Coupled fixed point theorems for generalized contractive mappings in partially ordered G-metric spaces

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R E S E A R C H

Open Access

Coupled ﬁxed point theorems for generalized

contractive mappings in partially ordered

G

-metric spaces

Rabian Wangkeeree

1,2*

_{and Thanatporn Bantaojai}

1

*_{Correspondence:} rabianw@nu.ac.th

1_{Department of Mathematics,} Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

2_{Centre of Excellence in} Mathematics, CHE, Si Ayutthaya Road, Bangkok, 10400, Thailand

Abstract

In this paper, we establish some coupled coincidence and coupled common ﬁxed point theorems for nonlinear contractive mappings having the mixed monotone property in partially orderedG-metric spaces. The results on ﬁxed point theorems are generalizations of the recent results of Choudhury and Maity (Math. Comput. Model. 54:73-79, 2011) and Luong and Thuan (Math. Comput. Model. 55:1601-1609, 2012).

Keywords: G-metric space; ordered set; coupled coincidence point; coupled ﬁxed point; mixedg-monotone property

1 Introduction and preliminaries

One of the simplest and the most useful results in the fixed point theory is the Banach-Caccioppoli contraction [] mapping principle, a power tool in analysis. This principle has been generalized in different directions in different spaces by mathematicians over the years (see [–] and references mentioned therein). On the other hand, fixed point theory has received much attention in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [] and they presented applica-tions of their results to matrix equaapplica-tions. Subsequently, Nieto and Rodríguez-López [] extended the results in [] for non-decreasing mappings and obtained a unique solution for a first-order ordinary differential equation with periodic boundary conditions (see also [–]).

In recent times, fixed point theory has developed rapidly in partially ordered metric spaces, that is, metric spaces endowed with a partial ordering. Some of these works are noted in [, , , ]. Bhaskar and Lakshmikantham [] introduced the concept of a coupled fixed point and the mixed monotone property. Furthermore, they proved some coupled fixed point theorems for mappings which satisfy the mixed monotone property and gave some applications in the existence and uniqueness of a solution for a periodic boundary value problem. After the publication of this work, several coupled fixed point and coincidence point results have appeared in the recent literature. Works noted in [– ] are some examples of these works.

Mustafa and Sims [, ] introduced a new structure of generalized metric spaces, which are calledG-metric spaces, as a generalization of metric spaces to develop and in-troduce a new ﬁxed point theory for various mappings in this new structure. Later, several ﬁxed point theorems inG-metric spaces were obtained by [–].

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To ﬁx the context in which we are placing our results, recall the following notions. Throughout this article, (X,) denotes a partially ordered set with the partial order. Byx≺y, we meanxybutx=y. A mappingg:X→Xis said to be non-decreasing (non-increasing) if for allx,y∈X,xyimpliesg(x)g(y) (g(y)g(x), respectively).

The concept of a mixed monotone property has been introduced by Bhaskar and Lak-shmikantham [].

Deﬁnition .[] Let (X,) be a partial ordered set. A mappingF:X×X→Xis said to be have themixed monotone propertyifF(x,y) is monotone non-decreasing inxand is monotone non-increasing iny, that is, for anyx,y∈X,

x,x∈X, xx =⇒ F(x,y)F(x,y)

and

y,y∈X, yy =⇒ F(x,y)F(x,y).

The following concepts were introduced in [].

Deﬁnition .[] Let (X,) be a partial ordered set andF:X×X→Xandg:X→Xbe two mappings. We say thatFhas themixed g-monotone propertyifF(x,y) isg-monotone non-decreasing inxand it isg-monotone non-increasing iny, that is, for anyx,y∈X

x,x∈X, g(x)g(x) =⇒ F(x,y)F(x,y) (.)

and

y,y∈X, g(y)g(y) =⇒ F(x,y)F(x,y). (.)

Deﬁnition .[, ] LetF:X×X→Xandg:X→Xbe mappings. An element (x,y)∈

X×Xis said to be:

(i) acoupled ﬁxed pointof a mappingFif

x=F(x,y) and y=F(y,x);

(ii) acoupled coincidence pointof mappingFandgif

g(x) =F(x,y) and g(y) =F(y,x);

(iii) acoupled common ﬁxed pointof mappingsFandgif

x=g(x) =F(x,y) and y=g(y) =F(y,x).

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Deﬁnition .(G-metric space []) LetXbe a non-empty set. LetG:X×X×X→R+ be a function satisfying the following properties:

(G) G(x,y,z) = ifx=y=z;

(G) G(x,x,y) > for allx,y∈Xwithx=y;

(G) G(x,x,y)≤G(x,y,z)for allx,y,z∈Xwithz=y;

(G) G(x,y,z) =G(x,z,y) =G(y,z,x) =· · · (symmetry in all three variables); (G) G(x,y,z)≤G(x,a,a) +G(a,y,z)for allx,y,z,a∈X(rectangle inequality).

Then the functionGis called aG-metric onXand the pair (X,G) is called aG-metric space.

Deﬁnition .[] LetXbe aG-metric space, and let{xn}be a sequence of points ofX, a pointx∈Xis said to be the limit of a sequence{xn}ifG(x,xn,xm)→ asn,m→ ∞and sequence{xn}is said to beG-convergent tox.

From this deﬁnition, we obtain that ifxn→xin aG-metric spaceX, then for anyε> , there exists a positive integerNsuch thatG(x,xn,xm) <εfor alln,m≥N.

It has been shown in [] that theG-metric induces a Hausdorﬀ topology and the con-vergence described in the above deﬁnition is relative to this topology. So, a sequence can converge, at the most, to one point.

Deﬁnition .[] LetXbe aG-metric space, a sequence{xn}is calledG-Cauchy if for everyε> , there is a positive integerNsuch thatG(xn,xm,xl) <εfor alln,m,l≥N, that is, ifG(xn,xm,xl)→, asn,m,l→ ∞.

We next state the following lemmas.

Lemma .[] If X is a G-metric space,then the following are equivalent:

() {xn}isG-convergent tox.

() G(xn,xn,x)→asn→ ∞.

() G(xn,x,x)→asn→ ∞.

() G(xm,xn,x)→asn,m→ ∞.

Lemma .[] If X is a G-metric space,then the following are equivalent:

() the sequence{xn}isG-Cauchy;

() for everyε> ,there exists a positive integerNsuch thatG(xn,xm,xm) <ε,for all

n,m≥N.

Lemma .[] If X is a G-metric space,then G(x,y,y)≤G(y,x,x)for all x,y∈X.

Lemma . If X is a G-metric space,then G(x,x,y)≤G(x,x,z) +G(z,z,y)for all x,y,z∈X.

Deﬁnition .[] Let (X,G), (X,G) be two generalized metric spaces. A mappingf :

X→XisG-continuous at a pointx∈Xif and only if it isGsequentially continuous atx, that is, whenever{xn}isG-convergent tox,{f(xn)}isG-convergent tof(x).

Deﬁnition .[] AG-metric spaceXis called a symmetricG-metric space if

G(x,y,y) =G(y,x,x)

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Deﬁnition .[] AG-metric spaceXis said to beG-complete (or a completeG-metric space) if everyG-Cauchy sequence inXis convergent inX.

Deﬁnition . LetXbe aG-metric space. A mappingF:X×X→Xis said to be con-tinuous if for any twoG-convergent sequences{xn}and{yn}converging toxandy, respec-tively,{F(xn,yn)}isG-convergent toF(x,y).

Deﬁnition . LetXbe a non-empty set andF:X×X→Xandg:X→Xtwo map-pings. We sayFandgare commutative (or thatFandgcommute) if

gF(x,y)=Fg(x),g(y), ∀x,y∈X.

Recently, Choudhury and Maity [] studied necessary conditions for the existence of a coupled ﬁxed point in partially orderedG-metric spaces. They obtained the following interesting result.

Theorem .[] Let(X,)be a partially ordered set such that X is a complete G-metric space and F:X×X→X be a mapping having the mixed monotone property on X.Suppose there exists k∈[, )such that

GF(x,y),F(u,v),F(w,z)≤k 

G(x,u,w) +G(y,v,z) (.)

for all x,y,z,u,v,w∈X for which xuw and yvz,where either u=w or v=z.If there exists x,y∈X such that

xF(x,y) and yF(y,x)

and either

(a) Fis continuous or

(b) Xhas the following property:

(i) if a non-decreasing sequence{xn}is such thatxn→x,thenxnxfor alln,

(ii) if a non-decreasing sequence{yn}is such thatyn→y,thenyynfor alln,

then F has a coupled ﬁxed point.

Letdenote the class of all functionsϕ: [,∞)×[,∞)→[,∞) satisfying the fol-lowing condition:

lim

t→r,t→rϕ(t,t) >  (.)

for all (r,r)∈[,∞)×[,∞) withr+r> .

Remark . If the functionϕ: [,∞)×[,∞)→[,∞) satisﬁes (.), then, for any

t,t∈[,∞) with eithert=  ort= ,ϕ(t,t) > . Indeed, suppose thatt= , we have

t+t> . Takingtn=tandtn=tfor alln∈N, we have, by (.), that

ϕ(t,t) = lim

tn→t,tn→ t

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Example . The following are some examples ofϕ, for all (t,t)∈[,∞)×[,∞),

() ϕ(t,t) =kmax{t,t}fork> ;

() ϕ(t,t) =atp+bt

q

fora,b,p,q> ;

() ϕ(t,t) =–_k(t+t)for somek∈[, ).

Using basically these concepts, Luong and Thuan [] proved the following coupled ﬁxed point theorem for nonlinear contractive mappings having the mixed monotone property in partially orderedG-metric spaces.

Theorem .[, Theorem .] Let(X,≤)be a partially ordered set and suppose that there exists a G-metric G on X such that(X,G)is a complete G-metric space.Let F:X× X→X be a mapping having the mixed monotone property on X.Suppose that there exists

ϕ∈such that

GF(x,y),F(u,v),F(w,z)≤G(x,u,w) +G(y,v,z)

 –ϕ

G(x,u,w),G(y,v,z) (.)

for all xuw and yvz.Suppose that either (a) Fis continuous or

(ii) if a non-decreasing sequence{yn}is such thatyn→y,thenyynfor alln.

If there exist x,y∈X such that xF(x,y)and yF(y,x),then F has a coupled

ﬁxed point in X.

Starting from the results in Choudhury and Maity [] and Luong and Thuan [], our main aim in this paper is to obtain more general coincidence point theorems and coupled common ﬁxed point theorems for mixed monotone operatorsF:X×X→Xsatisfying a contractive condition which is signiﬁcantly more general that the corresponding condi-tions (.) and (.) in [] and [], respectively, thus extending many other related results in literature. We also provide an illustrative example in support of our results.

2 Coupled coincidence points

The ﬁrst main result in this paper is the following coincidence point theorem which gen-eralizes [, Theorem .] and [, Theorem .].

Theorem . Let(X,)be a partially ordered set and G be a G-metric on X such that

(X,G)is a complete G-metric space.Let g:X→X be a mapping and F:X×X→X be a mapping having the mixed g-monotone property on X.Suppose that there existsϕ∈

such that

MG_F(x,u,w,y,v,z)≤Gg(x),g(u),g(w)+Gg(y),g(v),g(z)

– ϕGg(x),g(u),g(w),Gg(y),g(v),g(z) (.)

for all x,y,z,u,v,w∈X for which g(x)g(u)g(w)and g(y)g(v)g(z)where

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If there exists x,y∈X such that

g(x)F(x,y) and g(y)F(y,x),

and suppose F:X×X⊆g(X),g is continuous and commutes with F,and also suppose either

then F and g have a coupled coincidence point,that is,there exists(x,y)∈X×X such that g(x) =F(x,y)and g(y) =F(y,x).

Proof Letx,y∈X such thatg(x)F(x,y) andF(y,x)g(y). SinceF(X×X)⊆

g(X), we can choosex,y∈Xsuch thatg(x) =F(x,y) andg(y) =F(y,x). Again since

F(X×X)⊆g(X), we can choosex,y∈Xsuch thatg(x) =F(x,y) andg(y) =F(y,x). Continuing this process, we can construct sequences{xn}and{yn}inXsuch that

g(xn+) =F(xn,yn) and g(yn+) =F(yn,xn), for alln≥. (.)

Next, we show that

g(xn)g(xn+) and g(yn)g(yn+) for alln≥. (.)

Sinceg(x)F(x,y) =g(x) andg(y)F(y,x) =g(y), therefore, (.) holds forn= . Next, suppose that (.) holds for some ﬁxedn≥, that is,

g(xn)g(xn+) and g(yn)g(yn+). (.)

SinceFhas the mixedg-monotone property, from (.) and (.), we have

F(xn,y)F(xn+,y) and F(yn+,x)F(yn,x) (.)

for allx,y∈X, and from (.) and (.), we have

F(y,xn)F(y,xn+) and F(x,yn+)F(x,yn), (.)

for allx,y∈X. If we takey=ynandx=xnin (.), then we obtain

g(xn+) =F(xn,yn)F(xn+,yn) and F(yn+,xn)F(yn,xn) =g(yn+). (.)

If we takey=yn+andx=xn+in (.), then

F(yn+,xn)F(yn+,xn+) =g(yn+) and g(xn+) =F(xn+,yn+)F(xn+,yn). (.)

Now, from (.) and (.), we have

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Therefore, by the mathematical induction, we conclude that (.) holds for alln≥. Since

g(xn)g(xn+) andg(yn)g(yn+) for alln≥ so from (.), we have

Gg(xn+),g(xn+),g(xn)

+Gg(yn+),g(yn+),g(yn)

=GF(xn,yn),F(xn,yn),F(xn–,yn–)

+GF(yn,xn),F(yn,xn),F(yn–,xn–)

=MG_F(xn,xn,xn–,yn,yn,yn–)

≤Gg(xn),g(xn),g(xn–)

+Gg(yn),g(yn),g(yn–)

– ϕGg(xn),g(xn),g(xn–)

,Gg(yn),g(yn),g(yn–)

. (.)

Setting

ωx_n_+:=Gg(xn+),g(xn+),g(xn)

and

ωy_n_+:=Gg(yn+),g(yn+),g(yn)

for alln≥,

we have, by (.), that

ωx_n_++ω_ny_+≤ωx_n+ω_ny– ϕωx_n,ω_ny. (.)

Asϕ(t,t)≥ for all (t,t)∈[,∞)×[,∞), we have

ωx_n_++ω_ny_+≤ωx_n+ω_ny, for alln≥.

Then the sequence{ωx

n+ω

y

n}is decreasing. Therefore, there existsω≥ such that

lim

n→∞

ωx_n+ω_ny= lim

n→∞

=ω. (.)

Now, we show thatω= . Suppose, to contrary, thatω> . From (.), the sequences

{G(g(xn+),g(xn+),g(xn))} and {G(g(yn+),g(yn+),g(yn))} have convergent subsequences

{G(g(xn(j)+),g(xn(j)+),g(xn(j)))} and {G(g(yn(j)+),g(yn(j)+),g(yn(j)))}, respectively. Assume that

lim

j→∞ω

x

n(j)+=lim

j→∞G

g(xn(j)+),g(xn(j)+),g(xn(j))

=ω

and

lim

j→∞ω

y

n(j)+=_jlim_→∞G

g(yn(j)+),g(yn(j)+),g(yn(j))

=ω,

which gives thatω+ω=ω. From (.), we have

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Then taking the limit asj→ ∞in the above inequality, we obtain

ω≤ω– lim

j→∞ϕ

ω_nx₍_j₎,ωy_n₍_j₎<ω, (.)

which is a contradiction. Thusω= ; that is,

lim

n→∞

= lim

n→∞

ω_nx+ωy_n= . (.)

Next, we show that{g(xn)}and{g(yn)}areG-Cauchy sequences. On the contrary, assume that at least one of{g(xn)}or{g(yn)}is not aG-Cauchy sequence. By Lemma ., there is anε>  for which we can ﬁnd subsequences{g(xn(k))},{g(xm(k))}of{g(xn)}and{g(yn(k))},

{g(ym(k))}of{g(yn)}withn(k) >m(k)≥ksuch that

Gg(xn(k)),g(xn(k)),g(xm(k))

+Gg(yn(k)),g(yn(k)),g(ym(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

Gg(xn(k)–),g(xn(k)–),g(xm(k))

+Gg(yn(k)–),g(yn(k)–),g(ym(k))

<ε. (.)

By Lemma ., we have

≤Gg(xn(k)),g(xn(k)),g(xn(k)–)

+Gg(xn(k)–),g(xn(k)–),g(xm(k))

(.)

and

Gg(yn(k)),g(yn(k)),g(ym(k))

≤Gg(yn(k)),g(yn(k)),g(yn(k)–)

. (.)

In view of (.)-(.), we have

ε≤Gg(xn(k)),g(xn(k)),g(xm(k))

≤Gg(xn(k)),g(xn(k)),g(xn(k)–)

+Gg(yn(k)),g(yn(k)),g(yn(k)–)

+Gg(xn(k)–),g(xn(k)–),g(xm(k))

< Gg(xn(k)),g(xn(k)),g(xn(k)–)

+Gg(yn(k)),g(yn(k)),g(yn(k)–)

+ε.

Then lettingk→ ∞in the last inequality and using (.), we have

lim

k→∞

=ε. (.)

By Lemma . and Lemma ., we have

≤Gg(xn(k)),g(xn(k)),g(xn(k)+)

+Gg(xn(k)+),g(xn(k)+),g(xm(k))

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≤Gg(xn(k)+),g(xn(k)+),g(xn(k))

+Gg(xn(k)+),g(xn(k)+),g(xm(k)+)

+Gg(xm(k)+),g(xm(k)+),g(xm(k))

(.)

and

≤Gg(yn(k)+),g(yn(k)+),g(yn(k))

+Gg(yn(k)+),g(yn(k)+),g(ym(k)+)

+Gg(xm(k)+),g(xm(k)+),g(xm(k))

. (.)

It follows from (.) and (.) that

≤ωx_n₍_k_)++ωy_n₍_k_)++ωx_m₍_k_)++ω_my₍_k_)+

+Gg(xn(k)+),g(xn(k)+),g(xm(k)+)

+Gg(yn(k)+),g(yn(k)+),g(ym(k)+)

. (.)

Sincen(k) >m(k), we get

g(xn(k))g(xm(k)) and g(yn(k))g(ym(k)),

and also, from (.),

Gg(xn(k)+),g(xn(k)+),g(xm(k)+)

+Gg(yn(k)+),g(yn(k)+),g(ym(k)+)

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

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

=MG_F(xn(k),xn(k),xm(k),yn(k),yn(k),ym(k))

≤Gg(xn(k)),g(xn(k)),g(xm(k))

– ϕGg(xn(k)),g(xn(k)),g(xm(k))

,Gg(yn(k)),g(yn(k)),g(ym(k))

. (.)

From (.) and (.), we have

ωx_n₍_k_)++ωy_n₍_k_)++ωx_m₍_k_)++ω_my₍_k_)+

≥Gg(xn(k)),g(xn(k)),g(xm(k))

–Gg(xn(k)+),g(xn(k)+),g(xm(k)+)

–Gg(yn(k)+),g(yn(k)+),g(ym(k)+)

≥Gg(xn(k)),g(xn(k)),g(xm(k))

–Gg(xn(k)),g(xn(k)),g(xm(k))

+ ϕGg(xn(k)),g(xn(k)),g(xm(k))

= ϕGg(xn(k)),g(xn(k)),g(xm(k))

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This implies that

ωx_n₍_k_)++ωy_n₍_k_)++ωx_m₍_k_)++ω_my₍_k_)+

≥ϕGg(xn(k)),g(xn(k)),g(xm(k))

. (.)

From (.), the sequences{G(g(xn(k)),g(xn(k)),g(xm(k)))}and{G(g(yn(k)),g(yn(k)),g(ym(k)))} have subsequences converging to, say,εandε, respectively, andε+ε=ε> . By passing to subsequences, we may assume that

lim

k→∞G

g(xn(k)),g(xn(k)),g(xm(k))

=ε and lim

k→∞G

g(yn(k)),g(yn(k)),g(ym(k))

=ε.

Takingk→ ∞in (.) and using (.), we have

 = lim

k→∞

ω_nx₍_k_)++ωy_n₍_k_)++ωx_m₍_k₎+ωy_m₍_k₎

≥ lim

k→∞ϕ

= lim

G(g(xn(k)),g(xn(k)),g(xm(k)))→ε G(g(yn(k)),g(yn(k)),g(ym(k)))→ε

ϕGg(xn(k)),g(xn(k)),g(xm(k))

,

> ,

which is a contradiction. Therefore,{g(xn)}and{g(yn)}areG-Cauchy sequences. ByG -completeness ofX, there existsx,y∈Xsuch that

lim

n→∞g(xn) =x and nlim→∞g(yn) =y. (.)

This together with the continuity ofgimplies that

lim

n→∞g

g(xn)

=g(x) and lim

n→∞g

g(yn)

=g(y). (.)

Now, suppose that assumption (a) holds. From (.) and the commutativity ofFandg, we obtain

g(x) = lim

n→∞g

g(xn+)

= lim

n→∞g

F(xn,yn)

= lim

n→∞F

g(xn),g(yn)

=F

lim

n→∞g(xn),nlim→∞g(yn)

=F(x,y).

Similarly, we have

g(y) = lim

n→∞g

g(yn+)

= lim

n→∞g

F(yn,xn)

= lim

n→∞F

g(yn),g(xn)

=F

lim

n→∞g(yn),nlim→∞g(xn)

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Hence, (x,y) is a coupled coincidence point ofFandg.

Finally, suppose that assumption (b) holds. Since{g(xn)} is non-decreasing satisfying

g(xn)→xand{g(yn)}is non-increasing satisfyingg(yn)→y, we have

gg(xn)

g(x) and g(y)gg(yn)

for alln≥.

Using the rectangle inequality and (.), we get

GF(x,y),g(x),g(x)+GF(y,x),g(y),g(y)

≤GF(x,y),gg(xn+)

,gg(xn+)

+Ggg(xn+)

,g(x),g(x)

+GF(y,x),gg(yn+)

,gg(yn+)

+Ggg(yn+)

,g(y),g(y)

=GF(x,y),gF(xn,yn)

,gF(xn,yn)

+Ggg(xn+)

,g(x),g(x)

+GF(y,x),gF(yn,xn)

,gF(yn,xn)

+Ggg(yn+)

,g(y),g(y)

=GF(x,y),Fg(xn),g(yn)

,Fg(xn),g(yn)

+Ggg(xn+)

,g(x),g(x)

GF(y,x),Fg(yn),g(xn)

,Fg(yn),g(xn)

+Ggg(yn+)

,g(y),g(y)

≤Gg(x),gg(xn)

,gg(xn)

+Gg(y),gg(yn)

,gg(yn)

– ϕGg(x),gg(xn)

,gg(xn)

,Gg(y),gg(yn)

,gg(yn)

+Ggg(xn+)

,g(x),g(x)+Ggg(yn+)

,g(y),g(y)

< .

Lettingn→+∞in the above inequality, we obtain that

GF(x,y),g(x),g(x)+GF(y,x),g(y),g(y)= ,

which gives that G(F(x,y),g(x),g(x)) =G(F(y,x),g(y),g(y)) = ; that is,F(x,y) =g(x) and

F(y,x) =g(y). Therefore, (x,y) is a coupled coincidence point ofFandg. The proof is

com-plete.

Settingg(x) =xin Theorem ., we obtain the following new result:

Theorem . Let(X,)be a partially ordered set and G be a G-metric on X such that

(X,G)is a complete G-metric space.Let F:X×X→X be a mapping having the mixed monotone property on X.Suppose that there existsϕ∈such that

GF(x,y),F(u,v),F(w,z)+GF(y,x),F(v,u),F(z,w)

≤G(x,u,w) +G(y,v,z)– ϕG(x,u,w),G(y,v,z) (.)

for all xuw and yvz.Suppose that either (a) Fis continuous or

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xF(x,y) and yF(y,x),

then F has a coupled ﬁxed point in X.

Remark . Theorem . is more general than [, Theorem .] since the contractive condition (.) is weaker than (.), a fact which is clearly illustrated by the following example.

Example . LetX=Rbe a set endowed with orderxy⇔x≤y. Let the mapping

G:X×X×X→R+be deﬁned by

G(x,y,z) =|x–y|+|y–z|+|z–x|,

for allx,y,z∈X. ThenGis aG-metric onX. Deﬁne the mappingF:X×X→Xby

F(x,y) =x– y

 , for all (x,y)∈X _.

Then the following properties hold:

() Fis mixed monotone;

() Fsatisﬁes condition (.) butFdoes not satisfy condition (.).

Indeed, we ﬁrst show thatFdoes not satisfy condition (.). Assume to the contrary, that there existsϕ∈, such that (.) holds. This means

GF(x,y),F(u,v),F(w,z)=x– y

 –

u– v



+u– v

 –

w– z



+w– z

 –

x– y



≤ 



|x–u|+|u–w|+|w–x|+|y–v|+|v–z|

+|z–y|–ϕ|x–u|+|u–w|+|w–x|,

|y–v|+|v–z|+|z–y|.

Settingx=u=wandy=vorv=zorz=y, by Remark ., we get

 

|y–v|+|v–z|+|z–y|≤ 



|y–v|+|v–z|+|z–y|

–ϕ,|y–v|+|v–z|+|z–y|

<  

|y–v|+|v–z|+|z–y|,

which gives a contradiction. Hence,Fdoes not satisfy condition (.). Now, we prove that (.) holds. Indeed, we have

GF(x,y),F(u,v),F(w,z)+GF(y,x),F(v,u),F(z,w)

=x– y

 –

u– v



+u– v

 –

w– z



+w– z

 –

x– y

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+y– x

 –

v– u



+v– u

 –

z– w



+z– w

 –

y– x



≤ 

|x–u|+  |y–v|+



|u–w|+  |v–z|+



|w–x|+  |z–y|

+  |y–v|+



|x–u|+ 

|v–z|+ 

|u–w|+  |z–y|+

 |w–x|

=

|x–u|+ 

|u–w|+ 

|w–x|+  |y–v|+

 |v–z|+

 |z–y| =|x–u|+|u–w|+|w–x|+|y–v|+|v–z|+|z–y|

–  

|x–u|+|u–w|+|w–x|+|y–v|+|v–z|+|z–y|

=G(x,u,w) +G(y,v,z)– ϕG(x,u,w),G(y,v,z).

By the above, we get exactly (.) withϕ(t,t) =_(t+t).

By Theorem ., we also obtain the following new result for the coupled coincidence point theorem for mixedg-monotone operatorsFsatisfying a contractive condition.

Theorem . Let(X,)be a partially ordered set and suppose that there exists a G-metric G on X such that(X,G)is a complete G-metric space.Let F:X×X→X,g:X→X so that F is a mapping having the mixed g-monotone property on X.Suppose that there existsϕ∈

such that

GF(x,y),F(u,v),F(w,z)

≤G(g(x),g(u),g(w)) +G(g(y),g(v),g(z))



–ϕGg(x),g(u), (w),Gg(y),g(v),g(z) (.)

for all g(x)g(u)g(w)and g(y)g(v)g(z).Suppose that either (a) Fis continuous or

(ii) if a non-decreasing sequence{yn}is such thatyn→y,thenyynfor alln.

g(x)F(x,y) and g(y)F(y,x),

then F and g have a coupled coincidence point.

Letdenote the class of all functionsψ: [,∞)→[,∞) satisfying

lim

t→rψ(t) >  for allr> .

Corollary . Let(X,)be a partially ordered set and G be a G-metric on X such that

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such that

– ψmaxGg(x),g(u),g(w),Gg(y),g(v),g(z) (.)

MG_F(x,u,w,y,v,z) =GF(x,y),F(u,v),F(w,z)+GF(y,x),F(v,u),F(z,w).

g(x)F(x,y) and g(y)F(y,x),

and either

Proof In Theorem ., takingϕ(t,t) =ψ(max{t,t}) for all (t,t)∈[,∞), we get the

desired results.

Corollary . Let(X,)be a partially ordered set and G be a G-metric on X such that

(X,G)is a complete G-metric space.Let g:X→X be a mapping and F:X×X→X be a mapping having the mixed g-monotone property on X.Suppose that there existsψ∈

such that

– ψGg(x),g(u),g(w)+Gg(y),g(v),g(z) (.)

MG_F(x,u,w,y,v,z) =GF(x,y),F(u,v),F(w,z)+GF(y,x),F(v,u),F(z,w).

g(x)F(x,y) and g(y)F(y,x),

and either

Proof In Theorem ., takingϕ(t,t) =ψ(t+t) for all (t,t)∈[,∞), we get the desired

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3 Coupled common ﬁxed point

Now, we shall prove the existence and uniqueness theorem of a coupled common ﬁxed point. If (X,) is a partially ordered set, we endow the product setX×Xwith the partial order relation:

(x,y), (u,v)∈X×X, (x,y)(u,v) ⇔ xu,yv.

Theorem . In addition to the hypotheses of Theorem ., suppose that for all (x,y), (x*_,_y*₎_∈_X_×_X_,_{there exists}₍_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*_))._{Then F and g have a unique coupled common ﬁxed}

point.

Proof From Theorem ., the set of coupled coincidences is non-empty. Assume that (x,y) and (x*_,_y*_{) are coupled coincidence points of}_F_and_g_{. We shall show that}

g(x) =gx* and g(y) =gy*. (.)

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

=u,v=vand choosingu,v∈Xsuch that

g(u) =F(u,v) and g(v) =F(v,u).

Then, similarly as in the proof of Theorem ., we can inductively deﬁne sequences{g(un)} and{g(vn)}inXby

g(un+) =F(un,vn) and g(vn+) =F(vn,un), for alln≥.

Since (F(x*_,_y*_),_F₍_y*_,_x*_{)) = (}_g₍_x*_),_g₍_y*_{)) and (}_F₍_u_,_v_),_F₍_v_,_u_{)) = (}_g₍_u

),g(v)) are comparable, without restriction to the generality, we can assume that

Fx*,y*,Fy*,x*=gx*,gy*F(u,v),F(v,u)=g(u),g(v)

,

and

F(x,y),F(y,x)=g(x),g(y)F(u,v),F(v,u)=g(u),g(v)

.

This actually means that

gx*g(u) and g

y*g(v),

and

g(x)g(u) and g(y)g(v).

Using thatFis a mixedg-monotone mapping, we can inductively show that

gx*g(un) and g

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and

g(x)g(un) and g(y)g(vn) for alln≥.

Thus, from (.), we get

Gg(un+),g(x),g(x)

+Gg(vn+),g(y),g(y)

=GF(un,vn),F(x,y),F(x,y)

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

=MG_F(un,x,x,vn,y,y)

≤Gg(un),g(x),g(x)

+Gg(vn),g(y),g(y)

– ϕGg(un),g(x),g(x)

,Gg(vn),g(y),g(y)

, (.)

which implies that

Gg(un+),g(x),g(x)

+Gg(vn+),g(y),g(y)

≤Gg(un),g(x),g(x)

+Gg(vn),g(y),g(y)

;

that is, the sequence {G(g(un),g(x),g(x)) +G(g(vn),g(y),g(y))} is decreasing. Therefore, there existsα≥ such that

lim

n→∞

Gg(un),g(x),g(x)

+Gg(vn),g(y),g(y)

=α.

We shall show thatα= . Suppose, to the contrary, thatα> . Therefore,{G(g(un),g(x),

g(x))}and{G(g(vn),g(y),g(y))}have subsequences converging toα,α, respectively, with

α+α=α> .

Taking the limit up to subsequences asn→ ∞in (.), we have

α≤α–  lim

n→∞ϕ

Gg(un),g(x),g(x)

,Gg(vn),g(y),g(y)

<α,

which is a contradiction. Thus,α= ; that is,

lim

n→∞

Gg(un),g(x),g(x)

+Gg(vn),g(y),g(y)

= ,

which implies that

lim

n→∞G

g(un),g(x),g(x)

= lim

n→∞G

g(vn),g(y),g(y)

= . (.)

Similarly, one can show that

lim

n→∞G

g(un),g

x*,gx*= lim

n→∞G

g(vn),g

y*,gy*= . (.)

Therefore, from (.), (.) and the uniqueness of the limit, we getg(x) =g(x*) andg(y) =

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we have

gg(x)=gF(x,y)=Fg(x),g(y) and gg(y)=gF(y,x)=Fg(y),g(x). (.)

Denoteg(x) =zandg(y) =w, then by (.), we get

g(z) =F(z,w) and g(w) =F(w,z). (.)

Thus, (z,w) is a coincidence point. Then from (.) withx*₌_z_and_y*₌_w_{, we have}_g₍_x_{) =}

g(z) andg(y) =g(w), that is,

g(z) =z and g(w) =w. (.)

From (.) and (.), we get

z=g(z) =F(z,w) and w=g(w) =F(w,z).

Then, (z,w) is a coupled common ﬁxed point ofFandg. To prove the uniqueness, assume that (p,q) is another coupled common ﬁxed point. Then by (.), we havep=g(p) =g(z) =z

andq=g(q) =g(w) =w. The proof is complete.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this paper. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper. Finally, the ﬁrst author is supported by the ‘Centre of Excellence in Mathematics’ under the Commission on Higher Education, Ministry of Education, Thailand.

Received: 27 July 2012 Accepted: 17 September 2012 Published: 3 October 2012 References

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