Coupled common fixed point theorems for generalized nonlinear contraction mappings with the mixed monotone property in partially ordered metric spaces

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R E S E A R C H

Open Access

Coupled common ﬁxed point theorems for

generalized nonlinear contraction mappings

with the mixed monotone property in

partially ordered metric spaces

Jong Kyu Kim

1*

_{and Sumit Chandok}

2

*_{Correspondence:}

[email protected] 1_{Department of Mathematics}

Education, Kyungnam University, Changwon, 631-701, Korea Full list of author information is available at the end of the article

Abstract

The purpose of this paper is to establish some coupled coincidence point theorems for generalized nonlinear contraction mappings with the mixedg-monotone property in the framework of metric spaces endowed with partial order. The theorems presented in this paper are generalizations and improvements of the several well-known results in the literature.

MSC: 46N40; 47H10; 54H25; 46T99

Keywords: coupled ﬁxed point; ordered metric spaces; mixed monotone property; coupled coincidence point

1 Introduction and preliminaries

The Banach contraction principle is one of very popular tools in solving the existence in many problems of mathematical analysis. Due to its simplicity and usefulness, there are a lot of generalizations of this principle in the literature. Ran and Reurings [] extended the Banach contraction principle in partially ordered sets with some applications to linear and nonlinear matrix equations. While Nieto and López [] extended the result of Ran and Reurings and applied their main theorems to obtain a unique solution for a first-order or-dinary differential equation with periodic boundary conditions. Bhaskar and Lakshmikan-tham [] introduced the concept of mixed monotone mappings and obtained some cou-pled fixed point results. Also, they applied their results to a first-order differential equation with periodic boundary conditions. Recently, many researchers have obtained fixed point, common fixed point, coupled fixed point and coupled common fixed point results in cone metric spaces, partially ordered metric spaces and others (see [–]).

Deﬁnition  Let (X,d) be a metric space andF:X×X→Xandg:X→X,Fandgare said tocommuteifF(gx,gy) =g(F(x,y)) for allx,y∈X.

Deﬁnition  Let (X,) be a partially ordered set andF:X×X→X. The mappingFis said to benon-decreasingif forx,y∈X,xyimpliesF(x)F(y) andnon-increasingif for

x,y∈X,xyimpliesF(x)F(y).

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Deﬁnition  Let (X,) be a partially ordered set andF:X×X→Xandg:X→X. The mappingF is said to have themixed g-monotone propertyifF(x,y) is monotoneg -non-decreasing inxand monotoneg-non-increasing iny, that is, for anyx,y∈X,

x,x∈X, gxgx ⇒ F(x,y)F(x,y),

and

y,y∈X, gygy ⇒ F(x,y)F(x,y).

Ifg= identity mapping in Deﬁnition , then the mappingF is said to have themixed monotone property.

Deﬁnition  An element (x,y)∈X×Xis called acoupled coincidence pointof the map-pingsF:X×X→Xandg:X→XifF(x,y) =gxandF(y,x) =gy.

Ifgis the identity mapping in Deﬁnition , then (x,y)∈X×Xis called acoupled ﬁxed point.

Geraghty [] introduced an extension of the Banach contraction principle in which the contraction constant was replaced by a function having some speciﬁed properties.

Deﬁnition  Letbe the class of functionsβ:R+_→_{[, ) with}

(i) R+₌_{_R_/_t_{> }_}_;

(ii) β(tn)→impliestn→.

The method applied by Geraghty [] was utilized to obtain further new ﬁxed point result works like [, , ].

The purpose of this paper is to establish some coupled coincidence point results for a pair of mappings with the mixedg-monotone property satisfying a generalized contractive condition by using the ideas of Geraghty [] in partially ordered metric spaces. Also we give some examples to illustrate the main results.

2 Main results

Theorem  Let(X,)be a partially ordered set and suppose that there exists a metric d on X such that(X,d)is a complete metric space.Suppose that F:X×X→X and g:X→X are self-mappings on X such that F has the mixed g-monotone property on X such that there exist two elements x,y∈X with g(x)F(x,y)and g(y)F(y,x). Suppose that there existsθ∈such that

dF(x,y),F(u,v)≤θ

d(gx,gu) +d(gy,gv) 

(.)

for all x,y,u,v∈X with gxgu and gygv.Further suppose that F(X×X)⊆g(X),g is continuous and commutes with F,and also suppose that either

(a) Fis continuous,or

(b) Xhas the following properties:

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(ii) if{g(yn)} ⊂Xis a non-increasing sequence withgyn→gying(X),thengyngy for everyn.

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

Proof Letx,y∈Xbe such thatgxF(x,y) and gyF(y,x). SinceF(X×X)⊆ g(X), we can construct sequences{xn}and{yn}inXsuch that

gxn+=F(xn,yn) and gyn+=F(yn,xn), ∀n≥. (.)

We claim that for alln≥,

gxngxn+, (.)

and

gyngyn+. (.)

We use the mathematical induction. Letn= . SincegxF(x,y) andgyF(y,x),

in view of gx =F(x,y) and gy=F(y,x), we havegxgx andgygy, that is,

(.) and (.) hold for n= . Suppose that (.) and (.) hold for some n> . As F

has the mixed g-monotone property and gxngxn+ and gyngyn+, from (.), we

get

gxn+=F(xn,yn)F(xn+,yn)F(xn+,yn+) =gxn+ (.)

and

gyn+=F(yn,xn)F(yn+,xn)F(yn+,xn+) =gyn+. (.)

Now from (.) and (.), we obtain that gxn+ gxn+ andgyn+gyn+. Thus, by

the mathematical induction, we conclude that (.) and (.) hold for alln≥. There-fore

gxgxgx · · · gxngxn+ · · · (.)

and

gygygy · · · gyngyn+ · · ·. (.)

Assume that there is somer∈Nsuch thatd(gxr,gxr–) +d(gyr,gyr–) = , that is,gxr= gxr–andgyr=gyr–. Thengxr–=F(xr–,yr–) andgyr–=F(yr–,xr–), and hence we get

the result.

For simplicity, lettn+:=d(gxn+,gxn) +d(gyn+,gyn). Now, we assume that

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for alln. Sincegxngxn–andgyngyn–, from (.) and (.), we have tn+=d(gxn+,gxn) +d(gyn+,gyn)

=dF(xn,yn),F(xn–,yn–)

+dF(yn,xn),F(yn–,xn–)

≤θ

d(gxn–,gxn) +d(gyn–,gyn)



+θ



=θ



=θ



(tn)

≤tn, (.)

which implies thattn+≤tn. It follows that{tn}is a monotone decreasing sequence of

non-negative real numbers. Therefore, there is somet≥ such thatlimn→∞tn=t.

Now, we show thatt= . Assume to the contrary thatt> , then from (.) we have

tn+ tn ≤θ



< ,

which yields that limn→∞θ(tn_) = . This implies that d(gxn–,gxn)→  and d(gyn–, gyn)→. Thereforet= , that is,

lim

n→∞tn=nlim→∞

d(gxn+,gxn) +d(gyn+,gyn)

= . (.)

Next, we prove that{gxn}and{gyn}are Cauchy sequences. On the contrary, assume that at least one of{gxn}or{gyn}is not a Cauchy sequence. Then there exists an>  for which we can ﬁnd subsequences{gxm(k)}and{gxn(k)}of{gxn}, and{gym(k)}and{gyn(k)}of{gyn}

withn(k) >m(k) >ksuch that for everyk,

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

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)) <. (.)

Using (.) and (.), we have

≤rk:=d(gxn(k),gxm(k)) +d(gyn(k),gym(k))

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

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

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Lettingk→ ∞and using (.), we have

limrk=limd(gxm(k),gxn(k)) +d(gym(k),gyn(k))

=. (.)

Also, by the triangle inequality, we have

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

≤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))

=tn(k)+tm(k)+d(gxn(k)+,gxm(k)+) +d(gyn(k)+,gxm(k)+).

Sincen(k) >m(k),gxn(k)gxm(k)andgyn(k)gym(k), from (.) and (.), we have d(gxn(k)+,gxm(k)+) +d(gyn(k)+,gym(k)+)

=dF(xn(k),yn(k)),F(xm(k),ym(k))

+dF(yn(k),xn(k)),F(ym(k),xm(k))

≤θ

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



×d(gxn(k),gxm(k)) +d(gyn(k),gym(k))

=θ



rk.

Therefore, we have

rk≤tn(k)+tm(k)+θ



rk.

This implies that

rk–tn(k)–tm(k)

rk ≤θ



< .

Takingk→ ∞, we get

θ



= .

Sinceθ∈, we get

lim

n→∞d(gxn(k),gxm(k)) =nlim→∞d(gyn(k),gym(k)) = ,

which is a contradiction. This implies that{gxn} and{gyn} are Cauchy sequences inX. SinceXis a complete metric space, there is (x,y)∈X×Xsuch thatgxn→xandgyn→y. Sincegis continuous,g(gxn)→gxandg(gyn)→gy.

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gF(yn,xn) =g(gyn+)→gy. By the uniqueness of the limit, we getgx=F(x,y) andgy= F(y,x).

Second, suppose that (b) holds. Since {gxn} is a non-decreasing sequence such that

gxn→x and {gyn} is a non-increasing sequence such thatgyn→y, and g is a

non-increasing function, we getg(gxn)gxandg(gyn)gyhold for alln∈N. Hence, by (.),

we have

dg(gxn+),F(x,y)

+dg(gyn+),F(y,x)

=dF(gxn,gyn),F(x,y)+dF(gyn,gxn)F(y,x)

≤θ

d(g(gxn),gx) +d(g(gyn),gy)



dg(gxn),gx+dg(gyn),gy.

Takingn→ ∞, we getd(gx+F(x,y)) +d(gy,F(y,x)) = , and hencegx=F(x,y) andgy=

F(y,x). ThusFandghave a coupled coincidence point.

Ifθ(t) =k, wherek∈[, ), then we have the following result.

Corollary  Let(X,)be a partially ordered set and suppose that there exists a metric d on X such that(X,d)is a complete metric space.Suppose that F:X×X→X and g:X→X are self-mappings on X such that F has the mixed g-monotone property on X such that there exist two elements x,y∈X with g(x)F(x,y)and g(y)F(y,x).Suppose that there exists k∈[, )such that

dF(x,y),F(u,v)≤k

(.)

satisﬁes for all x,y,u,v∈X with gxgu and gygv.Further suppose that F(X×X)⊆g(X),

g is continuous non-decreasing and commutes with F and also suppose that either

(i) if{g(xn)} ⊂Xis a non-decreasing sequence withgxn→gxing(X),thengxngx for everyn;

Ifgis an identity mapping, we have the following result of Bhaskar and Lakshmikantham [].

Corollary  Let(X,)be a partially ordered set and suppose that there exists a metric d on X such that(X,d)is a metric space.Suppose that F:X×X→X is a mapping on X and has 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 that there exists k∈[, )such that

dF(x,y),F(u,v)≤k 

d(x,u) +d(y,v) (.)

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(i) if{xn} ⊂Xis a non-decreasing sequence withxn→xinX,thenxnxfor each n≥;

(ii) if{yn} ⊂Xis a non-increasing sequence withyn→yinX,thenynyfor each n≥.

Then there exist two elements x,y∈X such that F(x,y) =x and y=F(y,x),that is,F has a coupled ﬁxed point(x,y)∈X×X.

Example  LetX= [, ]. Then (X,≤) is a partially ordered set with the natural ordering of real numbers. Letd(x,y) =|x–y|for allx,y∈X. Deﬁne a mappingg:X→Xbyg(x) =x

and a mappingF:X×X→Xby

F(x,y) = ⎧ ⎨ ⎩

x–y

 , x≥y;

, x<y.

Then it is easy to prove that (X,d) is a complete metric space,g(X) is complete,F:X×X→ X⊆g(X) =X,Xsatisﬁes conditions () and () of Theorem  andFhas theg-monotone property. Letθ: (,∞)→[, ) be deﬁned by

θ(t) = ⎧ ⎨ ⎩

 –_t, t≤; α< , t> .

Now, we verify the inequality (.) of Theorem  for all x,y,u,v∈Xwith gxguand

gygv.

Now, we consider the following cases.

Case . (x,y) = (, ), (u,v) = (, ) or (x,y) = (, ), (u,v) = (, ), we have

dF(x,y),F(u,v)= .

Hence inequality (.) holds.

Case . (x,y) = (, ), (u,v) = (, ), we have

dF(x,y),F(u,v)=dF(, ),F(, )= 

and

θ

=θ

d(, ) +d(, ) 

=θ

 

=

 –  

 .

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Case . (x,y) = (, ), (u,v) = (, ), we have

dF(x,y),F(u,v)=dF(, ),F(, )=  

and

θ

=θ

d(, ) +d(, ) 

=θ()()

=

 –  

.

Case . (x,y) = (, ), (u,v) = (, ), we have

dF(x,y),F(u,v)=dF(, ),F(, )=  

and

θ

=θ

d(, ) +d(, ) 

=θ

 

=

 –  

 .

Thus, in all the cases, inequality (.) of Theorem  is satisﬁed. Hence, by Theorem , (, ) is a coupled coincidence point ofFandg.

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

dF(x,y),F(u,v)≤θM(x,y,u,v)M(x,y,u,v), (.)

where

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for all x,y,u,v∈X with gxgu and gygv.Further suppose that F(X×X)⊆g(X),g is continuous non-decreasing and commutes with F and also suppose that either

Proof Following the proof of Theorem , we have an increasing sequence{xn}and a de-creasing sequence{yn}inX. Now, we assume that

tn=d(gxn,gxn–) +d(gyn,gyn–)= 

for alln.

Sincegxngxn–andgyngyn–, from (.) and (.), we have tn+=d(gxn+,gxn) +d(gyn+,gyn)

=dF(xn,yn),F(xn–,yn–)

+dF(yn,xn),F(yn–,xn–)

≤θ

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



×



+θ

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



×

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



=θ



×



=θ

tn+tn+



tn+tn+



≤tn. (.)

It follows that {tn} is a monotone decreasing sequence of non-negative real numbers. Therefore, there is somet≥ such thatlimn→∞tn=t.

Next, we show thatt= . Assume to the contrary thatt> , then from (.) we have

tn+ tn+tn+



≤θ

tn+tn+



< ,

which yields thatlimn→∞θ(tn+tn+) = . This implies that d(gxn–,gxn)→ andd(gyn–, gyn)→. Thereforet= , that is,

lim

n→∞tn=nlim→∞

d(gxn+,gxn) +d(gyn+,gyn)

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Now, we prove that{gxn}and{gyn}are Cauchy sequences. On the contrary, assume that at least one of{gxn}or{gyn}is not a Cauchy sequence. Then there exists an>  for which we can ﬁnd subsequences{gxm(k)}and{gxn(k)}of{gxn}and{gym(k)}and{gyn(k)}of{gyn}with n(k) >m(k) >ksuch that for everyk,

rk=d(gxm(k),gxn(k)) +d(gym(k),gyn(k))≥. (.)

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)) <. (.)

Using (.) and (.), we have

≤rk:=d(gxn(k),gxm(k)) +d(gyn(k),gym(k))

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

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

<+tn(k).

Lettingk→ ∞and using (.), we have

limrk=limd(gxm(k),gxn(k)) +d(gym(k),gyn(k))

=. (.)

Also, by the triangle inequality, we have

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

≤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))

=tn(k)+tm(k)+d(gxn(k)+,gxm(k)+) +d(gyn(k)+,gxm(k)+).

Sincen(k) >m(k),gxn(k)gxm(k)andgyn(k)gym(k), from (.) and (.), we have d(gxn(k)+,gxm(k)+)

=dF(xn(k),yn(k)),F(xm(k),ym(k))

≤θ

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



×



,

and similarly,

d(gyn(k)+,gxm(k)+)

=dF(yn(k),xn(k)),F(ym(k),xm(k))

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≤θ d(gxn(k),gxn(k)+) +d(gyn(k),gyn(k)+) +d(gxm(k),gxm(k)+) +d(gym(k),gym(k)+) 

×



.

Therefore, we have

rk≤tn(k)+tm(k) + θ

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

×

_d₍_gx

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

.

Takingk→ ∞, we havetn(k),tm(k)→, and

 →.

Hence, we getrk== , which is a contradiction. This implies that{gxn}and{gyn}are

Cauchy sequences ing(X).

SinceXis a complete metric space, there is (x,y)∈X×Xsuch thatgxn→xandgyn→y.

Sincegis continuous,g(gxn)→gxandg(gyn)→gy.

First, suppose thatFis continuous. ThenF(gxn,gyn)→F(x,y) andF(gyn,gxn)→F(y,x). AsFcommutes withg, we have

F(gxn,gyn) =gF(xn,yn) =g(gxn+)→gx

and

F(gyn,gxn) =gF(yn,xn) =g(gyn+)→gy.

By the uniqueness of the limit, we getgx=F(x,y) andgy=F(y,x).

Second, suppose that (b) holds. Since {gxn} is a non-decreasing sequence such that gxn→x and {gyn} is a non-increasing sequence such thatgyn→y, and g is a non-decreasing function, we get thatg(gxn)gx andg(gyn)gyhold for alln∈N. Hence, by (.), we have

dg(gxn+),F(x,y)

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

≤θ

d(g(gxn),F(gxn,gyn)) +d(g(gyn),F(gyn,gxn)) +d(gx,F(x,y)) +d(gy,F(y,x))



×

d(g(gxn),F(gxn,gyn)) +d(g(gyn),F(gyn,gxn)) +d(gx,F(x,y)) +d(gy,F(y,x)) 

=θ

d(g(gxn),g(gxn+)) +d(g(gyn),g(gyn+)) +d(gx,F(x,y)) +d(gy,F(y,x))



×

d(g(gxn),g(gxn+)) +d(g(gyn),g(gyn+)) +d(gx,F(x,y)) +d(gy,F(y,x))



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and

dg(gyn+),F(y,x)

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

≤θ

d(g(gyn),F(gyn,gxn)) +d(g(gxn),F(gxn,gyn)) +d(gy,F(y,x)) +d(gx,F(x,y)) 

×

d(g(gyn),F(gyn,gxn)) +d(g(gxn),F(gxn,gyn)) +d(gy,F(y,x)) +d(gx,F(x,y))



=θ

d(g(gyn),g(gyn+)) +d(g(gxn),g(gxn+)) +d(gy,F(y,x)) +d(gx,F(x,y))



×

d(g(gyn),g(gyn+)) +d(g(gxn),g(gxn+)) +d(gy,F(y,x)) +d(gx,F(x,y))



.

Takingn→ ∞, we getd(gx+F(x,y)) +d(gy,F(y,x)) = , and hencegx=F(x,y) andgy=

F(y,x). ThusFandghave a coupled coincidence point.

Example  LetX= [, ]. Then (X,≤) is a partially ordered set with the natural ordering of real numbers. Letd(x,y) =|x–y|for allx,y∈X. Deﬁne a mappingg:X→Xbyg(x) =x

and a mappingF:X×X→Xby

F(x,y) = ⎧ ⎨ ⎩

x–y

, x≥y;

, x<y.

Then it is easy to prove that (X,d) is a complete metric space,g(X) is complete,F:X×X→ X⊆g(X) =X,Xsatisﬁes conditions () and () of Theorem  andFhas theg-monotone property. Letθ: (,∞)→[, ) be deﬁned as

θ(t) = ⎧ ⎨ ⎩

 –t, t≤; α< , t> .

Now, we verify inequality (.) of Theorem  for allx,y,u,v∈Xwithgxguandgygv. Now, we consider the following cases.

Case . (x,y) = (, ), (u,v) = (, ) or (x,y) = (, ), (u,v) = (, ), we have

dF(x,y),F(u,v)= .

Hence, inequality (.) holds.

Case . (x,y) = (, ), (u,v) = (, ), we have

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and

θ

d(gx,F(x,y)) +d(gy,F(y,x)) +d(gu,F(u,v)) +d(gv,F(v,u)) 

×

=θ

d(,F(, )) +d(,F(, )) +  +  

d(,F(, )) +d(,F(, )) +  +   =θ     =  –    .

Case . (x,y) = (, ), (u,v) = (, ), we have

dF(x,y),F(u,v)=dF(, ),F(, )=  

and

θ

×

=θ

d(,F(, )) +d(,F(, )) +d(,F(, )) +d(,F(, )) 

×

d(,F(, )) +d(,F(, )) +d(,F(, )) +d(,F(, ))  =θ     =  –   .

Case . (x,y) = (, ), (u,v) = (, ), we have

dF(x,y),F(u,v)=dF(, ),F(, )=  ;

and

θ

×

=θ

d(,F(, )) +d(,F(, )) +d(,F(, )) +d(,F(, )) 

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× d(,F(, )) +d(,F(, )) +d(,F(, )) +d(,F(, ))



=θ

 

=

 –  

 .

Thus, in all the cases, inequality (.) of Theorem  is satisﬁed. Hence, by Theorem , (, ) is a coupled coincidence point ofFandg.

Next, we prove the existence of a coupled coincidence point theorem, where we do not require thatFandgare commuting.

The following lemma proved by Haghiet al.[] is useful for our results.

Lemma [] Let X be a nonempty set,and let g:X→X be a mapping.Then there exists a subset E⊆X such that g(E) =g(X)and g:E→X is one-to-one.

Theorem  Let(X,)be a partially ordered set and suppose that there exists a metric d on X such that(X,d)is a metric space.Suppose that F:X×X→X and g:X→X are self-mappings on X such that F has the mixed g-monotone property on X such that there exist two elements x,y∈X with g(x)F(x,y)and g(y)F(y,x).Suppose that there existsθ∈such that

dF(x,y),F(u,v)≤θ

(.)

for all x,y,u,v∈X with gxgu and gygv.Further suppose that F(X×X)⊆g(X)and g(X)is a complete subspace of X.Also assume that either

Proof Using Lemma , there existsE⊆Xsuch thatg(E) =g(X) andg:E→Xis one-to-one. We deﬁne a mappingA:g(E)×g(E)→Xby

A(gx,gy) =F(x,y) for everygx,gy∈g(E). (.)

Asgis one-to-one ong(E), soAis well deﬁned.

SinceFhas the mixedg-monotone property, for allx,y∈X, we have

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and

y,y∈X, gygy ⇔ F(x,y)F(x,y). (.)

Thus, it follows from (.), (.) and (.) that for allgx,gy∈g(E),

gx,gx∈g(E), gxgx ⇔ A(gx,gy)A(gx,gy) (.)

and

gy,gy∈g(E), gygy ⇔ A(gx,gy)A(gx,gy), (.)

which implies thatAhas the mixed monotone property.

Suppose that the assumption (a) holds. SinceFis continuous,Ais also continuous. Using Theorem . of [] with the mappingA, it follows thatAhas a coupled ﬁxed point (u,v)∈

g(X)×g(X).

Suppose that the assumption (b) holds. We can conclude similarly to the proof of The-orem . of [] that the mappingAhas a coupled ﬁxed point (u,v)∈g(X)×g(X).

Finally, we prove thatF andg have a coupled coincidence point inX. Since (u,v) is a coupled ﬁxed point ofA, we get

u=A(u,v), v=A(v,u). (.)

Since (u,v)∈g(X)×g(X), there exists a point (uˆ,vˆ)∈X×Xsuch that

u=guˆ, v=gvˆ. (.)

Thus, it follows from (.) and (.) that

guˆ=A(guˆ,gˆv), gˆv=A(gˆv,guˆ). (.) Also, from (.) and (.), we get

guˆ=F(uˆ,ˆv), gˆv=F(ˆv,uˆ). (.) Therefore, (uˆ,vˆ) is a coupled coincidence point ofFandg. This completes the proof. Theorem  Let(X,)be a partially ordered set and suppose that there exists a metric d on X such that(X,d)is a metric space.Suppose that F:X×X→X and g:X→X are self-mappings on X such that F has the mixed g-monotone property on X such that there exist two elements x,y∈X with g(x)F(x,y)and g(y)F(y,x).Suppose that there existsθ∈such that

dF(x,y),F(u,v)≤θM(x,y,u,v)M(x,y,u,v), (.)

where

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for all x,y,u,v∈X with gxgu and gygv.Further suppose that F(X×X)⊆g(X)and g(X)is a complete subspace of X.Also assume that either

Proof Following similar arguments to those in Theorem  and using Theorem . of [],

we get the result.

Remark  Although Theorem  and Theorem  are an essential tool in the partially or-dered metric spaces to claim the existence of coupled coincidence points of two mappings, some mappings do not have the commutative property. For example, see the following.

Example  LetX= [, ]. Then (X,) is a partially ordered set with the natural ordering of real numbers. Letd(x,y) =x–yfor allx,y∈X. Deﬁne mappingsF:X×X→Xand

g:X→XbyF(x,y) =  for all (x,y)∈X×Xandg(x) =x– for eachx∈X. Sinceg(F(x,y)) =

g() = =  =F(gx,gy) for allx,y∈X, the mappingsFandgdo not satisfy the commutative condition. Hence, the above two theorems cannot be applied to this example. But, by a simple calculation, we see thatF(X×X)⊆g(X),gandFare continuous andFhas the mixedg-monotone property. Moreover, there existx=  andy=  withg() =  = F(, ) andg() =  =F(, ).

Therefore, it is very interesting to use Theorems  and  as another auxiliary tool to claim the existence of a coupled coincidence point.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The main idea of this paper was proposed by JKK. JKK and SC prepared the manuscript initially and performed all the steps of the proof in this research. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics Education, Kyungnam University, Changwon, 631-701, Korea.}2_{Department of}

Mathematics, Khalsa College of Engineering & Technology (Punjab Technical University), Ranjit Avenue, Amritsar, Punjab 143001, India.

Acknowledgements

This work was supported by the Kyungnam University Research Fund 2013. The authors are thankful to the referees for valuable suggestions.

Received: 1 April 2013 Accepted: 25 October 2013 Published:22 Nov 2013

References

1. Ran, ACM, Reurings, MCB: A ﬁxed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc.132(5), 1435-1443 (2004)

2. Nieto, JJ, López, RR: Contractive mapping theorems in partially ordered sets and applications to ordinary diﬀerential equations. Order22, 223-239 (2005)

(17)

4. Abbas, M, Sintunavarat, W, Kumam, P: Coupled ﬁxed point of generalized contractive mappings on partially ordered G-metric spaces. Fixed Point Theory Appl.2012, 31 (2012)

5. Agarwal, RP, El-Gebeily, MA, O’Regan, D: Generalized contractions in partially ordered metric spaces. Appl. Anal.87, 1-8 (2008)

6. 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. TMA72(5), 2238-2242 (2010)

7. Caballero, J, Harjani, J, Sadarangani, K: Contractive-like mapping principles in ordered metric spaces and application to ordinary diﬀerential equations. Fixed Point Theory Appl.2010, Article ID 916064 (2010). doi:10.1155/2010/916064 8. Chandok, S: Some common ﬁxed point theorems for generalizedf-weakly contractive mappings. J. Appl. Math.

Inform.29, 257-265 (2011)

9. Chandok, S: Some common ﬁxed point theorems for generalized nonlinear contractive mappings. Comput. Math. Appl.62, 3692-3699 (2011)

10. Chandok, S: Common ﬁxed points, invariant approximation and generalized weak contractions. Int. J. Math. Math. Sci.

2012, 102980 (2012)

11. Chandok, S, Khan, MS, Rao, KPR: Some coupled common ﬁxed point theorems for a pair of mappings satisfying a contractive condition of rational type without monotonicity. Int. J. Math. Anal.7(9), 433-440 (2013)

12. Chandok, S, Kim, JK: Fixed point theorem in ordered metric spaces for generalized contractions mappings satisfying rational type expressions. Nonlinear Funct. Anal. Appl.17, 301-306 (2012)

13. Chandok, S, Narang, TD, Taoudi, MA: Some common ﬁxed point results in partially ordered metric spaces for generalized rational type contraction mappings. Vietnam J. Math. (2013). doi:10.1007/s10013-013-0024-4 14. Chandok, S, Sintunavarat, W, Kumam, P: Some coupled common ﬁxed points for a pair of mappings in partially

orderedG-metric spaces. Math. Sci.7, 24 (2013). doi:10.1186/2251-7456-7-24

15. Choudhury, BS, Kundu, A: On coupled generalised Banach and Kannan type contractions. J. Nonlinear Sci. Appl.5, 259-270 (2012)

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

17. Haghi, RH, Rezapour, S, Shahzad, N: Some ﬁxed point generalizations are not real generalizations. Nonlinear Anal.74, 1799-1803 (2011). doi:10.1016/j.na.2010.10.052

18. Janos, L: On mappings contractive in the sense of Kannan. Proc. Am. Math. Soc.61(1), 171-175 (1976)

19. Karapinar, E, Kumam, P, Sintunavarat, W: Coupled ﬁxed point theorems in cone metric spaces with ac-distance and applications. Fixed Point Theory Appl.2012, 194 (2012)

20. Lakshmikantham, V, Ciric, L: Coupled ﬁxed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal.70, 4341-4349 (2009)

21. Sintunavarat, W, Cho, YJ, Kumam, P: Coupled ﬁxed point theorems for weak contraction mapping underF-invariant set. Abstr. Appl. Anal.2012, Article ID 324874 (2012)

22. Sintunavarat, W, Kumam, P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl.2012, 93 (2012)

23. Sintunavarat, W, Kumam, P: Coupled coincidence and coupled common ﬁxed point theorems in partially ordered metric spaces. Thai J. Math.10(3), 551-563 (2012)

24. Sintunavarat, W, Cho, YJ, Kumam, P: Coupled ﬁxed-point theorems for contraction mapping induced by cone ball-metric in partially ordered spaces. Fixed Point Theory Appl.2012, 128 (2012)

25. Sintunavarat, W, Petrusel, A, Kumam, P: Common coupled ﬁxed point theorems forw∗-compatible mappings without mixed monotone property. Rend. Circ. Mat. Palermo61, 361-383 (2012)

26. Sintunavarat, W, Kumam, P, Cho, YJ: Coupled ﬁxed point theorems for nonlinear contractions without mixed monotone property. Fixed Point Theory Appl.2012, 170 (2012)

27. Sintunavarat, W, Radenovic, S, Golubovic, Z, Kumam, P: Coupled ﬁxed point theorems forF-invariant set. Appl. Math. Inform. Sci.7(1), 247-255 (2013)

10.1186/1687-1812-2013-307