Coupled g-coincidence point theorems for a generalized compatible pair in complete metric spaces

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R E S E A R C H

Open Access

Coupled

g

-coincidence point theorems for a

generalized compatible pair in complete

metric spaces

Narawadee Na Nan

*

_{and Phakdi Charoensawan}

*_{Correspondence:} [email protected] Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

Abstract

In this work, we give the notions of coupledg-coincidence point andHg-increasing property ofFfor mappingsF,H:X×X→Xandg:X→Xand prove the existence and uniqueness of a coupledg-coincidence point theorem for mappings

F,H:X×X→Xandg:X→Xwith

ϕ

-contraction mappings in complete metric spaces without theHg-increasing property ofFand the mixed monotone property

ofG. Further, we apply our results to the existence and uniqueness of a coupled

g-coincidence point of the given mappings with theHg-increasing property ofFand

the mixed monotone property ofHin partially ordered metric spaces.

Keywords: coupled ﬁxed point; coupled coincidence point; coupledg-coincidence point;Hg-increasing; generalized compatible; invariant set; closed set; mixed

g-monotone; partially ordered set

1 Introduction

In , the study of a ﬁxed point in partially ordered metric spaces was initiated by Ran and Reurings [], and continued by Nieto and Lopez [, ]. Agarwalet al.[] and many other authors presented some new results for contractions in partially ordered metric spaces.

In , Guo and Lakshmikantham [] introduced the concept of coupled fixed point. Afterwards, Bhaskar and Lakshmikantham [] introduced the concept of mixed mono-tone property for contractive operators in partially ordered metric spaces and proved cou-pled fixed point theorems for mappings which satisfy the mixed monotone property. They also gave some applications on the existence and uniqueness of the coupled fixed point theorems for such mappings. As a continuation of this trend, Lakshimikantham and Ćirić [] extended the results in [] by defining the mixedg-monotonicity and studied the exis-tence and uniqueness of a coupled coincidence point for such mappings which satisfy the mixed monotone property in partially ordered metric spaces. For more work on the cou-pled fixed point theory and coucou-pled coincidence point theory in partially ordered metric spaces and different spaces, we refer to the reviews (see,e.g., [–]).

One of the interesting ways to develop coupled ﬁxed point theory in partially ordered metric spaces is to consider the mappingF:X×X→Xwithout the mixed monotone property. In , Sintunavaratet al.[, ] proved some coupled ﬁxed point theorems

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for nonlinear contractions without the mixed monotone property and extended some cou-pled ﬁxed point theorems of Bhaskar and Lakshmikantham [] by using the concept of

F-invariant set due to Samet and Vetro []. Later, Charoensawan and Klanarong [] proved the existence and uniqueness of a coupled coincidence point in partially ordered metric spaces without the mixedg-monotone property ofF:X×X→Xandg:X→X. Recently, Kutbiet al.[] introduced the concept ofF-closed set which is weaker than the concept ofF-invariant set and proved some coupled ﬁxed point theorems without the condition of mixed monotone property. Following this trend, many authors have studied ﬁxed point theorems for nonlinear contractions without the monotone property in several spaces (see,e.g., [, –]).

Very recently, Hussainet al.[] presented the new concept of generalized compatibility of a pair{F,G}of mappingsF,G:X×X→Xand proved some coupled coincidence point results of such mappings without the mixedG-monotone property ofF, which generalized some recent comparable results in the literature. They also gave some examples and an application to integral equations to support the result.

In this work, we give the notion of coupledg-coincidence point and theHg-increasing property ofFfor mappingsF,H:X×X→Xandg:X→X. We apply our results to the existence and uniqueness of a coupledg-coincidence point of the given mapping with the

Hg-increasing property ofFand the mixed monotone property ofHin partially ordered metric spaces which generalize and extend the coupled coincidence point theorem in [].

2 Preliminaries

In this section, we give some deﬁnitions, propositions, examples, and remarks which are useful for the main results in this paper. Throughout this paper, (X,) denotes a partially ordered set with the partial order. Byxy, we meanyx. Let (x,) be a partially ordered set, the partial orderfor the product setX×Xdeﬁned in the following way: for all (x,y), (u,v)∈X×X,

(x,y)(u,v) ⇒ H(x,y)H(u,v) and H(v,u)H(y,x), whereH:X×X→Xis one-one.

We say that (x,y) is comparable to (u,v) if either (x,y)(u,v) or (u,v)(x,y). Guo and Lakshmikantham [] introduced the concept of coupled ﬁxed point as follows.

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

F:X×X→XifF(x,y) =xandF(y,x) =y.

The concept of mixed monotone property was introduced by Bhaskar and Lakshmikan-tham in [].

Deﬁnition .[] Let (X,) be a partially ordered set andF:X×X→X. We say thatF

has the mixed monotone property if for anyx,y∈X,

x,x∈X, xx implies F(x,y)F(x,y)

and

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Lakshmikantham and Ćirić in [] introduced the concepts of mixedg-monotone map-ping and coupled coincidence point.

Deﬁnition .[] Let (X,) be a partially ordered set andF:X×X→Xandg:X→X. We say thatFhas the mixedg-monotone property if 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 mappingsF:X×X→Xandg:X→XifF(x,y) =gxandF(y,x) =gy.

Deﬁnition .[] LetXbe a non-empty set andF:X×X→Xandg:X→X. We say thatFandgare commutative ifgF(x,y) =F(gx,gy) for allx,y∈X.

Hussainet al.[] introduced the concept ofG-increasing and{F,G}generalized com-patibility as follows.

Deﬁnition .[] Suppose thatF,G:X×X→Xare two mappings.Fis said to beG -increasing with respect toif for allx,y,u,v∈X, withG(x,y)G(u,v), we haveF(x,y)

F(u,v).

Deﬁnition .[] An element (x,y)∈X×Xis called a coupled coincidence point of mappingsF,G:X×X→XifF(x,y) =G(x,y) andF(y,x) =G(y,x).

Deﬁnition .[] LetF,G:X×X→X. We say that the pair{F,G}is generalized com-patible if

⎧ ⎨ ⎩

d(F(G(xn,yn),G(yn,xn)),G(F(xn,yn),F(yn,xn)))→ asn→+∞,

d(F(G(yn,xn),G(xn,yn)),G(F(yn,xn),F(xn,yn)))→ asn→+∞, whenever (xn) and (yn) are sequences inXsuch that

⎧ ⎨ ⎩

limn→∞F(xn,yn) =limn→∞G(xn,yn) =t,

lim_n_→∞F(yn,xn) =limn→∞G(yn,xn) =t.

Deﬁnition .[] LetF,G:X×X→X be two maps. We say that the pair{F,G}is commuting if

FG(x,y),G(y,x)=GF(x,y),F(y,x) for allx,y∈X.

Letdenote the set of all functionsφ: [,∞)→[,∞) such that:

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(ii) φ(t) = if and only ift= ,

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

Let be the set of all functionsψ : [,∞)→[,∞) such thatlimt→rψ(t) >  for all

r>  andlim_t_→_+ψ(t) = .

Hussainet al.[] proved the coupled coincidence point for such mappings involving the (ψ,φ)-contractive condition as follows.

Theorem .[] Let(X,)be a partially ordered set and M be a non-empty subset of X,and let d be a metric on X such that(X,d)is a complete metric space.Assume that F,G:X×X→X are two generalized compatible mappings such that F is G-increasing with respect to,G is continuous and has the mixed monotone property.Suppose that for any x,y∈X,there exist u,v∈X such that F(x,y) =G(u,v)and F(y,x) =G(v,u).Suppose that there existφ∈andψ∈such that the following holds:

φdF(x,y),F(u,v)≤ φ

dG(x,y),G(u,v)+dG(y,x),G(v,u) –ψ

d(G(x,y),G(u,v)) +d(G(y,x),G(v,u)) 

for all x,y,u,v∈X with G(x,y)G(u,v)and G(y,x)G(v,u).

Also suppose that either

(a) Fis continuous,or

(b) Xhas the following properties:for any two sequences{xn}and{yn}

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

(ii) if a non-increasing sequence{yn} →y,thenyynfor alln.

If there exists(x,y)∈X×X with

G(x,y)F(x,y) and G(y,x)F(y,x),

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

Kutbiet al.[] introduced the notion ofF-closed set which extended the notion of

F-invariant set as follows.

Deﬁnition .[] LetF:X×X→Xbe a mapping, and letMbe a subset ofX_{. We}

say thatMis anF-closed subset ofX_{if, for all}_x_,_y_,_u_,_v_∈_X_,

(x,y,u,v)∈M ⇒ F(x,y),F(y,x),F(u,v),F(v,u)∈M.

Now, we give the notion of a coupledg-coincidence point and a (Hg,F)-closed set which is useful for our main results.

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Example . LetX=Nendowed with the natural ordering of real number≤. Deﬁne the mappingsF,H:X×X→Xandg:X→XbyF(x,y) = (x+y)_,_H₍_x_,_y_{) =}_x₊_y_and_gx₌_x_/

for allx,y∈X. Note thatFisHg-increasing with respect to≤.

Example . LetX=Rendowed with the natural ordering of real number≤. Deﬁne the mappingsF,H:X×X→Xandg:X→XbyF(x,y) =  +y–x,H(x,y) =x–yandgx=  –x

for allx,y∈X. Note thatF isHg-increasing with respect to≤but notH-increasing. If

H(x,y) =x–y≤H(u,v) =u–v, thenF(x,y) =  +y–x≥F(u,v) =  +v–u, but ifH(gx,gy) =

y–x≤H(gu,gv) =v–u, thenF(x,y) =  +y–x≤F(u,v) =  +v–u.

Deﬁnition . An element (x,y)∈X×Xis called a coupledg-coincidence point of map-pingsF,H:X×X→Xandg:X→XifF(x,y) =H(gx,gy) andF(y,x) =H(gy,gx). Example . LetX=Rendowed with the natural ordering of real number≤. Deﬁne the mappingsF,H:X×X→Xandg:X→XbyF(x,y) =x₊y

 ,H(x,y) =x+yandgx=

x



for allx,y∈X. Note that (, ) is a coupledg-coincidence point.

Deﬁnition . Let (X,d) be a metric space andF,H:X×X→Xbe two mappings and

g:X→X. LetMbe a subset ofX_{. We say that}_M_{is an (}_H

g,F)-closed subset ofXif for allx,y,u,v∈X,

H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu)∈M

⇒ F(x,y),F(y,x),F(u,v),F(v,u)∈M.

Deﬁnition . Let (X,d) be a metric space andH:X×X→Xbe a given mapping. Let

Mbe a subset ofX_{. We say that}_M_{satisﬁes the transitive property if and only if, for all}

x,y,u,v,a,b∈X,

H(x,y),H(y,x),H(u,v),H(v,u)∈M and

H(u,v),H(v,u),H(a,b),H(b,a)∈M

⇒ H(x,y),H(y,x),H(a,b),H(b,a)∈M.

Remark The setM=X_{is a trivially (}_H

g,F)-closed set, which satisﬁes the transitive prop-erty.

Example . LetX={, , , }endowed with the usual metric andF,H:X×X→Xbe deﬁned by

F(x,y) = ⎧ ⎨ ⎩

, x,y∈ {, }, , otherwise,

and

H(x,y) = ⎧ ⎨ ⎩

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Letg:X→Xbe deﬁned by

g(x) = ⎧ ⎨ ⎩

, x∈ {, }, , otherwise.

It is easy to see thatM={, }_⊂_X_{is an (}_H

g,F)-closed set but not anF-closed set. Deﬁnition . Let F,H:X×X→X andg:X→X. We say that the pair{F,H} is

g-generalized compatible if ⎧

⎨ ⎩

d(F(H(gxn,gyn),H(gyn,gxn)),H(gF(xn,yn),gF(yn,xn)))→ asn→+∞,

d(F(H(gyn,gxn),H(gxn,gyn)),H(gF(yn,xn),gF(xn,yn)))→ asn→+∞, whenever (xn),(gxn),(yn), and (gyn) are sequences inXsuch that

⎧ ⎨ ⎩

limn→∞F(xn,yn) =limn→∞H(gxn,gyn) =t,

limn→∞F(yn,xn) =limn→∞H(gyn,gxn) =t.

Example . Let (X,d) be a metric space endowed with a partial order. Letg:X→X

andF,H:X×X→Xbe two mappings such thatFisHg-increasing with respect to. Deﬁne a subsetM⊆X_by

M= (x,y,u,v)∈X:xu,yv.

Let (H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu))∈M. Since F isHg-increasing with respect to, we haveF(x,y)F(u,v) andF(y,x)F(v,u), and this implies that (F(x,y),F(y,x),

F(u,v),F(v,u))∈M. ThenMis an (Hg,F)-closed subset ofXwhich satisﬁes the transitive property.

3 Main results

Letdenote the set of functionsϕ: [,∞)→[,∞) satisfying

. ϕ(t) <tfor allt> , . limr→t+ϕ(r) <tfor allt> .

Now, we state our ﬁrst main theorem which guarantees a coupledg-coincidence point. Theorem . Let(X,d)be a complete metric space and M be a non-empty subset of X_.

Assume that g:X→X is continuous and F,H:X×X→X are two generalized compatible mappings such that H is continuous,and for any x,y∈X, there exist u,v∈X such that F(x,y) =H(gu,gv)and F(y,x) =H(gv,gu).Suppose that there existsϕ∈such that the following holds:

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

≤ϕdH(gx,gy),H(gu,gv)+dH(gy,gx),H(gv,gu) ()

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Also suppose that F is continuous.If there exists(x,y)∈X such that

H(gx,gy),H(gy,gx),F(x,y),F(y,x)

∈M

and M is(Hg,F)-closed,then there exists(x,y)∈X×X such that H(gx,gy) =F(x,y)and

H(gy,gx) =F(y,x),that is,F and H have a coupled g-coincidence point.

Proof Letx,y∈Xbe such that

∈M.

From the assumption, there exist (x,y)∈X×Xsuch that

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

Again from assumption, we can choosex,y∈Xsuch that

F(x,y) =H(gx,gy) and F(y,x) =H(gy,gx).

By repeating this argument, we can construct sequences such that for alln≥,

F(xn,yn) =H(gxn+,gyn+) and F(yn,xn) =H(gyn+,gxn+). ()

Since (H(gx,gy),H(gy,gx),F(x,y),F(y,x))∈MandMis (Hg,F)-closed, we get

=H(gx,gy),H(gy,gx),H(gx,gy),H(gy,gx)

∈M

⇒ F(x,y),F(y,x),F(x,y),F(y,x)

=H(gx,gy),H(gy,gx),H(gx,gy),H(gy,gx)

∈M.

Again, using the fact thatMis (Hg,F)-closed, we have

H(gx,gy),H(gy,gx),H(gx,gy),H(gy,gx)

∈M

⇒ F(x,y),F(y,x),F(x,y),F(y,x)

=H(gx,gy),H(gy,gx),H(gx,gy),H(gy,gx)

∈M.

Continuing this process, for alln≥, we get

H(gxn,gyn),H(gyn,gxn),H(gxn+,gyn+),H(gyn+,gxn+)

∈M. ()

For alln≥, denote

δn:=d

H(gxn,gyn),H(gxn+,gyn+)

+dH(gyn,gxn),H(gyn+,gxn+)

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We can suppose thatδn>  for alln≥. If not, (xn,yn) will be a coupledg-coincidence point and the proof is ﬁnished. From (), () and (), we have

dH(gxn+,gyn+),H(gxn+,gyn+)

+dH(gyn+,gxn+),H(gyn+,gxn+)

=dF(xn,yn),F(xn+,yn+)

+dF(yn,xn),F(yn+,xn+)

≤ϕdH(gxn,gyn),H(gxn+,gyn+)

+dH(gyn,gxn),H(gyn+,gxn+)

=ϕ(δn). ()

Therefore, the sequence{δn}∞n=satisﬁes

δn+≤ϕ(δn) for alln≥. ()

Using the property ofϕ, it follows that the sequence{δn}∞n= is decreasing. Therefore,

there exists someδ≥ such that

lim

n→∞δn=δ. ()

We shall prove thatδ= . Assume, to the contrary, thatδ> . Then by lettingn→ ∞in () and using the property ofϕ, we have

δ= lim

n→∞δn+≤nlim→∞ϕ(δn) =δnlim→δ+ϕ(δn) <δ,

a contradiction. Thusδ=  and hence

lim

n→∞δn= . ()

We now prove that{H(gxn,gyn)}∞n=and{H(gyn,gxn)}∞n=are Cauchy sequences in (X,d).

Suppose, to the contrary, that at least one of the sequences {H(gxn,gyn)}∞n= or {H(gyn,

gxn)}∞n= is not a Cauchy sequence. Then there exists >  for which we can ﬁnd

sub-sequences{H(gxm(k),gym(k))},{H(gxn(k),gyn(k))}of{H(gxn,gyn)}∞n= and{H(gym(k),gxm(k))}, {H(gyn(k),gxn(k))}of{H(gyn,gxn)}∞n=, respectively, withn(k) >m(k)≥ksuch that

Dk:=d

H(gxm(k),gym(k)),H(gxn(k),gyn(k))

+dH(gym(k),gxm(k)),H(gyn(k),gxn(k))

> . ()

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

dH(gxm(k),gym(k)),H(gxn(k)–,gyn(k)–)

+dH(ym(k),xm(k)),H(gyn(k)–,gxn(k)–)

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Using (), (), and the triangle inequality, we have

<Dk

≤dH(gxm(k),gym(k)),H(gxn(k)–,gyn(k)–)

+dH(gxn(k)–,gyn(k)–),H(gxn(k),gyn(k))

+dH(gym(k),gxm(k)),H(gyn(k)–,gxn(k)–)

+dH(gyn(k)–,gxn(k)–),H(gyn(k),gxn(k))

≤ +δn(k)–. ()

Lettingk→ ∞in () and using (), we get

lim

n→∞Dk= . ()

Again, for allk≥, we have

Dk≤d

H(gxm(k),gym(k)),H(gxm(k)+,gym(k)+)

+dH(gxm(k)+,gym(k)+),H(gxn(k)+,gyn(k)+)

+dH(gxn(k)+,gyn(k)+),H(gxn(k),gyn(k))

+dH(gym(k),gxm(k)),H(gym(k)+,gxm(k)+)

+dH(gym(k)+,gxm(k)+),H(gyn(k)+,gxn(k)+)

+dH(gyn(k)+,gxn(k)+),H(gyn(k),gxn(k))

≤δm(k)+δn(k)+d

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

+dH(gym(k)+,gxm(k)+),H(gyn(k)+,gxn(k)+)

. ()

From () andn(k) >m(k), we have

H(gxm(k),gym(k)),H(gym(k),gxm(k)),

H(gxm(k)+,gym(k)+),H(gym(k)+,gxm(k)+)

∈M

and

H(gxm(k)+,gym(k)+),H(gym(k)+,gxm(k)+),

H(gxm(k)+,gym(k)+),H(gym(k)+,gxm(k)+)

∈M.

UsingMhas the transitive property, we get

H(gxm(k)+,gym(k)+),H(gym(k)+,gxm(k)+)

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Continuing this process, we have

H(gxn(k),gyn(k)),H(gyn(k),gxn(k))

∈M. ()

From (),() and (), we have

dH(gxm(k)+,gym(k)+),H(gyn(k)+,gxn(k)+)

+dH(gxm(k)+,gym(k)+),H(gyn(k)+,gxn(k)+)

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

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

≤ϕdH(gxm(k),gym(k)),H(gxn(k),gyn(k))

+dH(gym(k),gxm(k)),H(gyn(k),gxn(k))

=ϕ(Dk), ()

which, by (), yields

Dk≤δm(k)+δn(k)+ϕ(Dk). ()

Lettingk→ ∞in the above inequality and using () and (), we get

= lim

k→∞Dk≤klim→∞

δm(k)+δn(k)+ϕ(Dk)

= lim Dk→+

ϕ(Dk) < ,

which is a contradiction. Hence, {H(gxn,gyn)}∞n= and {H(gyn,gxn)}∞n= are Cauchy

se-quences in (X,d). Since (X,d) is complete and (), there existx,y∈Xsuch that lim

n→∞H(gxn,gyn) =nlim→∞F(xn,yn) =x and

lim

n→∞H(gyn,gxn) =nlim→∞F(yn,xn) =y.

()

Since the pair{F,H}satisﬁes the generalized compatibility, from () we have

lim n→∞d

FH(gxn,gyn),H(gyn,gxn)

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

=  and

lim n→∞d

FH(gyn,gxn),H(gxn,gyn)

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

= .

()

SinceFis continuous, for alln≥, by the triangle inequality, we have

dH(gx,gy),FH(gxn,gyn),H(gyn,gxn)

≤dH(gx,gy),HgF(xn,yn),gF(yn,xn)

+dHgF(xn,yn),gF(yn,xn)

,FH(gxn,gyn),H(gyn,gxn)

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and

dH(gy,gx),FH(gyn,gxn),H(gxn,gyn)

≤dH(gy,gx),HgF(yn,xn),gF(xn,yn)

+dHgF(yn,xn),gF(xn,yn)

,FH(gyn,gxn),H(gxn,gyn)

. ()

Taking the limit asn→ ∞in () and (), using (),(), and the fact thatH,Fandgare continuous, we have

H(gx,gy) =F(x,y) and H(gy,gx) =F(y,x). ()

Therefore, (x,y) is a coupledg-coincidence point ofFandH. In our next theorem, we drop the continuity ofF.

Theorem . Let(X,d)be a complete metric space and M be a non-empty subset of X_.

Assume that g:X→X is continuous and F,H:X×X→X are two generalized compatible mappings such that H is continuous and for any x,y∈X,there exist u,v∈X such that F(x,y) =H(gu,gv)and F(y,x) =H(gv,gu).Suppose that there existsϕ∈such that the following holds:

≤ϕdH(gx,gy),H(gu,gv)+dH(gy,gx),H(gv,gu)

for all x,y,u,v∈X with(H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu))∈M.

Also suppose that(g(X),d)is a complete metric space,H(X×X)⊆g(X)and any two se-quences{xn}and{yn}with(xn,yn,xn+,yn+)∈M and{H(xn,yn)} →H(x,y),{H(yn,xn)} →

H(y,x)for all n≥implies

H(xn,yn),H(yn,xn),H(x,y),H(y,x)

∈M

for all n≥.If there exist x,y∈X such that

∈M

Proof As in the proof of Theorem .. Since (g(X),d) is complete, there existx,y∈Xsuch thatgx,gy∈g(X), and we have

lim

n→∞H(gxn,gyn) =nlim→∞F(xn,yn) =gx and

lim

n→∞H(gyn,gxn) =nlim→∞F(yn,xn) =gy.

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Since the pair{F,H}satisﬁes the generalized compatibility,His continuous and by (), we have

lim n→∞H

H(gxn,gyn),H(gyn,gxn)

=H(gx,gy) = lim

n→∞H

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

= lim n→∞F

H(gxn,gyn),H(gyn,gxn)

()

and

lim n→∞H

H(gyn,gxn),H(gxn,gyn)

=H(gy,gx) = lim

n→∞H

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

= lim n→∞F

H(gyn,gxn),H(gxn,gyn)

. ()

From (), (), (), by assumption, for alln≥, we have

HH(gxn,gyn),H(gyn,gxn)

,HH(gyn,gxn),H(gxn,gyn)

,

H(gx,gy),H(gy,gx)∈M. ()

Then, by (), (), (), and the triangle inequality, we have

dH(gx,gy),F(x,y)+dH(gy,gx),F(y,x)

≤dH(gx,gy),FH(gxn,gyn),H(gyn,gxn)

+dFH(gxn,gyn),H(gyn,gxn)

,F(x,y) +dH(gy,gx),FH(gyn,gxn),H(gxn,gyn)

+dFH(gyn,gxn),H(gxn,gyn)

,F(y,x)

≤ϕdHH(gxn,gyn),H(gyn,gxn)

,H(gx,gy) +dHH(gyn,gxn),H(gxn,gyn)

,H(gy,gx) +dH(gx,gy),FH(gxn,gyn),H(gyn,gxn)

+dH(gy,gx),FH(gyn,gxn),H(gxn,gyn)

.

Letting nown→ ∞in the above inequality and using the property ofϕthatlimr→+ϕ(r) = , we have

dH(gx,gy),F(x,y)+dH(gy,gx),F(y,x)= ,

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Example . LetX= [, ],d(x,y) =|x–y|andF,H:X×X→Xbe deﬁned by

F(x,y) =x

₊_y

 and H(x,y) =x+y.

Letg:X→Xbe deﬁned bygx=_x. Clearly,Hdoes not satisfy the mixed monotone prop-erty. Now, we prove that for anyx,y∈X, there existu,v∈Xsuch thatF(x,y) =H(gu,gv) andF(y,x) =H(gv,gu). It is easy to see that there existu=x

,v=

y

 ∈Xsuch that

F(x,y) =H

x , y 

=H(gu,gv), F(y,x) =H

y , x 

=H(gv,gu).

Now, we prove that the pair{F,H}satisﬁes the generalized compatibility hypothesis. Let

{xn}∞n=and{yn}∞n=be two sequences inXsuch that

t= lim

n→∞F(xn,yn) =nlim→∞H(gxn,gyn) and

t= lim

n→∞F(yn,xn) =nlim→∞H(gyn,gxn).

Then we must havet=  =t, and it is easy to prove that

⎧ ⎨ ⎩

limn→∞d(F(H(xn,yn),H(yn,xn)),H(gF(xn,yn),gF(yn,xn))) =  lim_n_→∞d(F(H(yn,xn),H(xn,yn)),H(gF(yn,xn),gF(xn,yn))) = .

Now, for allx,y,u,v∈Xwith (H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu))∈M=X, and let

ϕ: [, +∞)→[, +∞) be a function deﬁned byϕ(t) = _t, then we have

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

₊_y

 –

u+v

 +y

₊_x

 –

v+u



= (x

_–_u_{) + (}_y_–_v₎



= (x–u)(x+u)

 +

(y–v)(y+v) 

≤ 

(x+y) – (u+v) =ϕ(x+y) – (u+v) =ϕ(x+y)

 – (u+v)



+(y+x)  –

(v+u) 

=ϕdH(gx,gy),H(gu,gv)+dH(gy,gx),H(gv,gu).

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Example . LetX= [, ],d(x,y) =|x–y|andF,H:X×X→Xbe deﬁned by

F(x,y) = ⎧ ⎨ ⎩

x–y

 ifx≥y,

 ifx<y.

and

H(x,y) = ⎧ ⎨ ⎩

x+y ifx≥y,  ifx<y.

Letg:X→Xbygx=x_. Clearly,Hdoes not satisfy the mixed monotone property and ifx>y,u=v= , consider

H(gx,gy)≤H(gu,gv) ⇒ x+y  ≤

u+v



butF(x,y) =x

_–_y

 =

(x–y)(x+y)

 >  =F(u,v). ThenFis notHg-increasing.

Now, we prove that for anyx,y∈X, there existu,v∈Xsuch thatF(x,y) =H(gu,gv) and

F(y,x) =H(gv,gu). It is easy to see the following cases.

Case: Ifx=y, then we haveF(y,x) =F(x,y) =  =H(g,g) =H(, ).

Case: Ifx>y, then (x–y)x> (x–y)y, and we have

F(x,y) =x

_–_y

 =

(x–y)x+ (x–y)y

 =H

g(x–y)x

 ,g (x–y)y



and

F(y,x) =  =H

g(x–y)y

 ,g (x–y)x



.

Case: Ify>x, then (y–x)y> (y–x)x, and we have

F(y,x) =y

_–_x

 =

(y–x)y+ (y–x)x

 =H

g(y–x)y

 ,g (y–x)x



and

F(x,y) =  =H

g(y–x)x

 ,g (y–x)y



.

Now, we prove that the pair{F,G}satisﬁes the generalized compatibility hypothesis. Let

{xn}∞n=and{yn}∞n=be two sequences inXsuch that

t= lim

n→∞F(xn,yn) =nlim→∞G(gxn,gyn) and

t= lim

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Then we must havet=  =t, and it is easy to prove that

⎧ ⎨ ⎩

limn→∞d(F(H(xn,yn),H(yn,xn)),H(gF(xn,yn),gF(yn,xn))) = , limn→∞d(F(H(yn,xn),H(xn,yn)),H(gF(yn,xn),gF(xn,yn))) = .

Now, for allx,y,u,v∈Xwith (H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu))∈M=X_{, and let} ϕ: [, +∞)→[, +∞) be a function deﬁned byϕ(t) = _t, then we have

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

_–_y

 –

u_–_v

 +y

_–_x

 –

v_–_u



= x

_–_y

 –

u–v



= (x–y)(x+y)

 –

(u–v)(u+v) 

≤ 

(x+y) – (u+v) =ϕ(x+y) – (u+v) =ϕ(x+y)

 – (u+v)



+(y+x)  –

(v+u) 

=ϕdH(gx,gy),H(gu,gv)+dH(gy,gx),H(gv,gu).

Therefore, condition () is satisﬁed. Thus, all the requirements of Theorem . are satisﬁed and (, ) is a coupledg-coincidence point ofFandH.

Next, we show the uniqueness of the coupled coincidence point and the coupled ﬁxed point ofFandH.

Theorem . In addition to the hypotheses of Theorem ., suppose that for every

(x,y), (z,t)∈X×X,there exists(u,v)∈X×X such that

H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu)∈M and

H(gz,gt),H(gt,gz),H(gu,gv),H(gv,gu)∈M.

Then F and G have a unique coupled g-coincidence point.

Proof From Theorem ., we know thatF andH have a coupledg-coincidence point. Suppose that (x,y), (z,t) are coupledg-coincidence points ofFandH, that is,

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

F(z,t) =H(gz,gt), F(t,z) =H(gt,gz).

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Now, we show thatH(gx,gy) =H(gz,gt) andH(gy,gx) =H(gt,gz). By the hypothesis, there exists (u,v)∈X×Xsuch that

H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu)∈M and

We put gu=gu andgv=gv and deﬁne two sequences {H(gun,gvn)}∞n= and {H(gvn,

gun)}∞n=as follows:

F(un,vn) =H(gun+,gvn+) and F(vn,un) =H(gvn+,gun+)

for alln≥.

SinceMis (Hg,F)-closed and (H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu))∈M, we have

H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu)

=H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu)

∈M

⇒ F(x,y),F(y,x),F(u,v),F(v,u)

=H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu)

∈M.

From (H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu))∈M, if we use again the property of

(Hg,F)-closed, then

H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu)

∈M

⇒ F(x,y),F(y,x),F(u,v),F(v,u)

=H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu)

∈M.

By repeating this process, we get

H(gx,gy),H(gy,gx),H(gun,gvn),H(gvn,gun)

∈M for alln≥. ()

Using (), () and (), we have

dH(gx,gy),H(gun+,gvn+)

+dH(gy,gx),H(gvn+,gun+)

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

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

≤ϕdH(gx,gy),H(gun,gvn)

+dH(gy,gx),H(gvn,gun)

for alln. ()

Using the property thatϕ(t) <tand repeating this process, we get

≤ϕndH(gx,gy),H(gu,gv)

+dH(gy,gx),H(gv,gu)

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Fromϕ(t) <tandlimr→t+ϕ(r) <t, it follows thatlim_n_→∞ϕn(t) =  for eacht> . There-fore, from () we have

lim n→∞

= . ()

This implies that

lim n→∞d

H(gx,gy),H(gun+,gvn+)

=  and

lim n→∞d

H(gy,gx),H(gvn+,gun+)

= .

()

Similarly, we show that

lim n→∞d

H(gz,gt),H(gun+,gvn+)

=  and

lim n→∞d

H(gt,gz),H(gvn+,gun+)

= .

()

From () and (), we have

H(gx,gy) =H(gz,gt) and H(gy,gx) =H(gt,gz). ()

Next, we give some application of our results to coupled coincidence point theorems in partially metric spaces withFisHg-increasing with respect toandHhas the mixed monotone property.

Corollary . Let(X,)be a partially ordered set and M be a non-empty subset of X,and let d be a metric on X such that(X,d)is a complete metric space.Assume that g:X→X is continuous and F,H:X×X→X are two generalized compatible mappings such that F is Hg-increasing with respect to,H is continuous and has the mixed monotone property.

Suppose that for any x,y∈X,there exist u,v∈X such that F(x,y) =H(gu,gv)and F(y,x) =

H(gv,gu).Suppose that there existsϕ∈such that the following holds:

for all x,y,u,v∈X with H(gx,gy)H(gu,gv)and H(gy,gx)H(gv,gu).

Also suppose that F is continuous.If there exist x,y∈X such that

H(gx,gy)F(x,y) and H(gy,gx)F(y,x),

Proof We deﬁne the subsetM⊆X_by

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From Example ., M is an (Hg,F)-closed set which satisﬁes the transitive property. For all x,y,u,v∈ X with (H(gx,gy) H(gu,gv) and H(gy,gx) H(gv,gu)), we have (H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu))∈M. By (),we get

≤ϕdH(gx,gy),H(gu,gv)+dH(gy,gx),H(gv,gu).

Since (x,y)∈X×Xwith

H(gx,gy)F(x,y) and H(gy,gx)F(y,x), ()

we have

∈M.

For the assumption holds,Fis continuous. By the assumption of Theorem ., we have

H(gx,gy) =F(x,y) andH(gy,gx) =F(y,x). Corollary . Let(X,)be a partially ordered set and M be a non-empty subset of X_,_and

let d be a metric on X such that(X,d)is a complete metric space.Assume that g:X→X is continuous and F,H:X×X→X are two generalized compatible mappings such that F is Hg-increasing with respect to,H is continuous and has the mixed monotone property.

Suppose that for any x,y∈X,there exist u,v∈X such that F(x,y) =H(gu,gv)and F(y,x) =

H(gv,gu).Suppose that there existsϕ∈such that the following holds:

for all x,y,u,v∈X with H(gx,gy)H(gu,gv)and H(gy,gx)H(gv,gu).

Also suppose that(g(X),d)is a complete metric space and X has the following properties:

for any two sequences{xn}and{yn},

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

(ii) if a non-increasing sequence{yn} →y,thenyynfor alln.

If there exist x,y∈X such that

H(gx,gy)F(x,y) and H(gy,gx)F(y,x)

M= (x,y,u,v)∈X:xuandyv.

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(H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu))∈M. Let (x,y)∈X×Xwith

H(gx,gy)F(x,y) and H(gy,gx)F(y,x). ()

Using () andFisHg-increasing with respect to, we have

H(gxn,gyn)H(gxn+,gyn+) and H(gyn,gxn)H(gyn+,gxn+) for alln.

Therefore, (H(gxn,gyn),H(gyn,gxn),H(gxn+,gyn+),H(gyn+,gxn+))∈M.

From (g(X),d) is complete, as in Theorem ., we have two Cauchy sequences{H(gxn,

gyn)}∞n= and{H(gyn,gxn)}∞n= such that{H(gxn,gyn)}∞n= is a non-decreasing sequence in

X with H(gxn,gyn)→gx and {H(gyn,gxn)}∞n= is a non-increasing sequence in X with

H(gyn,gxn)→gy. Using assumption, we have

H(gxn,gyn)gx and H(gyn,gxn)gy for alln.

SinceHhas the mixed monotone property, we have

H(gx,gy),

HH(gyn,gxn),H(gxn,gyn)

H(gy,gx).

Therefore, we have

,HH(gyn,gxn),H(gxn,gyn)

,

H(gx,gy),H(gy,gx)∈M

for alln≥. Now, since all the hypotheses of Theorem . hold, thenF andHhave a coupledg-coincidence point. The proof is completed. Corollary . In addition to the hypotheses of Corollary ., suppose that for every

(x,y), (z,t)∈X×X,there exists(u,v)∈X×X such that(gu,gv)is comparable to(gx,gy)

and(gz,gt).Then F and H have a unique coupled g-coincidence point.

M= (x,y,u,v)∈X:xuandyv.

From Example .,Mis an (Hg,F)-closed set which satisﬁes the transitive property. Thus, the proof of the existence of a coupled coincidence point is straightforward by following the same lines as in the proof of Corollary ..

Next, we show the uniqueness of a coupledg-coincidence point ofFandH. Since for all (x,y), (z,t)∈X×X, there exists (u,v)∈X×Xsuch that

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and

H(gz,gt)H(gu,gv), H(gt,gz)H(gv,gu), we can conclude that

H(gx,gy),H(gy,gx),H(gu,gv),H(gv,gu)∈M and

Therefore, since all the hypotheses of Theorem . hold,FandHhave a uniqueg-coupled coincidence point. The proof is completed.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the ﬁnal manuscript.

Acknowledgements

This research was supported by Chiang Mai University and the authors would like to express sincere appreciation to Prof. Suthep Suantai for very helpful suggestions and many kind comments.

Received: 10 June 2014 Accepted: 26 August 2014 Published:26 Sep 2014 References

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