Almost contractive coupled mapping in ordered complete metric spaces

(1)

R E S E A R C H

Open Access

Almost contractive coupled mapping in

ordered complete metric spaces

Wasﬁ Shatanawi

1

_{, Reza Saadati}

2

_{and Choonkil Park}

3*

*_{Correspondence:}

[email protected]

3_{Department of Mathematics,}

Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea

Full list of author information is available at the end of the article

Abstract

In this paper, we introduce the notion of almost contractive mappingF:X×X→X

with respect to the mappingg:X→Xand establish some existence and uniqueness theorems of a coupled common coincidence point in ordered complete metric spaces. Also, we introduce an example to support our main results. Our results generalize several well-known comparable results in the literature.

MSC: 54H25; 47H10; 34B15

Keywords: coupled ﬁxed point; partially ordered set; mixed monotone property

1 Introduction and preliminaries

The existence and uniqueness theorems of a fixed point in complete metric spaces play an important role in constructing methods for solving problems in differential equations, matrix equations, and integral equations. Furthermore, the fixed point theory is a crucial method in numerical analysis to present a way for solving and approximating the roots of many equations in real analysis. One of the main theorems on a fixed point is the Banach contraction theorem []. Many authors generalized the Banach contraction theorem in different metric spaces in different ways. For some works on fixed point theory, we refer the readers to [–]. The study of a coupled fixed point was initiated by Bhaskar and Lakshmikantham []. Bhaskar and Lakshmikantham [] obtained some nice results on a coupled fixed point and applied their results to solve a pair of differential equations. For some results on a coupled fixed point in ordered metric spaces, we refer the reader to [–].

The following deﬁnitions will be needed in the sequel.

Deﬁnition . Let (X,) be a partially ordered set andF:X×X→X. The mappingFis

said to have the mixed monotone property ifF(x,y) is monotone non-decreasing inxand is monotone non-increasing iny, that is, for any

x,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).

(2)

Deﬁnition . We call an element (x,y)∈X×Xa coupled ﬁxed point of the mapping

F:X×X→Xif

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

Deﬁnition .[] Let (X,) be a partially ordered set andF:X×X→Xandg:X→X. The mappingFis said to have the mixedg-monotone property ifFis monotoneg -non-decreasing in its ﬁrst argument and is monotoneg-non-increasing in its second argument, 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 . An element (x,y)∈X×Xis called a coupled coincidence point of the mappingsF:X×X→Xandg:X→Xif

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

The main results of Bhaskar and Lakshmikantham in [] are the following.

Theorem .[] Let(X,)be a partially ordered set and d be a metric on X such that

(X,d)is a complete metric space.Let F:X×X→X be a continuous mapping having the mixed monotone property on X.Assume that there exists a k∈[, )with

dF(x,y),F(u,v)≤k 

d(x,u) +d(y,v) ∀xu and yv.

If there exist two elements x,y∈X with

xF(x,y) and yF(y,x),

then there exist x,y∈X such that

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

Theorem .[] Let(X,)be a partially ordered set and d be a metric on X such that

(X,d)is a complete metric space.Assume that X has the following property: (i) if a nondecreasing sequence{xn}inXconverges tox∈X,thenxnxfor alln,

(ii) if a nonincreasing sequence{yn}inXconverges toy∈X,thenynyfor alln. Let F:X×X→X be a mapping having the mixed monotone property on X.Assume that there exists k∈[, )with

dF(x,y),F(u,v)≤k 

(3)

If there exist two elements x,y∈X with

xF(x,y) and yF(y,x),

then there exist x,y∈X such that

Deﬁnition . Let (X,d) be a metric space andF:X×X→Xandg:X→Xbe mappings. We say thatFandgcommute if

Fg(x),g(y)=gF(x,y)

for allx,y∈X.

Nashine and Shatanawi [] proved the following coupled coincidence point theorems.

Theorem . [] Let (X,d,) be an ordered metric space. Let F :X×X →X and g:X→X be mappings 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 exist non-negative real numbersα,β,L withα+β< such that

dF(x,y),F(u,v)≤αmindF(x,y),g(x),dF(u,v),g(x)

+βmindF(x,y),g(u),dF(u,v),g(u)

+LmindF(x,y),g(u),dF(u,v),g(x) ()

for all(x,y), (u,v)∈X×X with g(x)g(u)and g(y)g(v).Further suppose that 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 nondecreasing sequence{xn}inXconverges tox∈X,thenxnxfor alln,

(ii) if a nonincreasing sequence{yn}inXconverges toy∈X,thenynyfor alln. Then there exist x,y∈X such that

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

that is,F and g have a coupled coincidence point(x,y)∈X×X.

Theorem .[] Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let F:X×X→X and g:X→X be mappings 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 exist non-negative real numbersα,β,L withα+β< such that

dF(x,y),F(u,v)≤αmindF(x,y),g(x),dF(u,v),g(x)

+βmindF(x,y),g(u),dF(u,v),g(u)

(4)

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

g(X),g is continuous nondecreasing and commutes with F,and also suppose that either

(a) Fis continuous,or

(b) Xhas the following property:

Berinde [–] initiated the concept of almost contractions and studied many inter-esting ﬁxed point theorems for a Ćirić strong almost contraction. So, it is fundamental to recall the following deﬁnition.

Deﬁnition .[] A single-valued mapping f :X×Xis called a Ćirić strong almost

contraction if there exist a constantα∈[, ) and someL≥ such that

d(fx,fy)≤αM(x,y) +Ld(y,fx)

for allx,y∈X, where

M(x,y) =max

d(x,y),d(x,fx),d(y,fy),d(x,fy) +d(y,fx)

 .

The aim of this paper is to introduce the notion of almost contractive mappingF:X× X→Xwith respect to the mappingg:X→Xand present some uniqueness and existence theorems of coupled ﬁxed and coincidence point. Our results generalize Theorems .-..

2 Main theorems

We start with the following deﬁnition.

Deﬁnition . Let (X,d,) be an ordered metric space. We say that the mappingF:X× X→Xis an almost contractive mapping with respect to the mappingg:X→Xif there exist a real numberα∈[, ) and a nonnegative numberLsuch that

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

≤αmaxdg(x),g(u),dg(y),g(v),dF(x,y),g(x),dF(u,v),g(u)

+LmindF(x,y),g(u),dF(u,v),g(x) ()

for all (x,y), (u,v)∈X×Xwithg(x)g(u) andg(y)g(v).

Theorem . Let(X,d,)be an ordered metric space.Let F:X×X→X and g:X→X

be mappings such that

(5)

() Fhas the mixedg-monotone property onX.

() There exist two elementsx,y∈Xwithg(x)F(x,y)andg(y)F(y,x).

() F(X×X)⊆g(X)andg(X)is a complete subspace ofX.

Also,suppose that X satisﬁes the following properties:

Proof Letx,y∈Xbe such thatg(x)F(x,y) andg(y)F(y,x). SinceF(X×X)⊆ g(X), we can choosex,y∈Xsuch thatg(x) =F(x,y) andg(y) =F(y,x).

In the same way, we constructg(x) =F(x,y) andg(y) =F(y,x).

Continuing in this way, we construct two sequences{xn}and{yn}inXsuch that

g(xn+) =F(xn,yn) and g(yn+) =F(yn,xn) ∀n∈N∪ {}. ()

SinceFhas the mixedg-monotone property, by induction we may show that

g(x)g(x)g(x) · · · g(xn+) · · ·

and

g(y)g(y)g(y) · · · g(yn+) · · ·.

If (g(xn+),g(yn+)) = (g(xn),g(yn)) for some n∈N, then F(xn,yn) =g(xn) andF(yn,xn) = g(yn), that is, (xn,yn) is a coincidence point ofF andg. So we may assume that (g(xn+), g(yn+))= (g(xn),g(yn)) for alln∈N. Letn∈N. Sinceg(xn)g(xn–) andg(yn)g(yn–),

from () and (), we have

dg(xn),g(xn+)

=dF(xn–,yn–),F(xn,yn)

≤αmaxdg(xn–),g(xn)

,dg(yn–),g(yn)

,dF(xn,yn),g(xn)

,

dF(xn–,yn–),g(xn–)

+LmindF(xn,yn),g(xn–)

,dF(xn–,yn–),g(xn)

=αmaxdg(xn–),g(xn)

,dg(xn+),g(xn)

,dg(xn),g(xn–)

+Lmindg(xn+),g(xn–)

,dg(xn),g(xn)

=αmaxdg(xn–),g(xn)

,dg(xn+),g(xn)

.

(6)

d(g(yn–),g(yn)) = . Therefored(g(xn–),g(yn–)) =d(g(xn),g(yn)), a contradiction. Thus

maxdg(xn–),g(xn)

,dg(xn+),g(xn)

=maxdg(xn–),g(xn)

.

Therefore

dg(xn+),g(xn)

. ()

Similarly, we may show that

dg(yn),g(yn+)

. ()

From () and (), we have

maxdg(xn+),g(xn)

,dg(yn),g(yn+)

. ()

Repeating ()n-times, we get

maxdg(xn+),g(xn)

,dg(yn),g(yn+)

≤αnmaxdg(x),g(x)

,dg(y),g(y)

. ()

Now, we shall prove that{g(xn)}and{g(yn)}are Cauchy sequences ing(X).

For eachm≥n, we have

dg(xm),g(xn)

≤dg(xn),g(xn+)

+dg(xn+),g(xn+)

+· · ·

+dg(xm–),g(xm)

≤αnmaxdg(x),g(x)

,dg(y),g(y)

+· · ·

+αm–maxdg(x),g(x)

,dg(y),g(y)

≤ αn

 –αmax

dg(x),g(x)

,dg(y),g(y)

.

Lettingn,m→+∞in the above inequalities, we get that{g(xn)}is a Cauchy sequence

ing(X). Similarly, we may show that{g(yn)}is a Cauchy sequence ing(X). Sinceg(X) is

a complete subspace ofX, there exists (x,y)∈X×Xsuch thatg(xn)→g(x) andg(yn)→ g(y). Since{g(xn)}is a non-decreasing sequence andg(xn)→g(x) and as{g(yn)}is a

non-increasing sequence andg(yn)→g(y), by the assumption we haveg(xn)g(x) andg(yn) g(y) for alln. Since

dg(xn+),F(x,y)

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

(7)

≤αmaxdg(xn),g(x)

,dg(yn),g(y)

,dg(xn+),g(xn)

,dF(x,y),g(x)

+Lmindg(xn+),g(x)

,dF(x,y),g(xn)

.

Lettingn→ ∞in the above inequality, we get d(g(x),F(x,y)) = . Henceg(x) =F(x,y). Similarly, one can show thatg(y) =F(y,x). Thus we proved thatFandg have a coupled

coincidence point.

Theorem . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let F:X×X→X and g:X→X be mappings such that

() Fis an almost contractive mapping with respect tog. () Fhas the mixedg-monotone property onX.

() There exist two elementsx,y∈Xwithg(x)F(x,y)andg(y)F(y,x).

() F(X×X)⊆g(X).

() gis continuous nondecreasing and commutes withF.

Also suppose that either

Proof As in the proof of Theorem ., we construct two Cauchy sequences (gxn) and (gyn)

inXsuch that (gxn) is a nondecreasing sequence inXand (gyn) is a nonincreasing sequence

inX. SinceXis a complete metric space, there is (x,y)∈X×Xsuch thatgxn→xand gyn→y. Sincegis continuous, we haveg(gxn)→gxandg(gyn)→gy.

Suppose that (a) holds. Since F is continuous, we have F(gxn,gyn) →F(x,y) and F(gyn,gxn)→F(y,x). Also, since g commutes with F and g is continuous, we have F(gxn,gyn) =gF(xn,yn) =g(gxn+)→gx andF(gyn,gxn) =gF(yn,xn) =g(gyn+)→gy. By

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

Second, suppose that (b) holds. Since g(xn) is a nondecreasing sequence such that g(xn)→x,g(yn) is a nonincreasing sequence such thatg(yn)→y, andgis a nondecreasing

function, we get thatg(gxn)gxandg(gyn)g(y) hold for alln∈N. By (), we have

dg(gxn+),F(x,y)

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

≤αmaxdg(gxn),g(x)

,dg(gyn),g(y)

,dg(gxn+),g(gxn)

,dF(x,y),g(x)

+Lmindg(gxn+),g(x)

,dF(x,y),g(gxn)

.

(8)

Corollary . Let(X,)be a partially ordered set and suppose that there is a metric d on X such that(X,d)is a complete metric space.Let F:X×X→X be a mapping such that F 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 exist a real numberα∈[, )and a nonnegative number L such that

dF(x,y),F(u,v)≤αmaxd(x,u),d(y,v),dF(x,y),x,dF(u,v),u

+LmindF(x,y),u,dF(u,v),x ()

for all(x,y), (u,v)∈X×X with xu and yv and also suppose that either

(ii) if a nonincreasing sequence{yn}inXconverges toy∈X,thenynyfor alln, then there exist x,y∈X such that

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

that is,F has a coupled ﬁxed point(x,y)∈X×X.

Proof Follows from Theorem . by takingg=I, the identity mapping.

Let (X,) be a partially ordered set. Then we deﬁne a partial orderon the product spaceX×Xas follows:

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

Now, we prove some uniqueness theorem of a coupled common ﬁxed point of mappings

F:X×X→Xandg:X→X.

Theorem . In addition to the hypotheses of Theorem.,suppose that L= ,α<  ,F and g commute and for every(x,y), (y∗,x∗)∈X×X,there exists(u,v)∈X×X such that

(F(u,v),F(v,u))is comparable to(F(x,y),F(y,x))and(F(x∗,y∗),F(y∗,x∗)).Then F and g have a unique coupled common ﬁxed point,that is,there exists a unique (x,y)∈X×X such that

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

Proof The existence of coupled coincidence points ofFandgfollows from Theorem .. To prove the uniqueness, let (x,y) and (x∗,y∗) be coupled coincidence points ofFandg; that is,g(x) =F(x,y),g(y) =F(y,x),g(x∗) =F(x∗,y∗) andg(y∗) =F(y∗,x∗). Now, we prove that

g(x) =gx∗ and g(y) =gy∗. ()

(9)

g(u) =F(u,v) andg(v) =F(v,u). Then as a similar proof of Theorem ., we construct

two sequences {g(un)}, {g(vn)}in g(X), whereg(un+) =F(un,vn) andg(vn+) =F(vn,un)

for all n∈N. Further, setx=x, y=y, x∗=x∗,y∗=y∗. Deﬁne the sequences{g(xn)}, {g(yn)} in the following way: deﬁnegx =F(x,y) =F(x,y) andgy=F(y,x) =F(y,x).

Also, definegx=F(x,y) andgy=F(y,x). For eachn∈N, definegxn+=F(xn,yn) and gyn+=F(yn,xn). In the same way, we define the sequences{g(x∗n)},{g(y∗n)}. Now, we prove

that

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

Since (x,y) is a coupled coincidence point ofFandg, we haveF(x,y) =g(x) andF(y,x) =

g(y). Thusg(x) =F(x,y) =F(x,y) =g(x) and g(y) =F(y,x) =F(y,x) =g(y).

There-foreg(x)g(x),g(x)g(x),g(y)g(y) andg(y)g(y). SinceFis monotoneg

-non-decreasing on its ﬁrst argument,g(x)g(x), andg(x)g(x), we haveF(x,y)F(x,y)

andF(x,y)F(x,y). Therefore,

F(x,y) =F(x,y). ()

Also, sinceFis monotoneg-non-increasing on its second argument,g(y)g(y) and g(y)g(y), we haveF(x,y)F(x,y) andF(x,y)F(x,y). Therefore,

F(x,y) =F(x,y). ()

From () and (), we have

g(x) =F(x,y) =F(x,y) =g(x).

g(y) =F(y,x) =F(y,x) =g(y).

Note thatg(x)g(x),g(x)g(x),g(y)g(y) andg(y)g(y). SinceFis monotone g-non-decreasing on its ﬁrst argument,g(x)g(x), andg(x)g(x), we haveF(x,y) F(x,y) andF(x,y)F(x,y). Therefore,

F(x,y) =F(x,y). ()

Also, sinceF is monotoneg-non-increasing on its second argument,g(y)g(y) and g(y)g(y), we haveF(x,y)F(x,y) andF(x,y)F(x,y). Therefore,

F(x,y) =F(x,y). ()

From () and (), we have

(10)

g(y) =F(y,x) =F(y,x) =g(y).

Continuing in the same way, we have that

g(xn) =F(x,y) =g(x) and g(yn) =F(y,x) =g(y)

hold for alln∈N. Similarly, we can show that

gx∗_n=Fx∗,y∗=gx∗ and gy∗_n=Fy∗,x∗=gy∗ ∀n∈N

hold for alln∈N. 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,g(x)g(u) andg(y)g(v). SinceFhas the mixedg-monotone

prop-erty ofX, we haveg(x)g(un) andg(y)g(vn) for alln∈N. Also, since (g(x∗),g(y∗)) and

(F(u,v),F(v,u)) = (g(u),g(v)) are comparable, andFhas theg-monotone property, then

we can show that forn∈N, we have that (g(x∗),g(y∗)) and (g(un),g(vn)) are comparable.

Now, if (g(x),g(y)) = (g(uk),g(vk)) for somek∈Nor (g(x∗),g(y∗)) = (g(uk),g(vk)) for some k∈N, then (g(x),g(y)) and (g(x∗),g(y∗)) are comparable, sayg(x)g(x∗) andg(y)g(y∗). Thus from () we have

dg(x),gx∗

=dF(x,y),Fx∗,y∗

≤αmaxdg(x),gx∗,dg(y),gy∗,dF(x,y),g(x),dFx∗,y∗,gx∗

=αmaxdg(x),gx∗,dg(y),gy∗ ()

and

dgy∗,g(y)

=dFy∗,x∗,F(y,x)

≤αmaxdg(y),gy∗,dg(x),gx∗,dFy∗,x∗,gy∗,dF(y,x),g(y)

=αmaxdg(y),gy∗,dg(x),g(y). ()

From () and (), we have

(11)

Sinceα< , we haved(g(x),g(x∗)) =  andd(g(y),g(y∗)) = . Therefore () is satisﬁed. Now, suppose that (g(x),g(y))= (g(un),g(vn)) for alln∈Nand (g(x∗),g(y∗))= (g(un),g(vn)) for all n∈N. Letn∈N. Sinceg(x)g(un) andg(y)g(vn), then from () we have

dg(x),g(un+)

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

≤αmaxdg(x),g(un)

,dg(y),g(vn)

,dF(x,y),g(x),dF(un,vn),g(un)

=αmaxdg(x),g(un)

,dg(y),g(vn)

,dg(un+),g(un)

≤αmaxdg(x),g(un)

,dg(y),g(vn)

,dg(un+),g(x)

+dg(x),g(un)

≤αmaxdg(x),g(un)

,dg(y),g(vn)

, dg(un+),g(x)

, dg(x),g(un)

=αmaxdg(x),g(un)

,dg(y),g(vn)

, dg(un+),g(x)

.

If

maxdg(x),g(un)

,dg(y),g(vn)

, dg(un+),g(x)

= dg(un+),g(x)

thend(g(un+),g(x))≤αd(g(un+),g(x)). Since α< , we haved(g(un+),g(x)) = .

There-fored(g(x),g(un)) =  andd(g(y),g(vn)) =  and hence (g(x),g(y)) = (g(un),g(vn)), a

contra-diction. Thus

dg(x),g(un+)

≤αmaxdg(x),g(un)

,dg(y),g(vn)

≤αmaxdg(x),g(un)

,dg(y),g(vn)

. ()

dg(vn+),g(y)

≤αmaxdg(x),g(un)

,dg(y),g(vn)

. ()

From () and (), we have

maxdg(x),g(un+)

,dg(vn+),g(y)

≤αmaxdg(vn),g(y)

,dg(un),g(x)

,d(g(un+)

. ()

By repeating ()n-times, we have

maxdg(x),g(un+)

,dg(vn+),g(y)

≤αmaxdg(vn),g(y)

,dg(un),g(x)

.. .

≤(α)n+maxdg(x),g(u)

,dg(v),g(y)

.

Lettingn→+∞in the above inequalities, we get that

lim

n→max

dg(x),g(un+)

,dg(vn+),g(y)

(12)

Hence

lim

n→∞d

g(x),g(un+)

=  ()

and

lim

n→∞d

g(y),g(vn+)

= . ()

lim

n→∞d

g(x),g(un+)

=  ()

and

lim

n→∞d

g(y),g(vn+)

= . ()

By the triangle inequality, (), (), () and (),

dg(x),gx∗≤dg(x),g(un+)

+dgx∗,g(un+)

→ asn→ ∞,

dg(y),gy∗≤dg(y),g(vn+)

+dgy∗,g(vn+)

→ asn→ ∞,

we haveg(x) =g(x∗) andg(y) =g(y∗). Thus we have (). This implies that (g(x),g(y)) = (g(x∗),g(y∗)).

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

gg(x)=gF(x,y)=Fg(x),g(y) and gg(y)=gF(y,x)=Fg(y),g(x). ()

Denoteg(x) =z,g(y) =w. Then from ()

g(z) =F(z,w) and g(w) =F(w,z). ()

Thus (z,w) is a coupled coincidence point. Then from () withx∗=zandy∗=wit follows

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

g(z) =z and g(w) =w. ()

From () and (),

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.

(13)

and(F(u,v),F(v,u))is comparable 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 exist a unique(x,y)∈X×X such that

Proof Follows from Theorem . by takingg=I, the identity mapping.

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 Follow the proof of Theorem . step by step until constructing two sequences{xn}

and{yn}inXsuch thatgxn→gxandgyn→gy, where (x,y) is a coincidence point ofF

andg. Supposegxgy, then it is an easy matter to show that

gxngyn and∀n∈N∪ {}.

Thus, by () we have

d(gxn,gyn)

=dF(xn–,yn–),F(yn–,xn–)

≤αmaxd(g(xn–),g(yn–),d

F(xn–,yn–),g(xn–)

,dF(yn–,xn–),g(yn–)

=αmaxdg(xn–),g(yn–)

,dg(xn),g(xn–)

,dg(yn),g(yn–)

.

On taking the limit asn→+∞, we getd(gx,gy) = . Hence

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

A similar argument can be used ifgygx.

Corollary . In addition to the hypotheses of Corollary.,if xand yare comparable and L= ,then F has a coupled ﬁxed point of the form(x,x).

Proof Follows from Theorem . by takingg=I, the identity mapping.

Now, we introduce the following example to support our results.

Example . LetX= [, ]. Then (X,≤) is a partially ordered set with the natural ordering of real numbers. Deﬁne the metricdonXby

d(x,y) =

max{x,y} ifx=y;  ifx=y.

Deﬁneg:X→Xbyg(x) =x_and_F_:_X_×_X_→_X_by

F(x,y) =

(x–y)

 , x>y;

(14)

Then

() g(X)is a complete subset ofX. () F(X×X)⊆g(X).

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

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

≤

max

dg(x),g(u),dg(y),g(v),dF(x,y),g(x),dF(u,v),g(u)

+LmindF(x,y),g(u),dF(u,v),g(x)

for allg(x)≤g(u)andg(y)≥g(v)holds for allx,y,u,v∈Xwithg(x)≤g(u)and

g(y)≥g(v).

Thus, by Theorem .,Fhas a coupled ﬁxed point. Moreover, (, ) is a coupled coinci-dence point ofF.

Proof The proof of ()-() is clear. We divide the proof of () into the following cases. Case : Ifg(x)≤g(y) andg(u)≤g(v), thenx≤yandu≤v. Hence

dF(x,y),F(u,v)=d(, ) = 

≤

max

+LmindF(x,y),g(u),dF(u,v),g(x).

Case : Ifg(x)≤g(y) andg(u) >g(v), thenx≤yandu>v. Hence

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

,(u

_–_v₎



=  

u–v

≤ 

u



=  max

 

u–v,u

=  max

F(u,v),g(u)

=  d

F(u,v),g(u)

≤ 

max

Case : Ifg(x) >g(y) and g(u)≤g(v), thenx>yandu≤v. Hencev≤y<x≤u≤v. Thereforev<v, which is impossible.

(15)

Subcase I:x=uandy=v. Here, we have

dF(x,y),F(u,v)=d(, ) = 

≤ 

max

Subcase II:x=uory=v. Here, we haveu_–_v_>_x_–_y_{. Therefore}

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

(x_–_y₎

 ,

(u_–_v₎



=  

u–v

≤ 

u



=  max

 

u–v,u

=  max

F(u,v),g(u)

=  d

F(u,v),g(u)

≤ 

max

Note that the mappings F andg satisfy all the hypotheses of Theorem . forα =_ and anyL≥. ThusFandg have a coupled coincidence point. Here (, ) is a coupled coincidence point ofFandg.

Remarks

() Theorem . is a special case of Corollary .. () Theorem . is a special case of Corollary .. () Theorem . is a special case of Theorem .. () Theorem . is a special case of Theorem ..

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa, 13115, Jordan.}2_{Department of Mathematics,}

Iran University of Science and Technology, Tehran, Iran.3_{Department of Mathematics, Research Institute for Natural}

Sciences, Hanyang University, Seoul, 133-791, Korea.

Acknowledgements

The authors would like to thank the editor and the referees for helpful comments.

(16)

References

1. Banach, S: Sur les opérations dans les ensembles absraites et leurs applications. Fundam. Math.3, 133-181 (1922) 2. Abdeljawad, T, Karapınar, E, Ta¸s, K: A generalized contraction principle with control functions on partial metric spaces.

Comput. Math. Appl.63, 716-719 (2012)

3. Abdeljawad, T, Karapınar, E, Ta¸s, K: Existence and uniqueness of a common ﬁxed point on partial metric spaces. Appl. Math. Lett.24, 1900-1904 (2011)

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

5. 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.72(5), 2238-2242 (2010)

6. Je Cho, Y, Saadati, R, Wang, S: Common ﬁxed point theorems on generalized distance in ordered cone metric spaces. Comput. Math. Appl.61, 1254-1260 (2011)

7. Drici, Z, McRae, FA, Vasundhara Devi, J: Fixed point theorems in partially ordered metric spaces for operators with PPF dependence. Nonlinear Anal.67(2), 641-647 (2007)

8. Ćirić, LB, Cakić, N, Rajović, M, Ume, JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl.2008, Article ID 131294 (2008)

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

10. Nieto, JJ, Lopez, RR: Existence and uniqueness of ﬁxed point in partially ordered sets and applications to ordinary diﬀerential equations. Acta Math. Sin. Engl. Ser.23(12), 2205-2212 (2007)

11. O’regan, D, Petrutel, A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl.341(2), 1241-1252 (2008)

12. 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)

13. Karapınar, E, Erhan, ˙I: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett.24, 1894-1899 (2011)

14. Shatanawi, W, Samet, B: On (ψ,φ)-weakly contractive condition in partially ordered metric spaces. Comput. Math. Appl. (2011). doi:10.1016/j.camwa.2011.08.033

15. Shatanawi, W: Fixed point theorems for nonlinear weaklyC-contractive mappings in metric spaces. Math. Comput. Model. (2011). doi:10.1016/j.mcm.2011.06.069

16. Wu, Y: New ﬁxed point theorems and applications of mixed monotone operator. J. Math. Anal. Appl.341(2), 883-893 (2008)

17. Wu, Y, Liang, Z: Existence and uniqueness of ﬁxed points for mixed monotone operators with applications. Nonlinear Anal.65(10), 1913-1924 (2006)

18. Bhaskar, TG, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal.65, 1379-1393 (2006)

19. Aydi, H, Damjanovi´c, B, Samet, B, Shatanawi, W: Coupled ﬁxed point theorems for nonlinear contractions in partially orderedG-metric spaces. Math. Comput. Model. (2011). doi:10.1016/j.mcm.2011.05.059

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

21. Karapınar, E: Coupled ﬁxed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl. 59(12), 3656-3668 (2010)

22. Nashine, HK, Shatanawi, W: Coupled common ﬁxed point theorems for pair of commuting mappings in partially ordered complete metric spaces. Comput. Math. Appl. (2011). doi:10.1016/j.camwa.2011.06.042

23. Samet, B: Coupled ﬁxed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal.72, 4508-4517 (2010)

24. Shatanawi, W: Some common coupled ﬁxed point results in cone metric spaces. Int. J. Math. Anal.4, 2381-2388 (2010)

25. Shatanawi, W: Partially ordered cone metric spaces and coupled ﬁxed point results. Comput. Math. Appl.60, 2508-2515 (2010)

26. Sintunavarat, W, Cho, YJ, Kumam, P: Common ﬁxed point theorems forc-distance in ordered cone metric spaces. Comput. Math. Appl.62(4), 1969-1978 (2011)

27. Berinde, V: Some remarks on a fixed point theorem for Ćirić-type almost contractions. Carpath. J. Math.25, 157-162 (2009)

28. Berinde, V: Approximation ﬁxed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum9, 43-53 (2004)

29. Berinde, V: On the approximation fixed points of weak contractive mappings. Carpath. J. Math.19, 7-22 (2003) 30. Berinde, V: General contractive fixed point theorems for Ćirić-type almost contraction in metric spaces. Carpath.

J. Math.24, 10-19 (2008) 10.1186/1029-242X-2013-565