Coupled common fixed point theorems for a pair of commuting mappings in partially ordered G-metric spaces

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R E S E A R C H

Open Access

Coupled common ﬁxed point theorems for a

pair of commuting mappings in partially

ordered

G

-metric spaces

Feng Gu

*

_{and Shuhang Zhou}

*_{Correspondence:}

[email protected] Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

Abstract

In this paper, we introduce a new contractive condition for a pair of commuting mappings in partially orderedG-metric spaces. Some new coupled coincidence point and coupled common ﬁxed point theorems are obtained. An example is given to support the usability of our results. The results presented in this paper generalize and improve the corresponding results of Nashine and Shatanawi from partially ordered metric spaces to partially orderedG-metric spaces.

MSC: 47H10; 54H25; 54E50

Keywords: G-metric space; mixedg-monotone property; coupled coincidence point; coupled common ﬁxed point; partial order

1 Introduction and preliminaries

In , Ran and Reurings [] showed the existence of fixed points of nonlinear contrac-tion mappings in metric spaces endowed with a partial ordering and presented applica-tions of their results to matrix equaapplica-tions. Subsequently, Nieto and Rodríguez-López [, ] extended the corresponding results in []. They proved the existence theorems of a unique fixed point for mappings in partially ordered sets and gave some applications to the ex-istence of a unique solution for a first-order ordinary differential equation with periodic boundary conditions.

In , the well-known Tarski theorem [] was used in [] to study the existence of solutions for fuzzy equations and to prove existence theorems for fuzzy diﬀerential equa-tions.

It is well known that fixed point theory in partially ordered metric spaces as one of the most important tools of nonlinear analysis has been widely applied to matrix equations (for more details, see [] and the references therein), ordinary differential equations (for more details, see [, , ] and [] and the references therein), fuzzy differential equations (for more details, see [] and the references therein), integral equations (for more details, see [] and [] and the references therein) and intermediate value theorems (for more details, see [] and the references therein). For more results on partially ordered metric spaces, one refers to the results in [–].

In , Bhaskar and Lakshmikantham [] introduced the notion of a coupled ﬁxed point and proved some interesting coupled ﬁxed point theorems for the mappings satis-fying a mixed monotone property; while Lakshmikantham and Ćirić [] introduced the

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concept of a mixedg-monotone mapping and proved coupled coincidence and coupled common ﬁxed point theorems that extended the theorems due to Bhaskar and Laksh-mikantham []. Subsequently, many authors obtained many coupled coincidence and coupled ﬁxed point theorems in ordered metric spaces; see [–] and the references therein.

In , Mustafa and Sims [] introduced the notion of generalized metric spaces or simply G-metric spaces as a generalization of the concept of a metric space. Based on the concept of aG-metric space, many authors obtained many fixed point and common fixed point theorems for the mappings satisfying different contractive conditions; see [– ] for more details. Fixed point problems also have been considered in partially ordered

G-metric spaces; see [–] and the references therein for more details.

Inspired and motivated by the facts said above, in this paper we establish coupled coin-cidence point and coupled common ﬁxed point theorems for a pair of commuting map-pings in partially orderedG-metric spaces. An example is given to support the usability of our results. The results presented in this paper generalize and improve the corresponding results of Nashine and Shatanawi [] from partially ordered metric spaces to partially orderedG-metric spaces.

First, we present some known deﬁnitions and propositions.

Deﬁnition .[] LetXbe a nonempty set, and letG:X×X×X–→R+be a function satisfying the following axioms:

(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 a generalized metric or, more speciﬁcally, aG-metric onX

and the pair (X,G) is called aG-metric space.

It is known that the functionG(x,y,z) on aG-metric spaceXis jointly continuous in all three of its variables, andG(x,y,z) =  if and only ifx=y=z; see [] and the references therein for more details.

Deﬁnition .[] Let (X,G) be aG-metric space, and let{xn}be a sequence of points

inX. A pointxinXis said to be the limit of the sequence{xn}iflimm,n→∞G(x,xn,xm) = ,

and one says that the sequence{xn}isG-convergent tox.

Thus, ifxn→xin aG-metric space (X,G), then for any> , there existsN∈Nsuch

thatG(x,xn,xm) <for alln,m≥N.

Proposition .[] Let(X,G)be a G-metric space.Then the following are equivalent: () {xn}isG-convergent tox;

() G(xn,xn,x)→asn→ ∞;

() G(xn,x,x)→asn→ ∞;

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Deﬁnition .[] Let (X,G) be aG-metric space. A sequence{xn}is called aG-Cauchy

sequence if, for each> , there exists a positive integerN∈Nsuch thatG(xn,xm,xl) <

for alln,m,l≥N; that is,G(xn,xm,xl)→ asn,m,l→ ∞.

Deﬁnition .[] AG-metric space (X,G) is said to beG-complete if everyG-Cauchy sequence in (X,G) isG-convergent inX.

Proposition .[] Let(X,G)be a G-metric space.Then the following are equivalent: () The sequence{xn}isG-Cauchy;

() For every> ,there existsk∈Nsuch thatG(xn,xm,xm) <for alln,m≥k.

Deﬁnition .[] Let (X,G) and (X,G) be twoG-metric spaces. Thenf :X→Xis

G-continuous atx∈Xif and only if it isG-sequentially continuous atx; that is, whenever {xn}isG-convergent tox,{f(xn)}isG-convergent tof(x).

Proposition . [] Let (X,G)be a G-metric space. Then, for any x,y∈X, we have G(x,y,y)≤G(y,x,x).

Deﬁnition .[] Let (X,G) be aG-metric space. A mappingF:X×X→Xis said to be continuous if for any twoG-convergent sequences{xn}and{yn}converging toxandy

respectively,{F(xn,yn)}isG-convergent toF(x,y).

Deﬁnition .[] An element (x,y)∈X×Xis called a coupled ﬁxed point of the map-pingF:X×X→XifF(x,y) =xandF(y,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 .[] LetXbe a nonempty set. Then we say that the mappingsF:X×X→ Xandg:X→Xare commutative ifgF(x,y) =F(gx,gy).

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

x,x∈X, x x ⇒ F(x,y) F(x,y)

and

y,y∈X, y y ⇒ F(x,y)F(x,y).

Deﬁnition . [] Let (X, ) be a partially ordered set and F:X×X→X andg :

X→X. The mappingF is said to have the mixedg-monotone property ifF is mono-toneg-nondecreasing in its ﬁrst argument and is monotoneg-nonincreasing in its second argument; that is, for anyx,y∈X, we have

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and

y,y∈X, g(y) g(y) ⇒ F(x,y)F(x,y). 2 Main results

Theorem . Let(X, )be a partially ordered set and (X,G)be a G-metric space.Let F:X×X→X and g:X→X be mappings such that F has the mixed g-monotone property on X.Suppose there exist non-negative real numbersα,β,γ and L withβ+γ < such that

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

≤αminGF(x,y),gx,gx,GF(u,v),gx,gx,GF(w,z),gx,gx

+βminGF(x,y),gu,gu,GF(u,v),gu,gu,GF(w,z),gu,gu

+γminGF(x,y),gw,gw,GF(u,v),gw,gw,GF(w,z),gw,gw

+LminGF(x,y),gu,gu,GF(u,v),gw,gw,GF(w,z),gx,gx (.)

for all(x,y), (u,v), (w,z)∈X×X with g(x) g(u) g(w)and g(y)g(v)g(z).Further suppose F(X×X)⊂g(X)and g(X)is a complete subspace of X.Also suppose that X satisﬁes the following properties:

(i) if a non-decreasing sequence{xn}inXconverges tox∈X,thenxn xfor alln∈N;

(ii) if a non-increasing sequence{yn}inXconverges toy∈X,thenynyfor alln∈N.

If there exist two elements x,y∈X such that g(x) F(x,y)and g(y)F(y,x),

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

Proof In view of the condition of the theorem, we see that there existx,y∈Xsuch that

g(x) F(x,y) andg(y)F(y,x). SinceF(X×X)⊂g(X), we can choosex,y∈X

such thatg(x) =F(x,y) andg(y) =F(y,x). Again sinceF(X×X)⊂g(X), we can choose

x,y∈Xsuch thatg(x) =F(x,y) andg(y) =F(y,x). Continuing in this way, we

con-struct two sequences{xn}and{yn}inXsuch that

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

Now we prove that

g(xn) g(xn+) and g(yn)g(yn+), ∀n≥. (.)

We will use the mathematical induction. Letn= . Sinceg(x) F(x,y) andg(y)

F(y,x), in view ofg(x) =F(x,y) andg(y) =F(y,x), we haveg(x) g(x) andg(y)

g(y); that is, (.) hold forn= . We assume that (.) hold for somen> . AsFhas the

mixedg-monotone property,g(xn) g(xn+) andg(yn)g(yn+), from (.) we get that

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

In the same way, we obtain that

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By combining (.) and (.), we obtaing(xn+) g(xn+) andg(yn+)g(yn+). Thus, by

the mathematical induction, we conclude that (.) hold for alln≥. We check easily that

g(x) g(x) g(x) · · · g(xn) g(xn+) · · ·

and

g(y)g(y)g(y) · · · g(yn)g(yn+) · · ·.

It follows from (.) and (.) that

G(gxn+,gxn,gxn)

=GF(xn,yn),F(xn–,yn–),F(xn–,yn–)

≤αminGF(xn,yn),gxn,gxn

,GF(xn–,yn–),gxn,gxn

+βminGF(xn,yn),gxn–,gxn–

,GF(xn–,yn–),gxn–,gxn–

,

GF(xn–,yn–),gxn–,gxn–

+γminGF(xn,yn),gxn–,gxn–

,

GF(xn–,yn–),gxn–,gxn–

+LminGF(xn,yn),gxn–,gxn–

,

GF(xn–,yn–),gxn,gxn

=αminG(gxn+,gxn,gxn),G(gxn,gxn,gxn),G(gxn,gxn,gxn)

+βminG(gxn+,gxn–,gxn–),G(gxn,gxn–,gxn–),G(gxn,gxn–,gxn–)

+γminG(gxn+,gxn–,gxn–),G(gxn,gxn–,gxn–),G(gxn,gxn–,gxn–)

+LminG(gxn+,gxn–,gxn–),G(gxn,gxn–,gxn–),G(gxn,gxn,gxn)

= (β+γ)minG(gxn+,gxn–,gxn–),G(gxn,gxn–,gxn–)

. (.)

Hence, we obtain

G(gxn+,gxn,gxn)≤(β+γ)G(gxn,gxn–,gxn–). (.)

Similarly, using (.) and (.), we can prove that

G(gyn+,gyn,gyn)≤(β+γ)G(gyn,gyn–,gyn–). (.)

Suppose thatρn=G(gxn+,gxn,gxn) +G(gyn+,gyn,gyn) andδ=β+γ, then ≤δ< . From

(.) and (.), we have

ρn=G(gxn+,gxn,gxn) +G(gyn+,gyn,gyn)

≤δG(gxn,gxn–,gxn–) +G(gyn,gyn–,gyn–)

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=δρn–≤δρn–≤ · · ·

≤δnρ, (.)

which implies that

lim

n→∞ρn=nlim→∞

G(gxn+,gxn,gxn) +G(gyn+,gyn,gyn)

= ,

and

lim

n→∞G(gxn+,gxn,gxn) =  and nlim→∞G(gyn+,gyn,gyn) = .

Next, we will prove that{gxn}and{gyn}areG-Cauchy sequences ing(X).

In fact, for eachn,m∈N,m>n, from (G) we have

G(gxm,gxn,gxn)≤G(gxm,gxm–,gxm–) +G(gxm–,gxm–,gxm–) +· · ·

+G(gxn+,gxn,gxn) (.)

and

G(gym,gyn,gyn)≤G(gym,gym–,gym–) +G(gym–,gym–,gym–) +· · ·

+G(gyn+,gyn,gyn). (.)

Therefore, combining inequalities (.), (.) and (.), we have

G(gxm,gxn,gxn) +G(gym,gyn,gyn)

≤ρm–+ρm–+· · ·+ρn

=δm–+δm–+· · ·+δnρ→ asn,m→ ∞. (.)

This implies that{gxn}and{gyn}areG-Cauchy sequences ing(X). Sinceg(X) is a complete

subspace ofX, there existx,y∈Xsuch thatgxn→gxandgyn→gyasn→ ∞. Since{gxn}

is a non-decreasing sequence and{gyn}is a non-increasing sequence, by the assumption

conditions (i) and (ii), we havegxn gxandgyngyfor alln≥.

It follows from (.) and (.) that

Ggxn+,F(x,y),F(x,y)

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

≤αminGF(xn,yn),gxn,gxn

,GF(x,y),gxn,gxn

+βminGF(xn,yn),gx,gx

,GF(x,y),gx,gx,GF(x,y),gx,gx

+γminGF(xn,yn),gx,gx

+LminGF(xn,yn),gx,gx

,GF(x,y),gx,gx,GF(x,y),gxn,gxn

=αminG(gxn+,gxn,gxn),G

F(x,y),gxn,gxn

,GF(x,y),gxn,gxn

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+ (β+γ)minG(gxn+,gx,gx),G

F(x,y),gx,gx

+LminG(gxn+,gx,gx),G

F(x,y),gx,gx,GF(x,y),gxn,gxn

. (.)

Taking the limit asn→ ∞in the above inequality, we obtainG(gx,F(x,y),F(x,y))≤. HenceG(gx,F(x,y),F(x,y)) = , and sogx=F(x,y).

Similarly, one can show thatgy=F(y,x). Thus we proved thatF andghave a coupled

coincidence point. This concludes the proof.

Theorem . Let(X, )be a partially ordered set and(X,G)be a complete G-metric space.

Let F:X×X→X and g:X→X be mappings such that F has the mixed g-monotone property on X.Suppose there exist non-negative real numbersα,β,γ and L withβ+γ < 

such that

+LminGF(x,y),gu,gu,GF(u,v),gw,gw,GF(w,z),gx,gx (.)

for all(x,y), (u,v), (w,z)∈X×X with g(x) g(u) g(w)and g(y)g(v)g(z).Further suppose F(X×X)⊂g(X)and g is continuous non-decreasing and commutes with F,and also suppose either

(a) Fis continuous or

(b) Xhas the following property:

(i) if a non-decreasing sequence{xn}inXconverges tox∈X,thenxn xfor all

n∈N;

(ii) if a non-increasing sequence{yn}inXconverges toy∈X,thenynyfor all

n∈N.

If there exist two elements x,y∈X such that g(x) F(x,y)and g(y)F(y,x),then

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

Proof Following the proof of Theorem ., we will get twoG-Cauchy sequences{gxn}and

{gyn}inXsuch that{gxn}is a non-decreasing sequence and{gyn}is a non-increasing

se-quence inX. Since (X,G) is a completeG-metric space, there is (x,y)∈X×Xsuch that

gxn→xandgyn→y.

Sincegis continuous, we haveggxn→gxandggyn→gyasn→ ∞.

First, suppose that (a) holds, that is,Fis continuous. Then

F(gxn,gyn)→F(x,y) and F(gyn,gxn)→F(y,x).

On the other hand, sincegcommutes withF, we have

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and

F(gyn,gxn) =gF(yn,xn) =ggyn+→gy asn→ ∞.

Therefore, by the uniqueness of 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,{gyn}is a non-increasing sequence such thatgyn→yandgis a non-decreasing

function, by (i) and (ii) we get thatg(gxn) g(x) andg(gyn)≥g(y) hold for alln∈N.

By (.), we have

Gg(gxn+),F(x,y),F(x,y)

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

≤αminGF(gxn,gyn),g(gxn),g(gxn)

,GF(x,y),g(gxn),g(gxn)

,

GF(x,y),g(gxn),g(gxn)

+βminGF(gxn,gyn),gx,gx

+γminGF(gxn,gyn),gx,gx

+LminGF(gxn,gyn),gx,gx

,GF(x,y),gx,gx,GF(x,y),g(xn),g(gxn)

=αminGg(gxn+),g(gxn),g(gxn)

,GF(x,y),g(gxn),g(gxn)

,

GF(x,y),g(gxn),g(gxn)

+ (β+γ)minGg(gxn+),gx,gx

,GF(x,y),gx,gx

+LminGg(gxn+),gx,gx

,GF(x,y),gx,gx,GF(x,y),g(xn),g(gxn)

. (.)

Lettingn→ ∞, we getG(gx,F(x,y),F(x,y))≤, which implies thatG(gx,F(x,y),F(x,y)) = , so thatgx=F(x,y).

Similarly, we can show thatgy=F(y,x). Thus we proved thatFandghave a coupled

coincidence point.

Remark . Theorems . and . generalize and extend the corresponding results in [, Theorems . and .] from partially ordered metric spaces to partially orderedG-metric spaces.

In Theorem ., if we takeg=I(Iis an identity mapping), then we have the following corollary.

Corollary . Let(X, )be a partially ordered set and(X,G) be a complete G-metric space.Let F:X×X→X be a mapping such that F has the mixed monotone property on X.Suppose there exist non-negative real numbersα,β,γ and L withβ+γ < such that

≤αminGF(x,y),x,x,GF(u,v),x,x,GF(w,z),x,x

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+γmin{GF(x,y),w,w,GF(u,v),w,w,GF(w,z),w,w

+LminGF(x,y),u,u,GF(u,v),w,w,GF(w,z),x,x (.)

for all(x,y), (u,v), (w,z)∈X×X with x u w and yvz,and also suppose either

n∈N;

n∈N.

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

Also, if we takeα=β=γ =  in Theorem ., then we get the following.

Corollary . Let(X, )be a partially ordered set and (X,G)be a complete G-metric space.Let F:X×X→X and g:X→X be mappings such that F has the mixed g-monotone property on X.Suppose there exists a non-negative real number L such that

≤LminGF(x,y),gu,gu,GF(u,v),gw,gw,GF(w,z),gx,gx (.)

n∈N;

n∈N.

Corollary . Let(X, )be a partially ordered set and (X,G)be a complete G-metric space.Let F:X×X→X and g:X→X be mappings such that F has the mixed g-monotone property on X.Suppose there exist non-negative real numbersα,β,γ and L withβ+γ < 

such that

≤(α+β+γ)minGF(x,y),gx,gx,GF(u,v),gx,gx,GF(w,z),gx,gx,

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GF(u,v),gw,gw,GF(w,z),gw,gw

+LminGF(x,y),gu,gu,GF(u,v),gw,gw,GF(w,z),gx,gx (.)

n∈N;

n∈N.

Proof By noting that ifα,βandγ are non-negative real numbers, from (.) we have

≤(α+β+γ)minGF(x,y),gx,gx,GF(u,v),gx,gx,GF(w,z),gx,gx,

GF(x,y),gu,gu,GF(u,v),gu,gu,GF(w,z),gu,gu,GF(x,y),gw,gw,

GF(u,v),gw,gw,GF(w,z),gw,gw

+LminGF(x,y),gu,gu,GF(u,v),gw,gw,GF(w,z),gx,gx

+LminGF(x,y),gu,gu,GF(u,v),gw,gw,GF(w,z),gx,gx. (.) From Theorem ., we see thatFandghave a coupled coincidence point (x,y)∈X×X;

that is,F(x,y) =gxandF(y,x) =gy.

Remark . Corollaries ., . and . generalize and extend the corresponding results in [, Corollaries ., . and .] from partially ordered metric spaces to partially ordered

G-metric spaces.

Now we will prove the existence and uniqueness theorem of a coupled common ﬁxed point. Note that if (X ) is a partially ordered set, then we endow the product spaceX×X

with the following partial order:

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

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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).

Proof From Theorem ., the set of coupled coincidence points ofFandgis nonempty. Suppose (x,y) and (x∗,y∗) are coupled coincidence points ofFandg, that is,gx=F(x,y),

gy=F(y,x),gx∗=F(x∗,y∗) andgy∗=F(y∗,x∗), then

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∗)). Putu=u, v=vand chooseu,v∈Xso that

g(u) =F(u,v) andg(v) =F(v,u). Then, similarly as in the proof of Theorem ., we

can inductively deﬁne sequences{g(un)}and{g(vn)}such that

g(un+) =F(un,vn) and g(vn+) =F(vn,un) ∀n∈N.

Further, setx=x,y=y,x∗=x∗ andy∗=y∗. In the same way, we deﬁne the sequences

{g(xn)},{g(yn)},{g(x∗n)}and{g(y∗n)}. It is easy to show that

g(xn)→F(x,y), g(yn)→F(y,x), g

x∗_n→Fx∗,y∗, gy∗_n→Fy∗,x∗.

Since

F(x,y),F(y,x)=g(x),g(y)

=g(x),g(y)

and

F(u,v),F(v,u)=g(u),g(v)

are comparable, we see thatg(x) g(u) andg(y)g(v). It is easy to show that (g(x),g(y))

and (g(un),g(vn)) are comparable, that is,g(x) g(un) andg(y)g(vn) for alln≥. Thus

from (.), we have

G(gx,gun+,gun+)

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

≤αminGF(x,y),gx,gx,F(un,vn),gx,gx),G

F(un,vn),gx,gx

+βminGF(x,y),gun,gun

,GF(un,vn),gun,gun

,F(un,vn),gun,gun)

+γminGF(x,y),gun,gun

,GF(un,vn),gun,gun

.

SinceF(x,y) =g(x) andg(un+) =F(un,vn), we have

G(gx,gun+,gun+)

≤(β+γ)minG(gx,gun,gun),G(gun+,gun,gun)

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Therefore, we have

G(gx,gun+,gun+)≤(β+γ)G(gx,gun,gun). (.)

Similarly, using (.),g(y) =yandg(vn+) =F(vn,un), we can prove that

G(gy,gvn+,gvn+)≤(β+γ)G(gy,gvn,gvn). (.)

From (.) and (.), we have

G(gx,gun+,gun+) +G(gy,gvn+,gvn+)

≤(β+γ)G(gx,gun,gun) +G(gy,gvn,gvn)

≤(β+γ)G(gx,gun–,gun–) +G(gy,gvn–,gvn–)

≤ · · ·

≤(β+γ)n+G(gx,gu,gu) +G(gy,gv,gv)

.

Taking the limit asn→ ∞, we get

lim

n→∞

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

= ,

which implies that

lim

n→∞G(gx,gun,gun) =nlim→∞G(gy,gvn,gvn) = . (.)

Similarly, we show that

lim

n→∞G

gx∗,gun,gun

= lim

n→∞G

gy∗,gvn,gvn

= . (.)

By the rectangle inequality (G), Proposition ., (.) and (.), we obtain

Ggx,gx∗,gx∗ ≤ G(gx,gun,gun) +G

gun,gx∗,gx∗

≤ (gx,gun,gun) + G

gx∗,gun,gun

→ asn→ ∞,

and

Ggy,gy∗,gx∗ ≤ G(gy,gvn,gvn) +G

gvn,gy∗,gy∗

≤ (gy,gvn,gvn) + G

gy∗,gvn,gvn

→ asn→ ∞.

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Sinceg(x) =F(x,y) andg(y) =F(y,x), by the commutativity ofFandg, we have

g(gx) =gF(x,y)=F(gx,gy) and g(gy) =gF(y,x)=F(gy,gx). (.)

Denoteg(x) =z,g(y) =w. Then from (.) we have

g(z) =F(z,w) and g(w) =F(w,z). (.)

Thus (z,w) is a coupled coincidence point ofFandg. Then from (.) withx∗=zand

y∗=wit followsg(z) =g(x) andg(w) =g(y), 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).

Therefore, (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) =zandq=g(q) =g(w) =w. This completes the proof of Theorem .. In Theorem ., if we takeg=I(Iis an identity mapping), then we have the following corollary.

Corollary . In addition to the hypotheses of Corollary.,suppose that L= ,and for every(x,y), (y∗,x∗)∈X×X,there exists a(u,v)∈X×X such that(F(u,v),F(v,u))is compa-rable to(F(x,y),F(y,x))and(F(x∗,y∗),F(y∗,x∗)).Then F has a unique coupled ﬁxed point;

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

Theorem . In addition to the hypotheses of Theorem.,if gxand gyare comparable

and L= ,then F and g have a coupled coincidence point(x,y)such that gx=F(x,y) =

F(y,x) =gy.

Proof By Theorem . we construct two sequences{xn}and{yn}inXsuch thatgxn→gx

andgyn→gy, where (x,y) is a coincidence point ofFandg. Supposegx gy, then it is

an easy matter to show thatgxn gynfor alln∈N. Thus, by (.) we have

G(gxn,gyn,gyn)

=GF(xn–,yn–),F(yn–,xn–),F(yn–,xn–)

≤αminGF(xn–,yn–),gxn–,gxn–

,GF(yn–,xn–),gxn–,gxn–

,

GF(yn–,xn–),gxn–,gxn–

+βminGF(xn–,yn–),gyn–,gyn–

,GF(yn–,xn–),gyn–,gyn–

,

GF(yn–,xn–),gyn–,gyn–

+γminGF(xn–,yn–),gyn–,gyn–

,GF(yn–,xn–),gyn–,gyn–

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GF(yn–,xn–),gyn–,gyn–

=αminG(gxn,gxn–,gxn–),G(gyn,gxn–,gxn–)

+βminG(gxn,gyn–,gyn–),G(gyn,gyn–,gyn–)

+γminG(gxn,gyn–,gyn–),G(gyn,gyn–,gyn–)

.

By taking the limit asn→ ∞, we getG(gx,gy,gy) = . HenceF(x,y) =gx=gy=F(y,x). A similar argument can be used ifgy gx. This completes the proof of Theorem ..

In Theorem ., if we takeg=I(Iis an identity mapping), then we have the following corollary.

Corollary . In addition to the hypotheses of Theorem.,if xand yare comparable

and L= ,then F has a coupled coincidence point of the form(x,x).

Remark . Theorems . and . generalize and extend the corresponding results in [, Theorems . and .] from partially ordered metric spaces to partially orderedG-metric spaces.

Remark . Corollaries . and . generalize and extend the corresponding results in [, Corollaries . and .] from partially ordered metric spaces to partially ordered

G-metric spaces.

Now, we introduce an example to support Theorem ..

Example . LetX= [, ], then (X,≤) is a partially ordered set with a natural ordering of real numbers. LetG(x,y,z) =|x–y|+|y–z|+|z–x|for allx,y,z∈X. SetFandgas

g:X→X, g(x) =x and

F:X×X→X, F(x,y) = ⎧ ⎨ ⎩

x_–_y

 , x≥y,

, x<y. Then

() (X,G)is a completeG-metric space; () g(X) =Xis complete;

() F(X×X)⊂g(X) =X;

() Xsatisﬁes (i) and (ii) of Theorem .; () Fhas the mixedg-monotone property; () For anyL∈[,∞),Fandgsatisfy

GF(x,y),F(u,v),F(w,z) ≤

min

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

+ min

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+ min

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

+LminGF(x,y),gu,gu,GF(u,v),gw,gw,GF(w,z),gx,gx (.)

for all (x,y), (u,v), (w,z)∈X×Xwithgx≤gu≤gwandgy≥gv≥gz. Thus by Theorem .,

Fandghave a coupled coincidence point. Moreover, (, ) is a coupled ﬁxed point ofF.

Proof The proofs of ()-() are clear. The proof of () is divided into the following cases. Case . Ifx≥y. In this case, we havez≤v≤y≤x≤u≤w, and so

F(x,y) =x

_–_y

 , F(u,v) =

u–v

 , F(w,z) =

w–z

 .

Hence, we get

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

x_–_y

 ,

u_–_v

 ,

w_–_z



=w

_–_z_{– (}_x_–_y₎

 ≤

w

 ≤

w

 ≤

w₊_w_–_x₊_y



= ·

w_–_x₊_y

 =

 G

x_–_y

 ,w

_,_w

= min G

x–y

 ,w

_,_w

,G

u–v

 ,w

_,_w

,G

w–z

 ,w

_,_w

= min

≤ min

+ min

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

+ min

+LminGF(x,y),gu,gu,GF(u,v),gw,gw,GF(w,z),gx,gx.

Case .x<y. We divide the study in two sub-cases: (a) Ifu≥v, thenz≤v≤u≤w, sow≥z. Therefore, we get

F(x,y) = , F(u,v) =u

_–_v

 , F(w,z) =

w_–_z

 .

Hence, we have

,u

_–_v

 ,

w_–_z



=w

_–_z

 ≤

w

 ≤

w

 ≤

w₊_w_–_u₊_v

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= ·

w_–_u₊_v

 =

 G

u_–_v

 ,w

_,_w

= min

G,w,w,G

u_–_v

 ,w

_,_w

,G

w_–_z

 ,w

_,_w

= min

≤ min

+ min

(b) Ifu<v, henceF(x,y) =  =F(u,v); the case wherew<zis obvious because we get

F(w,z) = . Ifw≥z, we haveF(w,z) =w_–z. Therefore

, ,w

_–_z



=w

_–_z

 ≤

w

 ≤

w

 ≤

w₊_z



= ·

w₊_z

 =

 G

w_–_z

 ,w

_,_w

= min

G,w,w,G,w,w,G

w–z

 ,w

_,_w

= min

≤ min

+ min

In all the above cases, inequality (.) of Theorem . is satisfied forα=β=γ =_ and anyL≥. All the required hypotheses of Theorem . are satisfied. Clearly,Fandghave a coupled coincidence point. Moreover, (, ) is a coupled fixed point ofF.

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

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Acknowledgements

The authors are grateful to the editor and the reviewer for suggestions which improved the contents of the article. This work is supported by the National Natural Science Foundation of China (11071169, 11271105) and the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030).

Received: 7 November 2012 Accepted: 26 February 2013 Published: 20 March 2013 References

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