Coupled common fixed point results involving a (φ,ψ) contractive condition for mixed g monotone operators in partially ordered metric spaces

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R E S E A R C H

Open Access

Coupled common ﬁxed point results

involving a

(

ϕ

,

ψ

)

-contractive condition for

mixed

g

-monotone operators in partially

ordered metric spaces

Manish Jain

1

_{, Kenan Tas}

2*

_{, Sanjay Kumar}

3

_{and Neetu Gupta}

4

*_{Correspondence:}

[email protected]

2_{Department of Mathematics and}

Computer Sciences, Cankaya University, Ankara, Turkey Full list of author information is available at the end of the article

Abstract

In the setting of partially ordered metric spaces, using the notion of compatible mappings, we establish the existence and uniqueness of coupled common ﬁxed points involving a (

ϕ

,

ψ

)-contractive condition for mixedg-monotone operators. Our results extend and generalize the well-known results of Berinde (Nonlinear Anal. TMA 74:7347-7355, 2011; Nonlinear Anal. TMA 75:3218-3228, 2012) and weaken the contractive conditions involved in the results of Alotaibiet al.(Fixed Point Theory Appl. 2011:44, 2011), Bhaskaret al.(Nonlinear Anal. TMA 65:1379-1393, 2006), and Luonget al.(Nonlinear Anal. TMA 74:983-992, 2011). The eﬀectiveness of the presented work is validated with the help of suitable examples.

MSC: 54H10; 54H25

Keywords: partially ordered set; compatible mappings;g-mixed monotone mappings; coupled coincidence point; coupled common ﬁxed point

1 Introduction and preliminaries

Bhaskar and Lakshmikantham [] introduced the notion of coupled fixed points and proved some coupled fixed point theorems for a mapping with the mixed monotone prop-erty in the setting of partially ordered metric spaces. These concepts are defined as follows.

Deﬁnition .[] Let (X,≤) be a partially ordered set andF:X×X→X. The mapping

Fis said to have the mixed monotone property ifF(x,y) is monotone non-decreasing inx

and monotone non-increasing iny; that is, for anyx,y∈X,

x,x∈X, x≤x implies F(x,y)≤F(x,y)

and

y,y∈X, y≤y implies F(x,y)≥F(x,y).

Deﬁnition .[] An element (x,y)∈X×Xis called a coupled ﬁxed point of the mapping

F:X×X→XifF(x,y) =xandF(y,x) =y.

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Bhaskar and Lakshmikantham [] proved the following results.

Theorem .[] Let(X,≤)be a partially ordered set and suppose there exists a metric d on X such that(X,d)is a complete metric space.Let F:X×X→X be a continuous mapping having the mixed monotone property on X.Assume that there exists a k∈[, )with

dF(x,y),F(u,v)≤k 

d(x,u) +d(y,v) (.)

for all x≥u and y≤v.

If there exist two elements x,y∈X with x≤F(x,y)and y≥F(y,x),then there

exist x,y∈X such that x=F(x,y)and y=F(y,x).

Theorem .[] Let(X,≤)be a partially ordered set and suppose there exists a metric d

on X such that(X,d)is a complete metric space.Assume that X has the following property:

(i) if a non-decreasing sequence{xn} →x,thenxn≤xfor alln, (ii) if a non-increasing sequence{yn} →y,theny≤ynfor alln.

Let F :X×X→X be a mapping having the mixed monotone property on X.Assume that there exists a k∈[, )with the condition(.).If there exist two elements x,y∈X

with x≤F(x,y)and y≥F(y,x),then there exist x,y∈X such that x=F(x,y)and

y=F(y,x).

Lakshmikantham and Ćirić [] extended the notion of mixed monotone property to mixedg-monotone property and generalized the results of Bhaskar and Lakshmikantham [] by establishing the existence of coupled coincidence point results using a pair of com-mutative maps.

Deﬁnition .[] Let (X,≤) be a partially ordered set andF:X×X→Xandg:X→X.

We sayFhas the mixedg-monotone property ifF(x,y) is monotoneg-nondecreasing in its ﬁrst argument and is monotoneg-nonincreasing in its second argument; that is, for anyx,y∈X,

x,x∈X, gx≤gx implies F(x,y)≤F(x,y)

and

y,y∈X, gy≤gy implies F(x,y)≥F(x,y).

Deﬁnition .[] An element (x,y)∈X×Xis called a coupled coincidence point of the mappingsF:X×X→Xandg:X→XifF(x,y) =gxandF(y,x) =gy.

Deﬁnition .[] An element (x,y)∈X×Xis called a coupled common ﬁxed point of

the mappingsF:X×X→Xandg:X→Xifx=gx=F(x,y) andy=gy=F(y,x).

Deﬁnition .[] LetXbe a non-empty set andF:X×X→Xandg:X→X. We sayF

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Later, Choudhury and Kundu [] introduced the notion of compatibility in the context of coupled coincidence point problems and used the notion to improve the results of Lak-shmikantham and Ćirić [].

Deﬁnition .[] The mappingsF:X×X→Xandg:X→Xare said to be compatible if

lim

n→∞d

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

=  and lim

n→∞d

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

= 

whenever{xn}and{yn}are sequences inXsuch thatlimn→∞F(xn,yn) =limn→∞gxn=x

andlimn→∞F(yn,xn) =limn→∞gyn=yfor somex,y∈X.

In recent years, following Bhaskar and Lakhsmikantham [], the existence and unique-ness of coupled ﬁxed points under more general contractive conditions were established by various authors. One can refer to [, –].

In order to generalize the results of Bhaskar and Lakshmikantham [], Luong and Thuan [] considered the following class of control functions.

Deﬁnition .[] Letdenote the class of functionsϕ: [,∞)→[,∞) which satisfy

(ϕ) ϕis continuous and non-decreasing;

(ϕ) ϕ(t) = if and only ift= ;

(ϕ) ϕ(t+s)≤ϕ(t) +ϕ(s), for allt,s∈[,∞).

Deﬁnition .[] Letdenote the class of functionsψ: [,∞)→[,∞) which satisfy

(iψ) limt→rψ(t) > for allr> andlimt→+ψ(t) = .

The contractive condition considered by Luong and Thuan [] is given below:

ϕdF(x,y),F(u,v)≤ ϕ

d(x,u) +d(y,v)–ψ

d(x,u) +d(y,v) 

, (.)

whereϕ∈,ψ∈andx≥u,y≤v.

On the other hand, Alotaibi and Alsulami [] extended the results of Luong and Thuan [] for a compatible pair (F,g), whereF:X×X→Xandg:X→Xare the maps satisfying the following contractive condition:

ϕdF(x,y),F(u,v)≤ ϕ

d(gx,gu) +d(gy,gv)–ψ

d(gx,gu) +d(gy,gv) 

, (.)

withϕ∈,ψ∈andgx≥gu,gy≤gv.

We consider the classredeﬁned by Berinde [] as follows.

Deﬁnition .[] Letdenote the class of functionsϕ: [,∞)→[,∞) which satisfy

(iϕ) ϕis continuous and (strictly) increasing;

(iiϕ) ϕ(t) <tfor allt> ;

(iiiϕ) ϕ(t+s)≤ϕ(t) +ϕ(s)for allt,s∈[,∞).

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Berinde [] weakened the contractive conditions (.) and (.) by considering the more general one

ϕ

d(F(x,y),F(u,v)) +d(F(y,x),F(v,u)) 

≤ϕ

d(x,u) +d(y,v) 

–ψ

d(x,u) +d(y,v) 

(.)

for a mixed monotone mappingF:X×X→X,x≥u,y≤v, whereϕ∈andψ∈. The present work extends and generalizes several results presented in the literature of ﬁxed point theory. Our theorems directly derive the main results of Berinde [, ]. We give suitable examples to show how our results extend the well-known results of Alotaibi

et al.[], Bhaskaret al.[] and Luonget al.[] by signiﬁcantly weakening the involved contractive condition.

2 Main results

Theorem . Let(X,≤)be a partially ordered set and suppose there exists a metric d on X such that(X,d)is a complete metric space.Let F:X×X→X,g:X→X be two maps with F having the mixed g-monotone property on X such that there exist two elements x,y∈X

with gx≤F(x,y)and gy≥F(y,x).Suppose there existϕ∈andψ∈such that

ϕ

d(F(x,y),F(u,v)) +d(F(y,x),F(v,u)) 

≤ϕ

–ψ

(.)

for all x,y,u,v∈X with gx≥gu and gy≤gv.

Suppose that F(X×X)⊆g(X),g is continuous and the pair(F,g)is compatible.

Also suppose either (a) Fis continuous,or

(b) Xhas the following property:

(i) if a non-decreasing sequence{xn} →x,thengxn≤gxfor alln; (ii) if a non-increasing sequence{yn} →y,thengy≤gynfor alln.

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

Proof Letx,y∈Xsuch thatgx≤F(x,y) andgy≥F(y,x). SinceF(X×X)⊆g(X), we can choosex,y∈Xsuch thatgx=F(x,y),gy=F(y,x). Again, we can choose

x,y∈Xsuch thatgx=F(x,y),gy=F(y,x).

Continuing this process, we can construct sequences{gxn}and{gyn}inXsuch that

gxn+=F(xn,yn), gyn+=F(yn,xn) for alln≥. (.)

We shall prove, for alln≥, that

gxn≤gxn+, (.)

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Sincegx≤F(x,y) andgy≥F(y,x),gx=F(x,y),gy=F(y,x), we havegx≤

gx,gy≥gy; that is, (.) and (.) hold forn= .

Suppose that (.) and (.) hold for somen> ,i.e.,gxn≤gxn+,gyn≥gyn+. AsFhas

the mixedg-monotone property, by (.), we have

gxn+=F(xn,yn)≤F(xn+,yn)≤F(xn+,yn+) =gxn+,

and

gyn+=F(yn,xn)≥F(yn+,xn)≥F(yn+,xn+) =gyn+;

that is,

gxn+≤gxn+ and gyn+≥gyn+.

Then, by mathematical induction, it follows that (.) and (.) hold for alln≥. If, for somen≥, we have (gxn+,gyn+) = (gxn,gyn), thenF(xn,yn) =gxnandF(yn,xn) = gyn; that is,Fandghave a coincidence point. So, now onwards, we suppose (gxn+,gyn+)= (gxn,gyn) for alln≥; that is, we suppose that eithergxn+=F(xn,yn)=gxn orgyn+=

F(yn,xn)=gyn.

Sincegxn≥gxn–andgyn≤gyn–, by (.) and (.), we have, for alln≥, that

ϕ

d(gxn+,gxn) +d(gyn+,gyn)



=ϕ

d(F(xn,yn),F(xn–,yn–)) +d(F(yn,xn),F(yn–,xn–))



≤ϕ

d(gxn,gxn–) +d(gyn,gyn–)



–ψ



. (.)

Sinceψis non-negative, we have

ϕ



≤ϕ



.

By the monotonicity ofϕ, we have

 ≤

 .

Let Rn = d(gxn+,gxn)+_d(gyn+,gyn), then {Rn} is a monotone decreasing sequence of

non-negative real numbers. Therefore, there exists someR≥ such that

lim

n→∞Rn=nlim→∞

 =R. (.)

We claim thatR= .

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Taking limit asn→ ∞on both sides of (.) and using the properties ofϕ andψ, we have

ϕ(R) = lim

n→∞ϕ(Rn)≤nlim→∞

ϕ(Rn–) –ψ(Rn–)

=ϕ(R) – lim

Rn–→R

ψ(Rn–) <ϕ(R),

a contradiction. Thus,R= ; that is,

lim

n→∞Rn=nlim→∞

 = . (.)

Next, we shall show that{gxn}and{gyn}are Cauchy sequences.

If possible, suppose that at least one of{gxn}and{gyn}is not a Cauchy sequence. Then

there exists an ε>  for which we can ﬁnd subsequences{gxn(k)}, {gxm(k)}of{gxn}and {gyn(k)},{gym(k)}of{gyn}withn(k) >m(k)≥ksuch that

rk=

d(gxn(k),gxm(k)) +d(gyn(k),gym(k))

 ≥ε. (.)

Further, corresponding tom(k), we can choosen(k) in such a way that it is the smallest integer withn(k) >m(k)≥kand satisﬁes (.). Then

d(gxn(k)–,gxm(k)) +d(gyn(k)–,gym(k))

 <ε. (.)

By (.), (.) and the triangle inequality, we have

ε≤rk=



≤{d(gxn(k),gxn(k)–) +d(gxn(k)–,gxm(k)) +d(gyn(k),gyn(k)–) +d(gyn(k)–,gym(k))}



< d(gxn(k),gxn(k)–) +d(gyn(k),gyn(k)–)

 +ε.

Lettingk→ ∞and using (.) in the last inequality, we have

lim

k→∞rk=klim→∞

 =ε. (.)

Again, by the triangle inequality

rk =



≤

d(gxn(k),gxn(k)+) +d(gxn(k)+,gxm(k)+) +d(gxm(k)+,gxm(k))

+d(gyn(k),gyn(k)+) +d(gyn(k)+,gym(k)+) +d(gym(k)+,gym(k))



=Rn(k)+Rm(k)+

d(gxn(k)+,gxm(k)+) +d(gyn(k)+,gym(k)+)

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By the monotonicity ofϕand the property (iiiϕ), we have

ϕ(rk)≤ϕ(Rn(k)) +ϕ(Rm(k)) +ϕ

d(gxn(k)+,gxm(k)+) +d(gyn(k)+,gym(k)+) 

. (.)

Sincen(k) >m(k),gxn(k)≥gxm(k)andgyn(k)≤gym(k). Then by (.) and (.), we have

ϕ

d(gxn(k)+,gxm(k)+) +d(gyn(k)+,gym(k)+)



=ϕ

d(F(xn(k),yn(k)),F(xm(k),ym(k))) +d(F(yn(k),xn(k)),F(ym(k),xm(k))) 

≤ϕ



–ψ



=ϕ(rk) –ψ(rk). (.)

By (.) and (.), we have

ϕ(rk)≤ϕ(Rn(k)) +ϕ(Rm(k)) +ϕ(rk) –ψ(rk).

Lettingk→ ∞, using (.), (.) and the properties ofϕandψin the last inequality, we have

ϕ(ε)≤ϕ() +ϕ() +ϕ(ε) – lim

k→∞ψ(rk) =ϕ(ε) – lim

rk→ε

ψ(rk) <ϕ(ε),

a contradiction.

Therefore, both{gxn}and{gyn}are Cauchy sequences inX. By the completeness ofX,

there existx,y∈Xsuch that

lim

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

SinceFandgare compatible mappings, we have from (.)

lim

n→∞d

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

= , (.)

lim

n→∞d

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

= . (.)

Let the condition (a) hold. For alln≥, we have

dgx,F(gxn,gyn)

≤dgx,gF(xn,yn)

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

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Takingn→ ∞in the last inequality, using the inequalities (.), (.) and the continu-ities ofFandg, we haved(gx,F(x,y)) = ; that is,gx=F(x,y). Again, for alln≥,

dgy,F(gyn,gxn)

≤dgy,gF(yn,xn)

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

.

Takingn→ ∞ in the last inequality, using the inequalities (.), (.) and the con-tinuities of F andg, we havedgy,F(y,x)= ; that is,gy=F(y,x). Hence, the element (x,y)∈X×Xis a coupled coincidence point of the mappingsF:X×X→Xandg:X→X.

Next, we suppose that the condition (b) holds.

By (.), (.) and (.), we have{gxn}is a non-decreasing sequence,gxn→xand{gyn}

is a non-increasing sequence,gyn→yasn→ ∞. Hence, by the assumption (b), we have

for alln≥,

ggxn≤gx and ggyn≥gy. (.)

SinceFandgare compatible mappings andgis continuous, by inequalities (.)-(.), we have

lim

n→∞ggxn=gx=nlim→∞g

F(xn,yn)

= lim

n→∞F(gxn,gyn), (.)

and

lim

n→∞ggyn=gy=nlim→∞g

F(yn,xn)

= lim

n→∞F(gyn,gxn). (.)

Now,

dF(x,y),gx≤dF(x,y),ggxn+

+d(ggxn+,gx);

that is,

dF(x,y),gx≤dF(x,y),gF(xn,yn)

+d(ggxn+,gx).

Takingn→ ∞in the last inequality and using (.), we have

dF(x,y),gx≤ lim

n→∞d

F(x,y),gF(xn,yn)

+ lim

n→∞d(ggxn+,gx) ≤ lim

n→∞d

F(x,y),F(gxn,gyn)

. (.)

Similarly,

dF(y,x),gy≤ lim

n→∞d

F(y,x),F(gyn,gxn)

. (.)

By (.), (.) and the property (iϕ), we have

ϕ

d(F(x,y),gx) +d(F(y,x),gy) 

≤ lim

n→∞ϕ

d(F(x,y),F(gxn,gyn)) +d(F(y,x),F(gyn,gxn))



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By (.) and (.), we have

ϕ

d(F(x,y),F(gxn,gyn)) +d(F(y,x),F(gyn,gxn))



≤ϕ

d(gx,ggxn) +d(gy,ggyn)



–ψ



. (.)

Inserting (.) in (.), we have

ϕ

≤ lim

n→∞

ϕ



–ψ



= lim

n→∞ϕ



– lim

n→∞ψ



.

By (.), (.), the continuity ofϕandlim_t_→_+ψ(t) = , we get

ϕ

≤ lim

n→∞ϕ



=ϕ() = .

Sinceϕis non-negative andϕ() = , we have

dF(x,y),gx=  and dF(y,x),gy= ;

that is,

F(x,y) =gx and F(y,x) =gy.

Hence, the element (x,y)∈X×Xis a coupled coincidence point of the mappingsF:X×

X→Xandg:X→X.

Now, we give an example in support of Theorem ..

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 complete metric space. Let :X→Xbe deﬁned as

g(x) =x, for allx∈X.

LetF:X×X→Xbe deﬁned as

F(x,y) = ⎧ ⎨ ⎩

x_–_y

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Let{xn}and{yn}be two sequences inXsuch that

lim

n→∞F(xn,yn) =a, nlim→∞g(xn) =a,

lim

n→∞F(yn,xn) =b and nlim→∞g(yn) =b.

Now, for alln≥,

g(xn) =xn, g(yn) =yn,

F(xn,yn) =

⎧ ⎨ ⎩

xn–yn

 , ifx,y∈[, ],xn≥yn, , ifxn<yn,

and

F(yn,xn) =

⎧ ⎨ ⎩

yn–xn

 , ifx,y∈[, ],yn≥xn, , ifyn<xn.

Obviously,a=  andb= . Then it follows that

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

→ asn→ ∞,

and

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

→ asn→ ∞.

Hence, the mappingsFandgare compatible inX. Clearly,Fobeys the mixedg-monotone property. Also,F(X×X)⊆g(X).

Letϕ,ψ: [,∞)→[,∞) be deﬁned asϕ(t) = t

,ψ(t) =

t

, fort∈[,∞).

Also,x=  andy=c(>) are two points inXsuch thatg(x) =g() =  =F(,c) =

F(x,y) andg(y) =g(c) =c≥c_ =F(c, ) =F(y,x).

Next, we verify inequality (.) of Theorem .. We takex,y,u,v∈Xsuch thatgx≥gu

andgy≤gv; that is,x_≥_u_and_y_≤ _v_{. We discuss the following cases.} Case :x≥y,u≥v.

Then

ϕ

d(F(x,y),F(u,v)) +d(F(y,x),F(v,u)) 

= 

d(F(x,y),F(u,v)) +d(, ) 

=

d

x_–_y  ,

u_–_v 

= 

x–_y –u _–_v

 =



(x–u) + (_ v–y)=  

(x_–_u₎  +

(v_–_y₎ 

≤ 



(x_–_u_{) + (}_v_–_y₎ 

= 



d(gx,gu) +d(gv,gy) 

=ϕ

–ψ

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Case :x≥y,u<v. Then

ϕ

d(F(x,y),F(u,v)) +d(F(y,x),F(v,u)) 

= 

d(F(x,y),F(u,v)) +d(F(y,x),F(v,u))  =  d

x_–_y  , 

+d

,v

_–_u



= 

x_–_y 

+

v_–_u 

= 

x_–_u 

+

v_–_y 

≤ 



x_–_u 

+

v_–_y 

= 



(x_–_u_{) + (}_v_–_y₎ 

= 

=ϕ

–ψ

.

Case :x<y,u≥v. Then

ϕ

d(F(x,y),F(u,v)) +d(F(y,x),F(v,u)) 

= 

d(F(x,y),F(u,v)) +d(F(y,x),F(v,u))  =  d ,u

_–_v



+d

y_–_x  , 

= 



u_–_v 

+

y_–_x 

= 

–(x–u) – (v–y) 

≤ 



(x–u) + (v–y) 

≤ 



(x_–_u_{) + (}_v_–_y₎ 

= 



=ϕ

–ψ

.

Case :x<y,u<v. Then

ϕ

d(F(x,y),F(u,v)) +d(F(y,x),F(v,u)) 

= 

d(, ) +d(F(y,x),F(v,u)) 

=

d

y_–_x  ,

v_–_u 

= 

y–_x –v _–_u

 =



–(x–u) – (_ v–y)=  

_|

(x–u) + (v–y)| 

= 

(x_–_u_{) + (}_v_–_y₎ 

≤ 



(x_–_u_{) + (}_v_–_y₎ 

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= 

=ϕ

–ψ

.

Hence, the inequality (.) of Theorem . is satisﬁed.

Thus, all the conditions of Theorem . are satisﬁed, and it can be easily seen that (, ) is the required coupled coincidence point ofFandginX.

Remark . If we choose the functionsϕ(t) =t/ andψ(t) =t/, fort∈[,∞), then with

this choice of functions, we can obtain the already existing contractive condition. Sinceϕ andψare actually contractions, this will be cleared in Corollary .. But if we chooseϕ(t) =

t/(t+ ) andψ(t) =t/, fort∈[,∞), then with this choice ofϕandψ, the contractive condition (.) does not turn to the existing contractive condition.

The next example shows that Theorem . is more general than Theorem . in [] since the contractive condition (.) is more general than (.).

Example . LetX=R. Then (X,≤) is a partially ordered set with the natural ordering of real numbers. Letd:X×X→R+_{be deﬁned by}

d(x,y) =|x–y| forx,y∈X.

Then (X,d) is a complete metric space.

DefineF:X×X→XbyF(x,y) =x–_y, (x,y)∈X×Xandg:X→Xbyg(x) = x_,x∈X. Clearly,F(X×X)⊆g(X),Fis continuous and has the mixedg-monotone property, the pair (F,g) is compatible and satisfies the condition (.) but does not satisfy the condition (.). Assume, to the contrary, that there existϕ∈(in accordance with Definition .) andψ∈such that (.) holds. Then we must have

ϕx– y  –

u– v



≤ 

ϕ

x_–u  +y

 –

v



–ψ

_|x

–

u

|+|

y

–

v

| 

=  ϕ

_|

x–u|+|y–v|



–ψ

_|

x–u|+|y–v|



for allx≥uandy≤v. Takex=u,y=vin the last inequality and letρ=|y–_v|, we obtain

ϕ(ρ)≤

ϕ(ρ) –ψ(ρ), ρ> .

But by (ϕ) we have _ϕ(ρ)≤ϕ(ρ) and hence we deduce that, for allρ> ,ψ(ρ)≤, that is,ψ(ρ) = , which contradicts (iψ). This shows thatFdoes not satisfy (.).

Now, we prove that (.) holds. Indeed, forx≥uandy≤v, we have

x– _y–u– v 

≤ 

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and

y– _x–v– u 

≤ 

|y–v|+  |x–u|.

By summing up the last two inequalities, we get exactly (.) withϕ(t) = _t,ψ(t) = _t. Also,x= –,y=  are the two points inXsuch thatgx≤F(x,y) andgy≥F(y,x).

F,g,ϕ,ψ satisfy all the conditions of Theorem .. So, by Theorem ., we obtain thatF

andghave a coupled coincidence point (, ), but Theorem . in [] cannot be applied toFandgin this example.

The following Corollary . is Theorem  in [].

Corollary .[] Let(X,≤)be a partially ordered set and suppose there exists a metric d on X such that(X,d)is a complete metric space.Let F:X×X→X be a mapping having the mixed monotone property on X such that there exist two elements x,y∈X with x≤

F(x,y)and y≥F(y,x).Suppose there existϕ∈andψ∈such that

ϕ

d(F(x,y),F(u,v)) +d(F(y,x),F(v,u)) 

≤ϕ

d(x,u) +d(y,v) 

–ψ

d(x,u) +d(y,v) 

(.)

for all x,y,u,v∈X with x≥u and y≤v.Suppose either (a) Fis continuous,or

(i) if a non-decreasing sequence{xn} →x,thenxn≤xfor alln; (ii) if a non-increasing sequence{yn} →y,theny≤ynfor all n. Then there exist x,y∈X such that x=F(x,y)and y=F(y,x).

Proof Takinggto be an identity mapping in Theorem ., we obtain Corollary .. The following example shows that Corollary . is more general than Theorem . (i.e., Theorem . in []) and Theorem . in [], since the contractive condition (.) is more general than (.) and (.).

Example . LetX=R. Then (X,≤) is a partially ordered set with the natural ordering of real numbers. Letd:X×X→R+be deﬁned by

d(x,y) =|x–y| forx,y∈X.

Then (X,d) is a complete metric space.

DeﬁneF:X×X→XbyF(x,y) =x–_y, (x,y)∈X×X.

Then F is continuous, has the mixed monotone property and satisﬁes the condition (.) but does not satisfy either the condition (.) or the condition (.). Indeed, assume there existsk∈[, ) such that (.) holds. Then we must have

x– _ y–u– v 

≤k



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by which, forx=u, we get

|y–v| ≤k|y–v|, y≤v,

which fory<vimplies ≤k, a contradiction, sincek∈[, ). Hence,F does not satisfy (.).

Further, (.) is also not satisﬁed. Assume, to the contrary, that there existϕ∈(in accordance with Deﬁnition .) andψ∈such that (.) holds. Then we must have

ϕx– y  –

u– v



≤

ϕ

|x–u|+|y–v|–ψ _|

x–u|+|y–v|



,

for allx≥uandy≤v. Takex=u,y=vin the last inequality and letα=|y–_v|, we obtain

ϕ(α)≤

ϕ(α) –ψ(α), α> .

But by (ϕ), we have_ϕ(α)≤ϕ(α) and hence we deduce that, for allα> ,ψ(α)≤, that is,ψ(α) = , which contradicts (iψ). This shows thatFdoes not satisfy (.).

Now, we prove that (.) holds. Indeed, forx≥uandy≤v, we have

x– _ y–u– v 

≤ 

|x–u|+  |y–v|,

and y– x

 –

v– u



≤ 

|y–v|+  |x–u|.

By summing up the last two inequalities, we get exactly (.) withϕ(t) =_t,ψ(t) = _t. Also,x= –,y=  are the two points inXsuch thatx≤F(x,y) andy≥F(y,x).

So, by Corollary ., we obtain thatFhas a coupled ﬁxed point (, ) but neither Theo-rem . in [] nor TheoTheo-rem . in [] can be applied toFin this example.

The following Corollary . is Corollary  in [].

Corollary .[] Let(X,≤)be a partially ordered set and suppose there exists a metric d on X such that(X,d)is a complete metric space.Let F:X×X→X be a mapping having the mixed monotone property on X such that there exist two elements x,y∈X with x≤

F(x,y)and y≥F(y,x).Suppose there existsψ∈such that

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

≤d(x,u) +d(y,v) – ψ

d(x,u) +d(y,v) 

(.)

for all x,y,u,v∈X with x≥u and y≤v.Suppose either (a) Fis continuous,or

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(i) if a non-decreasing sequence{xn} →x,thenxn≤xfor alln; (ii) if a non-increasing sequence{yn} →y,theny≤ynfor alln. Then F has a coupled ﬁxed point in X.

Proof Note that ifψ ∈, then for allr> ,rψ ∈. Now divide (.) by  and take ϕ(t) =_t,t∈[,∞), then the condition (.) reduces to (.) withψ=_ψandg(x) =x; and hence by Theorem ., we obtain Corollary .. Corollary . Let(X,≤)be a partially ordered set and suppose there exists a metric d on X such that(X,d)is a complete metric space.Let F:X×X→X,g:X→X be two maps with F having the mixed g-monotone property on X such that there exist two elements x,y∈X

with gx≤F(x,y)and gy≥F(y,x).Suppose there exists a real number k∈[, )such

that

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

for all x,y,u,v∈X with x≥u,y≤v.Suppose either (a) Fis continuous,or

(i) if a non-decreasing sequence{xn} →x,thengxn≤gxfor alln; (ii) if a non-increasing sequence{yn} →y,thengy≤gynfor alln.

Suppose that F(X×X)⊆g(X),g is continuous and the pair(F,g)is compatible,then there exist x,y∈X such that gx=F(x,y)and gy=F(y,x).

Proof Takingϕ(t) = _t andψ(t) = ( –k)t_, ≤k< , in Theorem ., we obtain

Corol-lary ..

Remark . (i) Corollary . is an extension of the recent coupled ﬁxed point result of

Berinde (Theorem  in []) to a coupled coincidence point theorem for a pair of compat-ible mappings having the mixedg-monotone property.

(ii) Again, the choice of functionsFandgin Example . shows that Corollary . is more general than Theorem . in [], since the contractive condition (.) is more general than (.). Indeed, the contractive condition (.) does not hold for the choice of functions

Fandg, but (.) holds exactly fork=_ withx= – andy=  and yields (, ) as the coupled coincidence point ofFandg.

Corollary . Let(X,≤)be a partially ordered set and suppose there exists a metric d on X such that(X,d)is a complete metric space.Let F:X×X→X,be a mapping having the mixed monotone property on X such that there exist two elements x,y∈X with x≤

F(x,y)and y≥F(y,x).Suppose there exists a real number k∈[, )such that

dF(x,y),F(u,v)+dF(y,x),F(v,u)≤kd(x,u) +d(y,v) (.)

for all x,y,u,v∈X with x≥u,y≤v.Suppose either (a) Fis continuous,or

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(i) if a non-decreasing sequence{xn} →x,thenxn≤xfor alln; (ii) If a non-increasing sequence{yn} →y,theny≤ynfor alln. Then F has a coupled ﬁxed point in X.

Proof Takinggto be the identity mapping in Corollary ., we obtain Corollary .. Remark . (i) By considering the condition of continuity of F in Corollary ., we obtain Theorem  in [].

(ii) Again, the choice of the functionFin Example . shows that Corollary . is more general than Theorem . (i.e., Theorem . in []) and Theorem . in [], since the tractive condition (.) is more general than (.) and (.). Indeed, the contractive con-ditions (.) and (.) do not hold for the choice of the functionF, but (.) holds exactly fork=_withx= – andy=  and yields (, ) as the coupled ﬁxed point ofF.

Now, in order to prove the existence and uniqueness of the coupled common ﬁxed point for our main results, we need the following lemma.

Lemma . Let F :X×X→X and g:X→X be compatible maps and let an element

(x,y)∈X×X such that gx=F(x,y)and gy=F(y,x)exist, then gF(x,y) =F(gx,gy)and gF(y,x) =F(gy,gx).

Proof Since the pair (F,g) is compatible, it follows that

lim

n→∞d

gF(xn,yn),F

g(xn),g(yn)

= ,

lim

n→∞d

gF(yn, xn),F

g(yn),g(xn)

= ,

whenever{xn}and{yn}are sequences inXsuch thatlimn→∞F(xn,yn) =limn→∞g(xn) =a,

limn→∞F(yn,xn) =limn→∞g(yn) =b for some a,b∈X. Takingxn=x,yn=yand using gx=F(x,y),gy=F(y,x), it follows that

dgF(x,y),F(gx,gy)=  and dgF(y,x),F(gy,gx)= .

Hence,gF(x,y) =F(gx,gy) andgF(y,x) =F(gy,gx). Theorem . In addition to the hypothesis of Theorem ., 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*)).Then F and g have a unique coupled common ﬁxed point;that is, there exists a unique(x,y)∈X×X such that x=g(x) =F(x,y)and y=g(y) =F(y,x).

Proof By Theorem ., the set of coupled coincidences is non-empty. In order to prove the theorem, we shall ﬁrst show that if (x,y) and (x*,y*) are coupled coincidence points, that is, ifgx=F(x,y),gy=F(y,x) andgx*₌_F₍_x*_,_y*_),_gy*₌_F₍_y*_,_x*_{), then}

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By assumption, there is (u,v)∈X× X such that (F(u,v),F(v,u)) is comparable with (F(x,y),F(y,x)) and (F(x*_,_y*_),_F₍_y*_,_x*_{)). Put}_u

=u,v=vand chooseu,v ∈X so that

gu=F(u,v),gv=F(v,u).

Then, similarly as in the proof of Theorem ., we can inductively deﬁne sequences{gun}

and{gvn}such thatgun+=F(un,vn) andgvn+=F(vn,un).

Further, set x=x,y=y,x*=x*,y*=y* and, in the same way, deﬁne the sequences

{gxn},{gyn}and{gx*n},{gy*n}. Then it is easy to show that

gxn+=F(xn,yn), gyn+=F(yn,xn)

and

gx*_n_+=Fx*_n,y*_n, gy*_n_+=Fy*_n,x*_n for alln≥.

Since (F(u,v),F(v,u)) = (gu,gv) and (F(x,y),F(y,x)) = (gx,gy) = (gx,gy) are comparable, thengu≥gxandgv≤gy. It is easy to show that (gun,gvn) and (gx,gy) are comparable,

that is,gun≥gxandgvn≤gyfor alln≥. Thus by (.),

ϕ

d(gun+,gx) +d(gvn+,gy) 

=ϕ

d(F(un,vn),F(x, y)) +d(F(vn,un),F(y,x))



≤ϕ

d(gun,gx) +d(gvn,gy)



–ψ



. (.)

Sinceψis non-negative, we have

ϕ

d(gun+,gx) +d(gvn+,gy) 

≤ϕ



.

By the monotonicity ofϕ, we have

d(gun+,gx) +d(gvn+,gy)

 ≤

 . (.)

Thus, the sequence{dn}deﬁned bydn=d(gun,gx)+_d(gvn,gy), is a monotonically decreasing

se-quence of non-negative real numbers, so there exists somed≥ such thatlimn→∞dn=d.

We shall show thatd= . Suppose, to the contrary, thatd> . Then taking limit as

n→ ∞, in (.) and using the continuity ofϕ, we have

ϕ(d)≤ϕ(d) – lim

dn→d

ψ(dn) <ϕ(d),

a contradiction. Thus,d= ; that is,limn→∞dn= .

Hence, it follows thatgun→gx,gvn→gy.

Similarly, one can show thatgun→gx*,gvn→gy*.

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Sincegx=F(x,y), gy=F(y,x) and the pair (F,g) is compatible, then by Lemma ., it follows that

ggx=gF(x,y) =F(gx,gy) and ggy=gF(y,x) =F(gy,gx). (.)

Denotegx=z,gy=w. Then by (.),

gz=F(z,w) and gw=F(w,z). (.)

Thus, (z,w) is a coupled coincidence point.

Then by (.) withx*₌_z_and_y*₌_w_{, it follows that}_gz₌_gx_and_gw₌_gy_{; that is,}

gz=z, gw=w. (.)

By (.) and (.),

z=gz=F(z,w) and w=gw=F(w,z).

Therefore, (z,w) is the coupled common ﬁxed point ofFandg.

To prove the uniqueness, assume that (p,q) is another coupled common ﬁxed point of

Fandg. Then by (.), we havep=gp=gz=zandq=gq=gw=w. Corollary . In addition to the hypothesis of Corollary ., 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*_))._{Then F and g have a unique coupled common}

ﬁxed point;that is, there exists a unique(x,y)∈X×X such that x=g(x) =F(x,y)and y=g(y) =F(y,x).

Proof Takingϕ(t) = t_ andψ(t) = ( –k)t_, ≤k<  in Theorem ., we obtain

Corol-lary ..

Remark . Indeed, (, ) is the unique coupled common ﬁxed point of the mapsFand

gin Example . in view of Theorem . and Corollary ..

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Ahir College, Rewari, 123401, India.}2_{Department of Mathematics and Computer Sciences,}

Cankaya University, Ankara, Turkey.3_{Department of Mathematics, DCRUST, Murthal, Sonepat, India.} 4_{HAS Department,}

YMCAUST, Faridabad, India.

Acknowledgements

The authors are thankful to the referees for their appreciation and valuable suggestions regarding this work.

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Cite this article as:Jain et al.:Coupled common ﬁxed point results involving a(ϕ,ψ)-contractive condition for mixed