We denote bythe set of all functionsψ: [,∞)−→[,∞) satisfying

(iψ) ψ is continuous and monotone non-decreasing,

(iiψ) ψ(t) = if and only ift= ;

and bywe denote the set of all functionsα: [,∞)−→[,∞) such that

(iα) αis bounded on any bounded interval in[,∞),

(iiα) αis continuous atandα() = .

3 Main results

Let (X,,d) be an ordered metric space.Xis calledregularif it has the following proper-ties:

(i) if a nondecreasing sequence{xn} −→xin(X,d), thenxnxfor alln≥;

(ii) if a nonincreasing sequence{yn} −→yin(X,d), thenyynfor alln≥.

Theorem . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let T,G:X−→X be two mappings such that G is continuous and nondecreasing,T(X)⊆G(X),T is G-nondecreasing with respect to and the pair(G,T)is compatible.Suppose that there existψ∈andϕ,θ∈such that

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for any sequence{xn}in[,∞)with xn−→t> ,

ψ(t) –limϕ(xn) +limθ(xn) > , (.)

and for all x,y∈X with GxGy

ψd(Tx,Ty)≤ϕd(Gx,Gy)–θd(Gx,Gy). (.)

Also,suppose that (a) Tis continuous,or

(b) Xis regular.

If there exists x∈X such that GxTx,then G and T have a coincidence point in X.

Proof Letx∈Xbe such thatGxTx. SinceT(X)⊆G(X), we can choosex∈Xsuch

thatGx=Tx. Again, we can choosex∈Xsuch thatGx=Tx. Continuing this process,

we construct a sequence{xn}inXsuch that

Gxn+=Txn for alln≥. (.)

SinceGxTxandGx=Tx, we haveGxGx, which implies thatTxTx. Now, TxTx, that is,GxGximplies thatTxTx. Again,TxTx, that is,GxGx

implies thatTxTx. Continuing this process, we have

GxGxGxGx · · · GxnGxn+ · · ·, (.)

and

TxTxTxTx · · · TxnTxn+ · · ·. (.)

LetRn=d(Gxn+,Gxn) for alln≥.

SinceGxn+Gxn, from (.) and (.), we have

ψd(Gxn+,Gxn+)

=ψd(Txn+,Txn)

≤ϕd(Gxn+,Gxn)

–θd(Gxn+,Gxn)

,

that is,

ψ(Rn+)≤ϕ(Rn) –θ(Rn), (.)

which, in view of the fact thatθ ≥, yieldsψ(Rn+)≤ϕ(Rn), which by (.) implies that

Rn+≤Rnfor all positive integern, that is,{Rn}is a monotone decreasing sequence. Hence there exists anr≥ such that

Rn=d(Gxn+,Gxn)−→r asn−→ ∞. (.)

Taking limit supremum on both sides of (.), using (.), the property (iα) ofϕandθ, and the continuity ofψ, we obtain

ψ(r)≤limϕ(Rn) +lim

–θ(Rn)

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Sincelim(–θ(Rn)) = –limθ(Rn), it follows that

ψ(r)≤limϕ(Rn) –limθ(Rn),

that is,

ψ(r) –limϕ(Rn) +limθ(Rn)≤,

which by (.) is a contradiction unlessr= . Therefore,

Rn=d(Gxn+,Gxn)−→ asn−→ ∞. (.)

Next we show that{Gxn}is a Cauchy sequence.

Suppose that{Gxn}is not a Cauchy sequence. Then there exists an>  for which we can ﬁnd two sequences of positive integers{m(k)}and{n(k)} such that for all positive integers k,n(k) >m(k) >kandd(Gxm(k),Gxn(k))≥. Assuming thatn(k) is the smallest

such positive integer, we get

n(k) >m(k) >k,

d(Gxm(k),Gxn(k))≥ and

d(Gxm(k),Gxn(k)–) <.

Now,

≤d(Gxm(k),Gxn(k))≤d(Gxm(k),Gxn(k)–) +d(Gxn(k)–,Gxn(k)),

that is,

≤d(Gxm(k),Gxn(k))≤+d(Gxn(k)–,Gxn(k)).

Lettingk−→ ∞in the above inequality and using (.), we have

lim

k→∞d(Gxm(k),Gxn(k)) =. (.)

Again,

d(Gxm(k)+,Gxn(k)+)≤d(Gxm(k)+,Gxm(k)) +d(Gxm(k),Gxn(k)) +d(Gxn(k)+,Gxn(k))

and

d(Gxm(k),Gxn(k))≤d(Gxm(k)+,Gxm(k)) +d(Gxm(k)+,Gxn(k)+) +d(Gxn(k)+,Gxn(k)).

Lettingk−→ ∞in the above inequalities, using (.) and (.), we have

lim

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Asn(k) >m(k),Gxn(k)Gxm(k), from (.) and (.), we have

ψd(Gxn(k)+,Gxm(k)+)

=ψd(Txn(k),Txm(k))

≤ϕd(Gxn(k),Gxm(k))

–θd(Gxn(k),Gxm(k))

.

Taking limit supremum on both sides of the above inequality, using (.), (.), the property (iα) ofϕandθ, and the continuity ofψ, we obtain

ψ()≤limϕd(Gxn(k),Gxm(k))

+lim–θd(Gxn(k),Gxm(k))

.

Sincelim(–θ(d(Gxn(k),Gxm(k)))) = –limθ(d(Gxn(k),Gxm(k))), it follows that

ψ()≤limϕd(Gxn(k),Gxm(k))

–limθd(Gxn(k),Gxm(k))

,

that is,

ψ() –limϕd(Gxn(k),Gxm(k))

+limθd(Gxn(k),Gxm(k))

≤,

which is a contradiction by (.). Therefore,{Gxn}is a Cauchy sequence inX. From the completeness ofX, there existsx∈Xsuch that

lim

n→∞Txn=nlim→∞Gxn=x. (.)

Since the pair (G,T) is compatible, from (.), we have

lim

n→∞d(GTxn,TGxn) = . (.)

Let the condition (a) hold.

By the triangular inequality, we have

d(Gx,TGxn)≤d(Gx,GTxn) +d(GTxn,TGxn).

Takingn−→ ∞in the above inequality, using (.), (.) and the continuities ofTand

G, we haved(Gx,Tx) = , that is,Gx=Tx, that is,xis a coincidence point of the mappings

GandT.

Next we suppose that the condition (b) holds.

By (.) and (.), we haveGxnxfor alln≥. Using the monotone property ofG, we obtain

GGxnGx. (.)

AsGis continuous and the pair (G,T) is compatible, by (.) and (.), we have

lim

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Then

d(Tx,Gx) = lim

n→∞d(Tx,TGxn).

Sinceψis continuous, from the above inequality, we obtain

ψd(Tx,Gx)=ψ

lim

n→∞d(Tx,TGxn)

= lim

n→∞ψ

d(Tx,TGxn)

,

which, by (.) and (.), implies that

ψd(Tx,Gx)≤ lim n→∞

ϕd(Gx,GGxn)

–θd(Gx,GGxn)

.

Using (.) and the property (iiα) ofϕandθ, we have

ψd(Tx,Gx)= ,

which, by the property ofψ, implies that d(Tx,Gx) = , that is,Gx=Tx, that is,x is a

coincidence point of the mappingsGandT.

Next we discuss some corollaries of Theorem .. By an example, we show that Theo-rem . properly contains all its corollaries.

Every commuting pair (G,T) is also a compatible pair. Then considering (G,T) to be the commuting pair in Theorem ., we have the following corollary.

Corollary . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let T,G:X−→X be two mappings such that G is continuous and nondecreasing,T(X)⊆G(X),T is G-nondecreasing with respect to and the pair(G,T)is commutative.Suppose that there existψ∈andϕ,θ∈such that

(.), (.)and(.)are satisﬁed.Also,suppose that (a) Tis continuous,or

(b) Xis regular.

Consideringψto be the identity mapping andθ(t) =  for allt∈[,∞) in Theorem ., we have the following corollary.

Corollary . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let T,G:X−→X be two mappings such that G is continuous and nondecreasing,T(X)⊆G(X),T is G-nondecreasing with respect toand the pair(G,T)is compatible.Suppose that there existsϕ∈such that for any sequence{xn}in[,∞)with xn−→t> ,

limϕ(xn) <t, (.)

and for all x,y∈X with GxGy,

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(b) Xis regular.

Consideringθ(t) =  for allt∈[,∞) andϕ(t) =kψ(t) withk∈[, ) in Theorem ., we have the following corollary.

Corollary . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let T,G:X−→X be two mappings such that G is continuous and nondecreasing,T(X)⊆G(X),T is G-nondecreasing with respect to and the pair(G,T)is compatible.Suppose that there existsψ∈and k∈[, )such that for all x,y∈X with GxGy,

ψd(Tx,Ty)≤kψd(Gx,Gy). (.)

(b) Xis regular.

Consideringϕto be the functionψin Theorem ., we have the following corollary.

Corollary . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let T,G:X−→X be two mappings such that G is continuous and nondecreasing,T(X)⊆G(X),T is G-nondecreasing with respect to and the pair(G,T)is compatible.Suppose that there existψ∈andθ∈such that for any sequence{xn}in[,∞)with xn−→t> ,

limθ(xn) > , (.)

and for all x,y∈X with GxGy,

ψd(Tx,Ty)≤ψd(Gx,Gy)–θd(Gx,Gy). (.)

Also,suppose that (a) Tis continuous or

(b) Xis regular.

Ifψandϕare the identity mappings in Theorem ., we have the following corollary.

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toand the pair(G,T)is compatible.Suppose that there existsθ ∈such that for any sequence{xn}in[,∞)with xn−→t> ,limθ(xn) > and for all x,y∈X with GxGy,

d(Tx,Ty)≤d(Gx,Gy) –θd(Gx,Gy). (.)

(b) Xis regular.

Consideringψandϕto be the identity mappings andθ(t) = ( –k)t, where ≤k<  in Theorem ., we have the following corollary.

Corollary . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let T,G:X−→X be two mappings such that G is continuous and nondecreasing,T(X)⊆G(X),T is G-nondecreasing with respect toand the pair(G,T)is compatible.Assume that there exists k∈[, )such that for all x,y∈X with GxGy,

d(Tx,Ty)≤kd(Gx,Gy). (.)

(b) Xis regular.

The condition (i), the continuity and the monotone property of the functionG, and (ii), the compatibility condition of the pairs (G,T), which were considered in Theorem ., are relaxed in our next theorem by takingG(X) to be closed in (X,d).

Theorem . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let T,G:X−→X be two mappings such that T(X)⊆G(X)and T is G-nondecreasing with respect toand G(X)is closed in X.

Suppose that there existψ∈andϕ,θ∈such that(.), (.)and(.)are satisﬁed.

Also,suppose that X is regular.

Proof We take the same sequence{xn}as in the proof of Theorem .. Then we have (.), that is,

lim

n→∞Txn=nlim→∞Gxn=x.

Since{Gxn}is a sequence inG(X) andG(X) is closed inX,x∈G(X). Asx∈G(X), there existsz∈Xsuch thatx=Gz. Then

lim

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Now,{Gxn}is nondecreasing and converges toGz. So, by the order condition of the metric spaceX, we have

GxnGz. (.)

Puttingx=zandy=xnin (.), by the virtue of (.), we get

ψd(Tz,Txn)

≤ϕd(Gz,Gxn)

–θd(Gz,Gxn)

.

Takingn→ ∞in the above inequality, using (.), the property (iiα) ofϕandθ and the continuity ofψ, we have

ψd(Tz,Gz)= ,

which, by the property ofψ, implies thatd(Tz,Gz) = , that is,Tz=Gz, that is,zis a

coincidence point of the mappingsGandT.

In the following, our aim is to prove the existence and uniqueness of the common ﬁxed point in Theorems . and ..

Theorem . In addition to the hypotheses of Theorems.and.,in both of the theo-rems,suppose that for every x,y∈X there exists u∈X such that Tu is comparable to Tx and Ty,and also the pair(G,T)is weakly compatible.Then G and T have a unique common ﬁxed point.

Proof From Theorem . or Theorem ., the set of coincidence points ofGandTis non-empty. Supposexandyare coincidence points ofGandT, that is,Gx=TxandGy=Ty. Now, we show

Gx=Gy. (.)

By the assumption, there existsu∈Xsuch thatTuis comparable withTxandTy. Putu=uand chooseu∈Xso thatGu=Tu. Then, similarly to the proof of

Theo-rem ., we can inductively deﬁne sequences{Gun}whereGun+=Tunfor alln≥. Hence

Tx=GxandTu=Tu=Guare comparable.

Suppose thatGuGx(the proof is similar to that in the other case).

We claim thatGunGxfor eachn∈N.

In fact, we will use mathematical induction. SinceGuGx, our claim is true forn= .

We presume thatGunGxholds for somen> . SinceTisG-nondecreasing with respect to, we get

Gun+=TunTx=Gx;

and this proves our claim.

LetRn=d(Gx,Gun). SinceGunGx, using the contractive condition (.), for alln≥, we have

ψd(Gx,Gun+)

=ψd(Tx,Tun)

≤ϕd(Gx,Gun)

–θd(Gx,Gun)

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that is,ψ(Rn+)≤ϕ(Rn) –θ(Rn), which, in view of the fact thatθ≥, yieldsψ(Rn+)≤

ϕ(Rn), which by (.) implies thatRn+≤Rn for all positive integern, that is,{Rn}is a monotone decreasing sequence.

Then, as in the proof of Theorem ., we have

lim

n→∞Rn=nlim→∞d(Gx,Gun) = . (.)

Similarly, we show that

lim

n→∞d(Gy,Gun) = . (.)

By the triangle inequality, using (.) and (.), we have

d(Gx,Gy)≤d(Gx,Gun) +d(Gun,Gy)

−→ asn−→ ∞.

HenceGx=Gy. Thus (.) is proved.

SinceGx=Tx, by weak compatibility ofGandT, we have

GGx=GTx=TGx. (.)

Denote

Gx=z. (.)

Then from (.) we have

Gz=Tz.

Thuszis a coincidence point ofGandT. Then from (.) withy=zit follows that

Gx=Gz.

By (.) it follows that

z=Gz. (.)

From (.) and (.), we getz=Gz=Tz.

Therefore,zis a common ﬁxed point ofGandT.

To prove the uniqueness, assume thatris another common ﬁxed point ofGandT. Then by (.) we haver=Gr=Gz=z. Hence the common ﬁxed point ofGandTis unique.

Example . LetX= [,∞). Then (X,≤) is a partially ordered set with the natural or-dering of real numbers. Letd(x,y) =|x–y|forx,y∈X. Then (X,d) is a complete metric space.

LetT,G: X→X be given respectively by the formulasTx= _x andGx=x for all

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Letψ,ϕ,θ: [,∞)−→[,∞) be given respectively by the formulas

ψ(t) =t, ϕ(t) =

 [t]

 _{if  <}_t_{< ,}

 t

 _otherwise, θ(t) =

 [t]

 _{if  <}_t_{< ,}

 otherwise.

Thenψ,ϕandθhave the properties mentioned in Theorem ..

It can be verified that (.) is satisfied for allx,y∈XwithGxGy. Hence the required conditions of Theorem . are satisfied and it is seen that  is a coincidence point ofG

andT. Also, the conditions of Theorem . are satisﬁed and it is seen that  is the unique common ﬁxed point ofGandT.

Remark . In the above example, the pair (G,T) is compatible but not commuting so that Corollary . is not applicable to this example and hence Theorem . properly contains its Corollary ..

Remark . In the above example,ψis not the identity mapping andθ(t)=  for alltin [,∞). Let us consider the sequence{xn}in [,∞), wherexn=  for alln. Thenϕ(xn) =  for alln. Nowxn−→t=  > , butlimϕ(xn) = ≮t= . Therefore, Corollary . is not applicable to this example, and hence Theorem . properly contains its Corollary ..

Remark . The above exampleθ(t)=  for allt∈[,∞), and hence Corollary . is not applicable to the example, and so Theorem . properly contains its Corollary ..

Remark . In the above example, ϕ is not identical to the function ψ, and also limθ(xn)≯ for any sequence{xn} in [,∞) with xn−→t> . Therefore, Corollaries . and . are not applicable to this example, and hence Theorem . properly contains its Corollaries . and ..

Remark . In the above example,ψandϕare not the identity functions andθ(t)= ( –

k)twith ≤k< . Therefore, Corollary . is not applicable to the above example. Hence Theorem . properly contains its Corollary ..

Remark . Theorem . generalizes the results in [–, , –].

Example . LetX= [, ]. Then (X,≤) is a partially ordered set with the natural ordering of real numbers. Letd(x,y) =|x–y|forx,y∈X. Then (X,d) is a metric space with the required properties of Theorem ..

LetT,G:X→Xbe given respectively by the formulas

Tx=  for allx∈X, Gx=

 ifxis rational,



 ifxis irrational.

ThenT andGhave the properties mentioned in Theorem .. Letψ,ϕ,θ: [,∞)−→[,∞) be given respectively by the formulas

ψ(t) =t, ϕ(t) =



[t] if  <t< , 

t otherwise,

θ(t) =



[t] if  <t< ,

 otherwise.

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All the required conditions of Theorem . are satisfied. It is seen that every rational numberx∈Xis a coincidence point ofGandT. Also, the conditions of Theorem . are satisfied and it is seen that  is the unique common fixed point ofGandT.

Remark . In the above example, the functiong is not continuous. Therefore, Theo-rem . is not applicable to the above example.

4 Applications to coupled coincidence point results

In this section, we use the results of the previous section to establish new coupled coinci-dence point results in partially ordered metric spaces. Our results are extensions of some existing results.

Let (X,) be a partially ordered set. Now, we endow the product spaceX×Xwith the following partial order:

for (x,y), (u,v)∈X×X, (u,v)(x,y) ⇔ xu,yv.

Let (X,d) be a metric space. Thendgiven by the law

d

(x,y), (u,v)=d(x,u) +d(y,v) for (x,y), (u,v)∈X×X

is a metric onX×X.

Letg: X−→XandF: X×X−→Xbe two mappings. Then we deﬁne two functions

G:X×X−→X×XandT:X×X−→X×Xrespectively as follows:

G(x,y) = (gx,gy) forx,y∈X, T(x,y) =F(x,y),F(y,x) forx,y∈X.

Lemma . Let(X,)be a partially ordered set,F:X×X−→X and g:X−→X.If F has the mixed g-monotone property,then T is G-nondecreasing.

Proof Let (x,y), (x,y)∈X×Xsuch thatG(x,y)G(x,y). Then, by the deﬁnition

ofG, it follows that (gx,gy)(gx,gy), that is,gxgxandgygy. SinceFhas the

mixedg-monotone property, we have

F(x,y)F(x,y)F(x,y),

and

F(y,x)F(y,x)F(y,x).

It follows that (F(x,y),F(y,x))(F(x,y),F(y,x)), that is,T(x,y)T(x,y).

There-fore,T isG-nondecreasing.

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Proof Let (x,y)∈X×X. SincegandFare commutative, by the deﬁnition ofGandT, we have

GT(x,y) =GF(x,y),F(y,x)=gF(x,y),gF(y,x)=F(gx,gy),F(gy,gx)

=T(gx,gy) =TG(x,y),

which shows thatGandT are commutative.

Lemma . Let(X,d)be metric space and g: X−→X,F:X×X−→X.If g and F are compatible,then the mappings G and T are also compatible.

Proof Let{(xn,yn)}be a sequence inX×Xsuch thatlimn→∞G(xn,yn) =limn→∞T(xn,yn) = (x,y) for some (x,y)∈X×X. By the deﬁnition ofGandT, we havelim_n_→∞(gxn,gyn) = limn→∞(F(xn,yn),F(yn,yn)) = (x,y), which implies that

lim

n→∞F(xn,yn) =nlim→∞gxn=x, nlim→∞F(yn,xn) =nlim→∞gyn=y.

Now

d

GT(xn,yn),TG(xn,yn)

=d

GF(xn,yn),F(yn,xn)

,T(gxn,gyn)

=d

gF(xn,yn),gF(yn,xn)

,F(gxn,gyn),F(gyn,gxn)

=dgF(xn,yn),F(gxn,gyn)

+dgF(yn,xn),F(gyn,gxn)

.

SincegandFare compatible, we have

lim n→∞d

GT(xn,yn),TG(xn,yn)

= .

It follows thatGandTare compatible.

Lemma . Let X be a nonempty set,g:X−→X and F: X×X−→X.If g and F are weak compatible,then the mappings G and T are also weak compatible.

Proof Let (x,y)∈X×Xbe a coincidence pointGandT. ThenG(x,y) =T(x,y), that is, (gx,gy) = (F(x,y),F(y,x)), that is,gx=F(x,y) andgy=F(y,x). SincegandFare weak com-patible, by the deﬁnition ofGandT, we have

GT(x,y) =GF(x,y),F(y,x)=gF(x,y),gF(y,x)

=F(gx,gy),F(gy,gx)=T(gx,gy) =TG(x,y),

which shows thatGandTcommute at their coincidence point, that is,GandT are weak

compatible.

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andϕ,θ∈such that(.)and(.)are satisﬁed and for all x,y,u,v∈X with gxgu and gygv,

ψdF(x,y),F(u,v)+dF(y,x),F(v,u)

≤ϕd(gx,gu) +d(gy,gv)–θd(gx,gu) +d(gy,gv). (.)

Also,suppose that (a) Fis continuous,or

(b) Xis regular.

If there exist x,y∈X such that gxF(x,y)and gyF(y,x),then there exist x,y∈X such that gx=F(x,y)and gy=F(y,x);that is,g and F have a coupled coincidence point in X.

Proof We consider the product space (X×X,), the metricdonX×Xand the functions

G:X×X−→X×XandT: X×X−→X×Xas mentioned above. DenoteS=X×X.

Then (S,d) is a complete metric space. By the deﬁnition ofGandT, we have that (i) Gis continuous and nondecreasing; andT is continuous,

(ii) T(S)⊆G(S),

(iii) TisG-nondecreasing with respect to, (iv) the pair(G,T)is compatible.

Letp= (x,y),q= (u,v)∈S=X×Xsuch thatgxguandgygv, that is, (gx,gy)(gu,gv), that is,GpGq. Then (.) reduces to

ψd(Tp,Tq)

≤ϕd(Gp,Gq)

–θd(Gp,Gq)

.

Now, the existence of x,y∈X such thatgxF(x,y) andgyF(y,x) implies

the existence of a point p= (x,y)∈Ssuch that (gx,gy)(F(x,y),F(y,x)), that

is, GpTp. Therefore, the theorem reduces to Theorem ., and hence there exists w= (x,y)∈S=X×X such that Gw=Tw, that is, G(x,y) =T(x,y), that is, (gx,gy) = (F(x,y),F(y,x)), that is,gx=F(x,y) andgy=F(y,x), that is, (x,y) is a coupled coincidence

point ofgandF.

The following corollary is a consequence of Corollary ..

Corollary . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let F: X×X−→X and g:X−→X be two mappings such that g is continuous and nondecreasing,F(X×X)⊆g(X),F has the mixed g-monotone property on X and the pair(g,F)is commutative.Suppose that there existψ∈,ϕ,θ∈such that(.), (.)and(.)are satisﬁed.Also,suppose that

(a) Fis continuous,or

(b) Xis regular.

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Corollary . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let F: X×X−→X and g:X−→X be two mappings such that g is continuous and nondecreasing,F(X×X)⊆g(X),F has the mixed g-monotone property on X and the pair(g,F)is compatible.Suppose that there existsϕ∈

such that for any sequence{xn}in[,∞)with xn−→t> ,

limϕ(xn) <t,

and for all x,y,u,v∈X with gxgu and gygv,

dF(x,y),F(u,v)+dF(y,x),F(v,u)≤ϕd(gx,gu) +d(gy,gv).

(b) Xis regular.

The following corollary is a consequence of Corollary ..

Corollary . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let F: X×X−→X and g:X−→X be two mappings such that g is continuous and nondecreasing,F(X×X)⊆g(X),F has the mixed g-monotone property on X and the pair(g,F)is compatible.Suppose that there existψ∈

and k∈[, )such that for all x,y,u,v∈X with gxgu and gygv,

ψdF(x,y),F(u,v)+dF(y,x),F(v,u)≤kψd(gx,gu) +d(gy,gv).

(b) Xis regular.

The following corollary is a consequence of Corollary ..

Corollary . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let F: X×X−→X and g:X−→X be two mappings such that g is continuous and nondecreasing,F(X×X)⊆g(X),F has the mixed g-monotone property on X and the pair(g,F)is compatible.Suppose that there existψ∈

andθ∈such that for any sequence{xn}in[,∞)with xn−→t> ,limθ(xn) > and for

all x,y,u,v∈X with gxgu and gygv,

ψdF(x,y),F(u,v)+dF(y,x),F(v,u)

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(b) Xis regular.

Remark . The above result is also true if the arguments ofψandθin (.) are replaced by their half values, that is, when (.) is replaced by

ψ d(F(x,y),F(u,v)) +d(F(y,x),F(v,u))



≤ψ d(gx,gu) +d(gy,gv)



–θ d(gx,gu) +d(gy,gv)



.

In this case, we can writeψ(t) =ψ(t_) andθ(t) =θ(_t) and proceed with the same proof

by replacingψ,θbyψ,θrespectively. Then we obtain a generalization of Theorem  of

Berinde in [].

The following corollary is a consequence of Corollary ..

Corollary . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that (X,d)is a complete metric space.Let F: X×X−→X and g: X−→X be two mappings such that g is continuous and nondecreasing, F(X×X)⊆g(X),F has the mixed g-monotone property on X and the pair(g,F)is compatible.Suppose that there existsθ∈such that for any sequence{xn}in[,∞)with xn−→t> ,limθ(xn) > and

for all x,y,u,v∈X with gxgu and gygv,

dF(x,y),F(u,v)+dF(y,x),F(v,u)≤d(gx,gu) +d(gy,gv) –θd(gx,gu) +d(gy,gv).

(b) Xis regular.

The following corollary is a consequence of Corollary ..

Corollary . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that (X,d)is a complete metric space.Let F: X×X−→X and g: X−→X be two mappings such that g is continuous and nondecreasing,F(X×X)⊆g(X),F has the mixed g-monotone property on X and the pair(g,F)is compatible.Assume that there exists k∈[, )such that for all x,y,u,v∈X with gxgu,gygv,

dF(x,y),F(u,v)+dF(y,x),F(v,u)≤kd(gx,gu) +d(gy,gv).

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(a) Fis continuous,or

(b) Xis regular.

The following theorem is a consequence of Theorem ..

Theorem . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Consider the mappings F:X×X−→X and g: X−→X such that F(X×X)⊆g(X),F has the mixed g-monotone property on X and G(X)is closed in X.Suppose that there existψ∈andϕ,θ∈such that(.), (.)and

(.)are satisﬁed.Also,suppose that X is regular.

The following theorem is a consequence of Theorem ..

Theorem . In addition to the hypotheses of Theorems.and.,in both of the theo-rems,suppose that for every(x,y), (x*_,_y*₎_∈_X_×_X_,_{there exists a}₍_u_,_v₎_∈_X_×_{X such that}

(F(u,v),F(v,u))is comparable to(F(x,y),F(y,x))and(F(x*,y*),F(y*,x*)),and also the pair

(g,F)is weakly compatible.Then g and F have a unique coupled common ﬁxed point;that is,there exists a unique(x,y)∈X×X such that x=gx=F(x,y)and y=gy=F(y,x).

Example . LetX= [,∞). Then (X,≤) is a partially ordered set with the natural or-dering of real numbers. Letd(x,y) =|x–y|forx,y∈X. Then (X,d) is a complete metric space.

Letg: X→Xbe given bygx=x_{for all}_x_∈_X_{. Also, consider}

F:X×X→X, F(x,y) =



(x–y) ifx,y∈X,x≥y,

 ifx≤y,

which obeys the mixedg-monotone property. Let{xn}and{yn}be two sequences inXsuch that

lim

n→∞F(xn,yn) =nlim→∞gxn=a,nlim→∞F(yn,xn) =nlim→∞gyn=b.

Then, obviously,a=  andb= .

Now, for alln≥,gxn=xn,gyn=yn, while

F(xn,yn) =



(xn–yn) ifxn≥yn,

 ifxn≤yn,

F(yn,xn) =



(yn–xn) ifyn≥xn,

 ifyn≤xn.

Then it follows that

lim n→∞d

gF(xn,yn),F(gxn,gyn)

= , lim

n→∞d

gF(yn,xn),F(gyn,gxn)

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Hence, the pair (g,F) is compatible inX.

Letx=  andy=c(> ) be two points inX. Then

g(x) =g() =  =F(,c) =F(x,y), g(y) =g(c) =c≥ c

 =F(c, ) =F(y,x).

Letψ,ϕ,θ: [,∞)−→[,∞) be given respectively by the formulas

ψ(t) =t, ϕ(t) =



[t] if  <t< , 

t

 _otherwise, θ(t) =



[t] if  <t< ,

 otherwise.

Thenψ,ϕandθhave the properties mentioned in Theorem .. We now verify inequality (.) of Theorem ..

We takex,y,u,v∈Xsuch thatgx≥guandgy≤gv, that is,x≥uandy≤v. LetM=d(gx,gu) +d(gy,gv) =|x_–_u_|₊_|_y_–_v_|_{= (}_x_–_u_{) + (}_v_–_y_).

The following are the four possible cases. Case-:x≥yandu≥v. Then

dF(x,y),F(u,v)+dF(y,x),F(v,u)

=d x

_–_y

 ,

u–v



+d(, ) =(x

_–_y_{) – (}_u_–_v₎



=(x

_–_u_{) + (}_v_–_y₎



=(x

_–_u_{) + (}_v_–_y₎

 =

 M.

Case-:x<yandu<v. Then

=d(, ) +d y _–_x

 ,

v_–_u



=(y

_–_x_{) – (}_v_–_u₎



=(x

_–_u_{) + (}_v_–_y₎



=(x

_–_u_{) + (}_v_–_y₎

 =

 M.

Case-:x≥yandu≤v. Then

=d x

_–_y

 , 

+d ,v

_–_u



=x

_–_y

 +

v_–_u

 =

(x_–_u_{) + (}_v_–_y₎

 =

 M.

Case-: The case ‘x<yandu>v’ is not possible. Under this condition,x_<_y_and_u_> v_{. Then by the condition}_y_≤_v_{, we have}_x_<_y_≤_v_<_u_{, which contradicts that}_x_≥_u_.

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Remark . In the above example, the pair (g,F) is compatible but not commuting so that Corollary . is not applicable to this example, and hence Theorem . properly contains its Corollary ..

Remark . As discussed in Remarks .-., Theorem . properly contains its Corol-laries .-..

Remark . Theorem . properly contains its Corollary ., which is an extension of Theorem  of Berinde [], and Theorems . and . of Bhaskar and Lakshmikantham []. Therefore, Theorem . is an actual extension over Theorem  of Berinde [] and Theorems . and . of Bhaskar and Lakshmikantham [].

Example . LetX= [, ]. Then (X,≤) is a partially ordered set with the natural ordering of real numbers. Letd(x,y) =|x–y|forx,y∈X. Then (X,d) is a metric space with the required properties of Theorem ..

LetF: X×X→X,F(x,y) =  for (x,y)∈X×X. Letg:X→Xbe given by the formula

gx=

 ifxis rational,



 ifxis irrational.

ThenFandghave the properties mentioned in Theorems .. Letψ,ϕ,θ: [,∞)−→[,∞) be given respectively by the formulas

ψ(t) =t, ϕ(t) =



[t] if  <t< , 

t otherwise,

θ(t) =



[t] if  <t< ,

 otherwise.

Thenψ,ϕandθhave the properties mentioned in Theorem ..

All the required conditions of Theorem . are satisﬁed. It is seen that every (x,y)∈

X×X, where bothxandyare rational, is a coupled coincidence point ofgandFinX. Also, the conditions of Theorem . are satisﬁed and it is seen that (, ) is the unique coupled common ﬁxed point ofgandFinX.

Remark . In the above example, the functiong is not continuous. Therefore, Theo-rem . is not applicable to the above example.

Remark . In some recent papers [, ] it has been proved that some of the contractive conditions involving continuous control functions are equivalent. Here two of our control functions are discontinuous. Therefore, the contraction we use here is not included in the class of contractions addressed by Aydiet al.[] and Jachymski [].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors completed the paper together. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah, West Bengal 711103, India.} 2_{Department of Mathematics, Bengal Institute of Technology, Kolkata, West Bengal 700150, India.}3_{Faculty of Applied}

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Acknowledgements

The work is partially supported by the Council of Scientiﬁc and Industrial Research, India (No. 25(0168)/09/EMR-II). Professor BS Choudhury gratefully acknowledges the support.

Received: 12 March 2013 Accepted: 24 May 2013 Published: 11 June 2013

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doi:10.1186/1687-1812-2013-152