A coupled coincidence point theorem in partially ordered metric spaces with an implicit relation

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R E S E A R C H

Open Access

A coupled coincidence point theorem in

partially ordered metric spaces with an

implicit relation

Selma Gülyaz

1

_{, Erdal Karapınar}

2*

_{and ˙Ilker Savas Yüce}

3

*_{Correspondence:}

[email protected]; [email protected] 2_{Department of Mathematics,}

Atilim University, ˙Incek, Ankara 06836, Turkey

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

Abstract

In this manuscript, we discuss the existence of a coupled coincidence point for mappingsF:X×X→Xandg:X→X, whereFhas the mixedg-monotone property, in the context of partially ordered metric spaces with an implicit relation. Our main theorem improves and extends various results in the literature. We also state some examples to illustrate our work.

MSC: 47H10; 54H25; 46J10; 46J15

Keywords: coupled coincidence point; coupled ﬁxed point; mixed monotone

property; implicit relation; ordered partial metric space;O-compatible

1 Introduction and preliminaries

It is well known that fixed point theory is one of the crucial and very efficient tools in nonlinear functional analysis. This is because its ever-growing use in this field is very ex-tensive in applications. In particular, the effects of fixed point theory are most apparent in fields like economy, computer sciences and engineering including many branches of mathematics. Historically, in , Poincaré initiated first fixed point results. Then, in , Brouwer published a result in this field, which was equivalent to Poincaré’s theorem, which in the simplest terms states that a continuous function from a disk D to itself has a fixed point. But most of the substantial advances in fixed point theory started after the celebrated fixed point result of Banach, known as Banach’s contraction mapping principle, in . This principle can be stated as follows: any contraction in a complete metric space has a unique fixed point. When compared to Browder’s fixed point theorem, the power of Banach’s principle comes from the fact that it guarantees the uniqueness of a fixed point and gives a method to determine the fixed point. These two strengths of Banach’s contrac-tion mapping principle have attracted attencontrac-tion of many prominent mathematicians who aim to broaden the applications of nonlinear functional analysis via fixed point theory in various quantitative sciences.

In the light of these developments, Guo and Lakshmikantham [] defined the notion of a coupled fixed point in . Later, Gnana-Bhaskar and Lakshmikantham [] improved the idea of a coupled fixed point in the category of partially ordered metric spaces by introducing the notion of a mixed monotone mapping and presented certain applications on the solution of periodic boundary value problems. Interested readers may refer to [–

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] and the references therein to follow the development of ﬁxed point theory on partially ordered metric spaces.

For the sake of completeness, we will review the basic deﬁnitions and fundamental re-sults.

Deﬁnition (See []) Let (X,) be a partially ordered set andF:X×X→X. The map-pingFis said to have the mixed monotone property ifF(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).

Deﬁnition (See []) An element (x,y)∈X×Xis called a coupled ﬁxed point of the mappingF:X×X→Xif

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

We state now the main results of Gnana-Bhaskar and Lakshmikantham in [].

Theorem (See []) Let(X,)be a partially ordered set and suppose there exists a metric d on X such that(X,d)is a complete metric space.Assume that there exists a k∈[, )with

dF(x,y),F(u,v)≤k 

d(x,u) +d(y,v)

for all xu and yv.Let either

(a) F:X×X→Xbe a continuous mapping having the mixed monotone property onX,

or

(b) Xhave the following property:

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

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

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).

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Deﬁnition (See []) Let (X,) be a partially ordered set and letF:X×X→Xand

g:X→Xbe two mappings. We sayFhas the mixedg-monotone property ifF(x,y) isg -non-decreasing in its ﬁrst argument and isg-non-increasing in its second argument, that is, for anyx,y∈X,

x,x∈X, gxgx ⇒ F(x,y)F(x,y)

and

y,y∈X, gygy ⇒ F(x,y)F(x,y).

Deﬁnition (See []) An element (x,y)∈X×Xis called a coupled coincident point of the mappingsF:X×X→Xandg:X→Xif

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

Deﬁnition [] The mappingsFandg, whereF:X×X→X,g:X→X, are said to be compatible if

lim

n→∞d

gF(xn,yn),F(gxn,gyn)

=  and

lim

n→∞d

gF(yn,xn),F(gyn,gxn)

= ,

where{xn}and{yn}are sequences inXsuch thatF(xn,yn) =gxn→xandF(yn,xn) =gyn→

yasn→ ∞for allx,y∈Xare satisﬁed.

Deﬁnition (See []) Two self-mappingsAandBare said to be weakly compatible if they commute at their coincidence points,i.e.,ABu=BAuwheneverAu=Bu,u∈X.

Choudhury and Kundu in [] deﬁned the notion of compatibility and showed the re-sult established in [] with a diﬀerent set of conditions. In other words, the authors con-structed their result by assuming thatFandgare compatible mappings. Later, Luong and Thuan [] slightly improved the notion of compatible mappings on partially ordered met-ric spaces, namelyO-compatible mappings. In this paper [], the authors proved some coupled coincidence point theorems forO-compatible type mappings in the context of partially ordered generalized metric spaces.

We recall the concept ofO-compatible mappings as follows.

Deﬁnition (cf.[]) Let (X,,d) be a partially ordered metric space. Two mappings

F:X×X→Xandg:X→Xare said to beO-compatible if

lim

n→∞d

=  and

lim

n→∞d

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where{xn}and{yn}are sequences inXsuch that{gxn},{gyn}are monotone and

lim

n→∞F(xn,yn) =n→∞lim gxn=x

and

lim

n→∞F(yn,xn) =n→∞lim gyn=y

are satisﬁed for somex,y∈X.

Let (X,,d) be a partially ordered metric space. IfF:X×X→Xandg:X→Xare compatible, then they areO-compatible. However, the converse is not true. The following example shows that there exist mappings which areO-compatible but not compatible.

Example  LetX={} ∪[/, ] with the usual metricd(x,y) =|x–y|for allx,y∈X. We consider the following order relation onX:

x,y∈X, xy ⇔ x=y or (x,y)∈(, ), (, ), (, ).

LetF:X×X→Xbe given by

F(x,y) =

⎧ ⎨ ⎩

 ifx,y∈ {} ∪[/, ],  otherwise

andg:X→Xbe deﬁned by

gx=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

 ifx= ,  if /≤x≤,

a––x

(a–) if  <x≤a, 

 ifa<x≤.

ThenFandgareO-compatible but not compatible. To discern this, let{xn}and{yn}be two sequences inXsuch that{gxn}and{gyn}are monotone and

lim

n→∞F(xn,yn) =n→∞lim gxn=x

and

lim

n→∞F(yn,xn) =n→∞lim gyn=y

for somex,y∈X. It is easy to see thatx=y∈ {, }sinceF(xn,yn) =F(yn,xn)∈ {, }. It is

not possible to havex=y= . Assume otherwise. There exists a positive integerN such thatgxn=  andgyn=  for alln>N, since bothgxnandgynare monotone. Then we have

xn,yn∈[/, ] for alln>N, which implies thatF(xn,yn) =  andF(yn,xn) =  for alln>N,

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for some positive integerM. Then, we getxn=  andyn=  for alln>M. As a result, we

obtain

lim

n→∞d

= , lim

n→∞d

= ,

d(gF(xn,yn),F(gxn,gyn)) = , andd(gF(yn,xn),F(gyn,gxn)) = . Therefore,F andg areO

-compatible.

On the other hand, let

xn=  +

(a– )

(n+ ) and yn=  +

(a– ) (n+ ) forn= , , , . . . . Observe that

F(xn,yn) =F

 +(a– ) (n+ ) ,  +

(a– ) (n+ )

= →

and

gxn=g

 +(a– ) (n+ )

=  – 

n+ →

asnapproaches to∞, whereF(yn,xn) =F(xn,yn) andgxn=gyn. But we have

dgF(xn,yn),F(gxn,gyn)

=d

F

 – 

n+ ,  – 

n+ 

,g

=d(, ) = 

which does not approach to  asnapproaches to∞. Hence,Fandgare not compatible.

Remark  In Example , if we leta= /, we obtain the example presented in []. In nonlinear analysis, especially in ﬁxed point theory, implicit relations on metric spaces have been investigated heavily in many articles (see,e.g., [–] and references therein). In this paper, by using the following implicit relation, we examine the existence of a cou-pled coincidence point theorem for mappingsF:X×X→Xandg:X→Xin the con-text of a partial metric space, whereFhas the mixedg-monotone property andF,gare

O-compatible.

LetR+denote the set of all nonnegative real numbers. Also, letdenote the collection

of all functionsϕ:R+→R+which satisfy

(i) ϕis continuous and non-decreasing, (ii) ϕ(t) <tfor eacht> andϕ() = .

We reserveHfor the class of all continuous functionsH:R

+→Rsatisfying

(H) H(t,t,t,t,t,t)is non-increasing intandt,

(H) ifH(z,u,u+v,v,w,u+v)≤, thenz+u≤h(v+w), whereh∈[, ). Example  The following functions lie inH:

• H(t,t,t,t,t,t) =t+t–αt–βt–γt–θt, whereα,β,γ,θare nonnegative

real numbers satisfyingα+β+γ +θ< .

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• H(t,t,t,t,t,t) =t–k{t+t}, wherek∈(,_).

• H(t,t,t,t,t,t) =t+t–h{t+t}, whereh∈(, ).

• H(t,t,t,t,t,t) =t+t–α(t+t), whereα∈[, ).

• H(t,t,t,t,t,t) =t+t–αt–β(t+t) –γ(t+t), whereα,β,γ are

nonnegative real numbers satisfyingα+β+ γ < .

• H(t,t,t,t,t,t) =t+t–αmax{t_,t+_t,t+t–t}, whereα∈[, ).

In this paper, we prove a coupled coincidence point theorem for mappings satisfying such implicit relations.

2 Main result

We start by stating our primary theorem:

Theorem  Let (X,d,) be a partially ordered complete metric space. Suppose that F:X×X→X and g:X→X are two mappings such that F has the mixed g-monotone property.Assume that there exists H∈Hsuch that

H

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

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

≤ (.)

for all x,y,u,v∈X with gxgu and gygv.Suppose also that F(X×X)⊆g(X)and g is continuous on X and O-compatible with F.Additionally,suppose that either

(a) Fis continuous,or

(b) Xhas the properties

(i) if a non-decreasing sequencexn→x,thengxngxfor alln,and

(ii) if a non-increasing sequenceyn→y,thengygynfor alln.

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

gxF(x,y) and gyF(y,x),

then F and g have a coupled coincidence point in X.

Proof Letx,y∈Xbe such thatgxF(x,y) andgyF(y,x). We will construct the

iterative sequences{xn}and{yn}inXas follows:

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

The sequences{xn}and{yn}are well deﬁned sinceF(X×X)⊆g(X). Using the mathemat-ical induction and the fact thatFhas the mixedg-monotone property, we obtain

gxngxn+ and gyngyn+ (.)

for all n≥. If there is a number n∈N∗ ={, , , , . . .} such that gxn =gxn+ and

gyn=gyn+, thengxn=gxn+=F(xn,yn) andgyn=gyn+=F(yn,xn). In this case,

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Assume thatgxn=gxn+ orgyn=gyn+for alln∈N*. Sincegxn+gxnandgyn+gyn,

we have

H

⎛ ⎜ ⎝

d(F(yn+,xn+),F(yn,xn)),d(F(xn+,yn+),F(xn,yn)),

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

d(gyn+,gyn),d(F(xn+,yn+),gxn) +d(F(xn,yn),gxn+) ⎞ ⎟ ⎠≤

by (.). Using the deﬁnitions of{xn}and{yn}in ., we obtain

H

d(gyn+,gyn+),d(gxn+,gxn+),d(gxn+,gxn+) +d(gxn+,gxn),

d(gxn+,gxn),d(gyn+,gyn),d(gxn+,gxn)

≤. (.)

By combining the property (H) ofHwith the triangle inequality, inequalities in (.) turn into

H

d(gyn+,gyn+),d(gxn+,gxn+),d(gxn+,gxn+) +d(gxn+,gxn),

d(gxn+,gxn),d(gyn+,gyn),d(gxn+,gxn+) +d(gxn+,gxn)

≤.

Then, the property (H) ofHimplies that

d(gyn+,gyn+) +d(gxn+,gxn+)≤h

d(gxn+,gxn) +d(gyn+,gyn)

, (.)

whereh∈[, ).

Setdn:=d(gxn+,gxn) +d(gyn+,gyn). Thus, the inequality (.) can be written as

dn+≤hdn≤hn+d for alln= , , , . . . .

We will show that the sequences{gxn}and{gyn}are Cauchy. Without loss of generality, we may assume thatm>n. Then, by the triangle inequality, we have

d(gym,gyn) +d(gxm,gxn)≤ m–

i=n

d(gyi+,gyi) +d(gxi+,gxi)

≤ _m_–

i=n

hi

d

=

_m_–

i=

hi_– n–

i=

hi

d. (.)

Lettingn,m→ ∞in (.), we have

lim

n,m→∞d(gym,gyn) +d(gxm,gxn) = .

Consequently, lim_n_,_m→∞d(gxm,gxn) =  and limn,m→∞d(gym,gyn) = . Hence the

se-quences{gxn}and{gyn}are Cauchy. SinceXis complete, there existx,y∈Xsuch that

lim

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and

lim

n,m→∞d(gyn,gym) =n→∞limd(gyn,y) = . (.)

Thus, we obtain

F(xn,yn) =gxn+→x and F(yn,xn) =gyn+→y (.)

asn→ ∞. SinceFandgareO-compatible, we have

lim

n→∞d

=  (.)

and

lim

n→∞d

=  (.)

by (.).

Now, assume that (a) holds,i.e.,F is continuous. By taking the limit in the following inequality:

dgx,F(gxn,gyn)

≤dgx,gF(xn,yn)

+dgF(xn,yn),F(gxn,gyn)

asn→ ∞, we obtain

dgx,F(x,y)= 

by (.), (.) and the continuity ofFandg. Similarly, we can show thatd(gy,F(y,x)) = . As a result,Fandghave a coupled coincidence point inX,i.e.,gx=F(x,y) andgy=F(y,x). Assume that (b) holds. Since{gxn}is a non-decreasing sequence andgxn→x, we have

ggxngxfor alln∈N*by (i). Similarly, since{gyn}is a non-increasing sequence andgyn→

y, we also haveggyngyfor alln∈N*. Sincegis continuous, using (.), (.) and (.),

we derive that

lim

n→∞d(ggxn,gx) =d(gx,gx) =n→∞limd

gF(xn,yn),gx

= lim

n→∞d

F(gxn,gyn),gx

(.) and

lim

n→∞d(ggyn,gy) =d(gy,gy) =n→∞limd

gF(yn,xn),gy

= lim

n→∞d

F(gyn,gxn),gy

. (.) By the hypothesis of the theorem, we know that

H

⎛ ⎜ ⎝

d(F(gyn,gxn),F(y,x)),d(F(gxn,gyn),F(x,y)),

d(F(gxn,gyn),ggxn) +d(F(x,y),gx),d(ggxn,gx),

d(ggyn,gy),d(F(gxn,gyn),gx) +d(F(x,y),ggxn) ⎞ ⎟ ⎠≤

for everyn∈N*. In the inequality above, we letn→ ∞and use (.) and (.) to obtain

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which implies thatd(gy,F(y,x)) +d(gx,F(x,y))≤h( + ) =  by (H) forh∈[, ). Finally, we ﬁnd thatgx=F(x,y) andgy=F(y,x) completing the proof.

Example  Let (X,d,),Fandgbe deﬁned as in Example . We see that

• Xis complete andXhas the properties

(i) if a non-decreasing sequencexn→x, thengxngxfor alln∈N*,

(ii) if a non-increasing sequenceyn→y, thengygynfor alln∈N*.

• F(X×X) ={, } ⊂ {} ∪[/, ] =g(X). • gis continuous andgandFareO-compatible.

• There existx= andy= such thatgxF(x,y)andgyF(y,x).

• Fhas the mixedg-monotone property, which can be proved as follows: Let y,x,x∈Xsuch thatgxgx. There are two cases to consider:

() Ifgx=gx, thenx= andx= orx,x∈[/, ]orx,x∈(,a]or

x,x∈(a, ]. Thus,F(x,y) =  =F(x,y)ify∈ {} ∪[/, ]andx= and

x= orx,x∈[/, ]. Otherwise,F(x,y) =  =F(x,y).

() Ifgx≺gx, thengx= andgx= ,i.e.,x= andx∈[/, ]. Thus,

F(x,y) =  =F(x,y)ify∈ {} ∪[/, ]andF(x,y) =  =F(x,y)ify∈(, ].

Therefore,Fisg-non-decreasing in its ﬁrst argument. Similarly, it can be shown that Fisg-non-increasing in its second argument.

• There existsH(t,t,t,t,t,t) =t–max{t,t}/∈Hsuch that (.) holds. To

discern this, for anyx,y,u,v∈Xwithgxguandgygv, we need to show that d(F(y,x),F(v,u)) = . Indeed,

() ifgxguandgy≺gv, theny=u= andx,v∈[/, ]. Thus dF(y,x),F(v,u)=d(F(,x),F(v, )=d(, ) = ;

() ifgx=guandgy≺gv, theny= andv∈[/, ]. Eitherx=u= , or x,u∈[/, ]. In any case,

dF(,x),F(v,u)=d(, ) = .

Otherwise, we get

dF(,x),F(v,u)=d(, ) = .

Similarly, ifgxguandgy=gv, thend(F(y,x),F(v,u)) = ;

() ifgx=guandgy=gv, then bothx,uare in one of the sets{},[/, ],(,a]or

(a, ]and bothy,vare also in one of the sets{},[/, ],(,a]or(a, ]. Thus dF(y,x),F(v,u)=d(, ) = .

Ifx=u= orx,u∈[/, ]andy=v= ory,v∈[/, ], otherwise, dF(y,x),F(v,u)=d(, ) = .

Therefore, all of the conditions of Theorem  are satisﬁed. Therefore, we conclude that

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Note that we cannot apply the result of Choudhury and Kundu [], the result of Choud-hury, Metiya and Kundu [] as well as the result of Lakshmikantham and Ciric [] to this example.

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, Cumhuriyet University, Sivas, Turkey.}2_{Department of Mathematics, Atilim University, ˙Incek,}

Ankara 06836, Turkey. 3_{TED University, Ankara, Turkey.}

Received: 7 September 2012 Accepted: 6 February 2013 Published: 25 February 2013 References

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doi:10.1186/1687-1812-2013-38