Common coupled fixed point theorems for Geraghty-type contraction mappings using monotone property

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R E S E A R C H

Open Access

Common coupled ﬁxed point theorems for

Geraghty-type contraction mappings using

monotone property

Zoran Kadelburg

1

_{, Poom Kumam}

2,3*

_{, Stojan Radenovi´c}

4

_{and Wutiphol Sintunavarat}

5*

*_{Correspondence:}

[email protected]; [email protected]; [email protected]

2_{Department of Mathematics,}

Faculty of Science, King Mongkut’s University of Technology Thonburi, 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok, 10140, Thailand

5_{Department of Mathematics and}

Statistics, Faculty of Science and Technology, Thammasat University, Rangsit Center, Pathumthani, 12121, Thailand

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

Abstract

In this paper, we prove common coupled fixed point theorems under a Geraghty-type condition using monotone instead of the often used mixed monotone property. Also we give some sufficient conditions for the uniqueness of a common coupled fixed point. An example illustrating our results is provided.

MSC: 47H10; 54H25

Keywords: partially ordered metric space; coupled ﬁxed point; mixed monotone property; Geraghty-type condition

1 Introduction

As the Banach contraction principle (BCP) is a power tool for solving many problems in applied mathematics and sciences, it has been improved and extended in many ways. In particular, Geraghty proved in  [] an interesting generalization of BCP which has had a lot of applications.

The notion of a coupled fixed point was firstly introduced and studied by Opoitsev [, ] and then by Guo and Lakshmikantham in []. In , Bhaskar and Lakshmikantham [] were the first to introduce the notion of mixed monotone property. They also studied and proved the following classical coupled fixed point theorems for mappings by using this property under contractive type conditions.

Theorem .([]) Let(X,d,)be a partially ordered complete metric space and let F:

X×X→X be a continuous mapping having the mixed monotone property,i.e.,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)

hold.Assume that there exists k∈[, )such that

dF(x,y),F(u,v)≤k 

d(x,u) +d(y,v) (.)

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holds for all x,y,u,v∈X with xu and yv.If there exist x,y∈X such that

xF(x,y), yF(y,x),

then there exist x,y∈X such that x=F(x,y)and y=F(y,x),i.e.,F has a coupled ﬁxed point.

Theorem .([]) If the continuity assumption for the mapping F is replaced by the as-sumption that the space(X,d,)has the following properties:

. if{xn}is a non-decreasing sequence with{xn} →x,thenxnxfor alln≥, . if{yn}is a non-increasing sequence with{yn} →y,thenynyfor alln≥. Then the same conclusion as in the previous theorem holds.

Due to the important role of Theorems . and . for the investigation of solutions of nonlinear diﬀerential and integral equations, several authors have studied various gener-alizations of these results (see,e.g., papers [–] and the references cited therein). Al-most all of them used the mixed monotone property or, in the case of additional mapping

g:X→X, the so-calledg-mixed monotone property.

Recently, in [–], the author established common coupled fixed point theorems by using (g-)monotone property instead of (g-)mixed monotone property. These kinds of results can be applied in another type of situations, so they give an opportunity to widen the field of applications. In particular the so-called tripled fixed point results (and, more generally,n-tupled results) can be more easily handled using monotone property instead of mixed monotone property (see,e.g., [–]).

The aim of this work is to prove some common coupled ﬁxed point theorems for

Geraghty-type contraction mappings by using monotone and g-monotone property

in-stead of mixed monotone and g-mixed monotone property. An illustrative example is

presented in this work showing how our results can be used in proving the existence of a common coupled ﬁxed point, while the results of many other papers cannot.

2 Preliminaries

In this section, we give some deﬁnitions that are useful for our main results in this paper. Throughout this paper (X,) denotes a partially ordered set. Byxy, we meanyx. Letf,g:X→Xbe mappings. A mappingf is said to beg-non-decreasing (resp.,g -non-increasing) if, for all x,y∈X,gxgyimplies fxfy(resp., fyfx). Ifg is an identity mapping, thenf is said to be non-decreasing (resp., non-increasing).

Deﬁnition . Let (X,) be a partially ordered set and letF:X×X→Xandg:X→X

be two mappings. The mappingFis said to have theg-monotone propertyifFis monotone

g-non-decreasing in both of its arguments, 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)

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Deﬁnition .([, ]) LetXbe a nonempty set andF:X×X→X,g:X→Xbe two mappings. An element (x,y)∈X×Xis called:

(C) acoupled ﬁxed pointofFifx=F(x,y)andy=F(y,x);

(C) acoupled coincidence pointof mappingsgandFif

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

and in this case(gx,gy)is called acoupled point of coincidence; (C) acommon coupled ﬁxed pointof mappingsgandFif

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

Deﬁnition .([]) Let (X,d) be a metric space and letg:X→X,F:X×X→X. The mappingsgandFare said to becompatibleif

lim n→∞d

gF(xn,yn),F(gxn,gyn)=  and lim n→∞d

gF(yn,xn),F(gyn,gxn)= 

hold whenever{xn}and{yn}are sequences inXsuch thatlimn→∞F(xn,yn) =limn→∞gxn

andlim_n_→∞F(yn,xn) =limn→∞gyn.

3 Main results

Letdenote the class of all functionsθ: [,∞)×[,∞)→[, ) which satisfy the follow-ing conditions:

(θ) θ(s,t) =θ(t,s)for alls,t∈[,∞);

(θ) for any two sequences{sn}and{tn}of nonnegative real numbers, θ(sn,tn)→ ⇒ sn,tn→.

The following are examples of some functions belonging to.

() θ(s,t) =kfors,t∈[,∞), wherek∈[, ). () θ(s,t) =ln(+ks+kslt+lt), s> ort> ,

r∈[, ), s= ,t= , wherek,l∈(, ). () θ(s,t) =ln(+maxmax{s,t{s},t}), s> ort> ,

r∈[, ), s= ,t= .

Now, we will prove our main result.

Theorem . Let(X,d,)be a complete partially ordered metric space and let g:X→X and F:X×X→X be such that F has the g-monotone property.Suppose that the following hold:

(i) gis continuous andg(X)is closed;

(ii) F(X×X)⊂g(X)andgandFare compatible;

(iii) there existx,y∈Xsuch thatgxF(x,y)andgyF(y,x);

(iv) there existsθ∈such that for allx,y,u,v∈Xsatisfyinggxguandgygvor

gxguandgygv,

dF(x,y),F(u,v)≤θd(gx,gu),d(gy,gv)maxd(gx,gu),d(gy,gv) (.)

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(v) (a)Fis continuous or(b)if,for an increasing sequence{xn}inX,xn→x∈Xas

n→ ∞,thenxnxfor alln∈N.

Then there exist u,v∈X such that gu=F(u,v)and gv=F(v,u),i.e.,g and F have a cou-pled coincidence point.

Proof Starting fromx,y(condition (iii)) and using thatF(X×X)⊂g(X) (condition (ii)),

we can construct sequences{xn}and{yn}inXsuch that

gxn=F(xn–,yn–) and gyn=F(yn–,xn–) forn= , , . . . .

Ifgxn=gxn–andgyn=gyn–for somen∈N, then (gxn–,gyn–) is a coupled point

of coincidence forgandF. Therefore, in what follows, we will assume that for eachn∈N, gxn=gxn–orgyn=gyn–holds.

By (iii),gxF(x,y) =gxandgyF(y,x) =gy, hence theg-monotone property

ofFimplies thatgx=F(x,y)F(x,y) =gxandgy=F(y,x)F(y,x) =gy.

Pro-ceeding by induction we get thatgxn–gxnandgyn–gynhold for eachn∈N. Hence,

the contractive condition (.) can be used to conclude that

d(gxn,gxn+) =d

F(xn–,yn–),F(xn,yn)

≤θd(gxn–,gxn),d(gyn–,gyn)

maxd(gxn–,gxn),d(gyn–,gyn)

(.)

and

d(gyn,gyn+) =d

F(yn–,xn–),F(yn,xn)

≤θd(gyn–,gyn),d(gxn–,gxn)

maxd(gyn–,gyn),d(gxn–,gxn)

=θd(gxn–,gxn),d(gyn–,gyn)

(.)

for alln∈N. From (.) and (.), we get

maxd(gxn,gxn+),d(gyn,gyn+)

≤θd(gxn–,gxn),d(gyn–,gyn)

≤maxd(gxn–,gxn),d(gyn–,gyn)

(.)

for alln∈N. Thus the sequencedn:=max{d(gxn–,gxn),d(gyn–,gyn)}is decreasing. It

fol-lows thatdn→dasn→ ∞for somed≥. Next, we claim thatd= .

Assume on the contrary thatd> ; then from (.) we obtain that

max{d(gxn,gxn+),d(gyn,gyn+)}

max{d(gxn–,gxn),d(gyn–,gyn)}≤

θd(gxn–,gxn),d(gyn–,gyn)

< .

On taking limit asn→ ∞, we get

θd(gxn–,gxn),d(gyn–,gyn)

→ asn→ ∞.

Sinceθ∈, we have

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asn→ ∞and hence

dn=maxd(gxn–,gxn),d(gyn–,gyn)

→ asn→ ∞, (.)

which contradicts the assumptiond> . Therefore, we can conclude thatdn=max{d(gxn–,

gxn),d(gyn–,gyn)} → asn→ ∞.

Next, we show that{gxn}and{gyn}are Cauchy sequences. On the contrary, assume that

at least one of{gxn}or{gyn}is not a Cauchy sequence. Then there is>  for which we

can ﬁnd subsequences{gxnk},{gxmk}of{gxn}and{gynk},{gymk}of{gyn}withnk>mk≥k such that

maxd(gxn_k,gxm_k),d(gyn_k,gym_k)≥ (.)

and

maxd(gxnk–,gxmk),d(gynk–,gymk)

<. (.)

Using (.), (.) and the triangle inequality, we have

≤rk:=maxd(gxnk,gxmk),d(gynk,gymk)

≤maxd(gxnk,gxnk–),d(gynk,gynk–)

+maxd(gxnk–,gxmk),d(gynk–,gymk)

<maxd(gxnk,gxnk–),d(gynk,gynk–)

+.

On taking limit ask→ ∞, we have

rk=maxd(gxn_k,gxm_k),d(gyn_k,gym_k)→. (.)

By the triangle inequality, we get

rk =maxd(gxnk,gxmk),d(gynk,gymk)

≤maxd(gxnk,gxnk+),d(gynk,gynk+)

+maxd(gxnk+,gxmk+),d(gynk+,gymk+)

+maxd(gxmk+,gxmk),d(gymk+,gymk)

=maxd(gxnk,gxnk+),d(gynk,gynk+)

+maxd(gxnk+,gxmk+),d(gynk+,gymk+)

≤maxd(gxnk,gxnk+),d(gynk,gynk+)

+θd(gxnk,gxmk),d(gynk,gymk)

maxd(gxnk,gxmk),d(gynk,gymk)

=dnk++dmk++θ

d(gxnk,gxmk),d(gynk,gymk)

rk

≤dnk++dmk++rk

(the usage of contractive condition (.) was possible since the sequences{gxn}and{gyn}

are increasing). Now, we have

rk≤dnk++dmk++θ

d(gxnk,gxmk),d(gynk,gymk)

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On taking limit ask→ ∞and using (.) and (.), we get

θd(gxn_k,gxm_k),d(gyn_k,gym_k)→.

Using the properties of functionθ, we obtain

d(gxnk,gxmk)→ and d(gynk,gymk)→

ask→ ∞, which imply that

lim

k→∞rk=klim→∞

maxd(gxnk,gxmk),d(gynk,gymk)

= ,

which contradicts with> .

Therefore, we get that{gxn}and{gyn}are Cauchy sequences. Sinceg(X) is a closed subset of a complete metric space, there existu,v∈g(X) such that

lim

n→∞gxn=nlim→∞F(xn,yn) =u and nlim→∞gyn=nlim→∞F(yn,xn) =v.

By condition (ii), the compatibility ofgandFimplies that

lim n→∞d

gF(xn,yn),F(gxn,gyn)=  and lim n→∞d

gF(yn,xn),F(gyn,gxn)= . (.)

Consider the two possibilities given in condition (v).

(a) Suppose thatFis continuous. Using the triangle inequality we get that

dgu,F(gxn,gyn)

≤dgu,gF(xn,yn)

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

.

Passing to the limit asn→ ∞and using (.) and the continuity ofgandF, we get that

d(gu,F(u,v)) = ,i.e.,gu=F(u,v). In a similar way,gv=F(v,u) is obtained.

(b) In this casegxnu=gxandgynv=gyfor somex,y∈Xand eachn∈N. Using

(.) we get

dF(x,y),gx≤dF(x,y),gxn+

+d(gxn+,gx)

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

+d(gxn+,gx)

≤θd(gx,gxn),d(gy,gyn)

maxd(gx,gxn),d(gy,gyn)

+d(gxn+,gx)

→

asn→ ∞. Hence,gx=F(x,y) and similarlygy=F(y,x).

Note that in this case continuity and compatibility assumptions were not needed in the

proof.

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Puttingg =IX, whereIX is an identity mapping onX in Theorem ., we obtain the following.

Corollary . Let(X,d,)be a partially ordered complete metric space and let F:X× X→X have the monotone property.Suppose that the following hold:

(i) there existx,y∈Xsuch thatxF(x,y)andyF(y,x);

(ii) there existsθ∈such that for allx,y,u,v∈Xsatisfying(xuandyv)or(ux

andvy),

dF(x,y),F(u,v)≤θd(x,u),d(y,v)maxd(x,u),d(y,v)

holds true;

(iii) (a)Fis continuous or(b)if{xn}is an increasing sequence inXandxn→xas

n→ ∞,thenxnxfor alln.

Then there exist x,y∈X such that x=F(x,y)and y=F(y,x),i.e.,F has a coupled ﬁxed point.

Takingθ(t,t) =kwithk∈[, ) for allt,t∈[,∞) in Theorem . and Corollary .,

we obtain the following corollary.

Corollary . Let(X,d,)be a complete partially ordered metric space and let g:X→X and F:X×X→X be such that F has the g-monotone property.Suppose that the following hold:

(i) gis continuous andg(X)is closed;

(ii) F(X×X)⊂g(X)andgandFare compatible;

(iii) there existx,y∈Xsuch thatgxF(x,y)andgyF(y,x);

(iv) there existsk∈[, )such that for allx,y,u,v∈Xsatisfying(gxguandgygv)or

(gugxandgvgy),

dF(x,y),F(u,v)≤kmaxd(gx,gu),d(gy,gv) (.)

holds true;

(v) (a)Fis continuous or(b)if{xn}is an increasing sequence inXandxn→xas

Then there exist u,v∈X such that gu=F(u,v)and gv=F(v,u),i.e.,g and F have a coupled coincidence point.

Corollary . Let(X,d,)be a partially ordered complete metric space and let F:X× X→X.Suppose that the following hold:

(i) Fsatisﬁes the monotone property;

(ii) there existx,y∈Xsuch thatxF(x,y)andyF(y,x);

(iii) there existsk∈[, )such that for allx,y,u,v∈Xsatisfying(xuandyv)or

(uxandvy),

dF(x,y),F(u,v)≤kmaxd(x,u),d(y,v)

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(iv) (a)Fis continuous or(b)if{xn}is an increasing sequence inXandxn→xas

Then there exist x,y∈X such that x=F(x,y)and y=F(y,x),i.e.,F has a coupled ﬁxed point.

Remark . Since, fork,l≥,k+l≤,

kd(gx,gu) +ld(gy,gv)≤maxd(gx,gu),d(gy,gv),

Corollary . remains valid if the right-hand side of condition (.) is replaced bykd(gx,

gu) +ld(gy,gv) for somek,l≥,k+l< .

Now, reasoning on Theorem ., some questions arise naturally. To be precise, one can ask the following.

Question Is it possible to find a sufficient condition for the existence and uniqueness of the coupled common fixed point ofgandF?

Motivated by the interest in this research, we give a positive answer to this question adding to Theorem . some hypotheses.

For the given partial orderon the setX, we shall denote also bythe order onX×X

given by

(x,y)(x,y) ⇔ xx and yy.

Theorem . In addition to the hypotheses of Theorem.assume that

(vi) for any two elements(x,y), (u,v)∈X×X,there exists(w,z)∈X×Xsuch that (F(w,z),F(z,w))is comparable to both(F(x,y),F(y,x))and(F(u,v),F(v,u)).

Then g and F have a unique common coupled ﬁxed point,i.e.,there exists unique(p,q)∈

X×X such that p=gp=F(p,q)and q=gq=F(q,p).

Proof Theorem . implies that there exists a coupled coincidence point (x,y)∈X×X, that is,gx=F(x,y) andgy=F(y,x). Suppose that there exists another coupled coincidence point (u,v)∈X×Xand hencegu=F(u,v) andgv=F(v,u). We will prove thatgx=guand

gy=gv.

From condition (vi) we get that there exists (w,z)∈X×Xsuch that (F(w,z),F(z,w)) is comparable to both (F(x,y),F(y,x)) and (F(u,v),F(v,u)). Putw=w,z=zand,

analo-gously to the proof of Theorem ., choose sequences{wn},{zn}inXsatisfying

gwn=F(wn–,zn–) and gzn=F(zn–,wn–)

forn∈N. Starting fromx=x, y=yandu=u, v=v, choose sequences {xn},{yn}

and{un},{vn}satisfyinggxn=F(xn–,yn–),gyn=F(yn–,xn–) andgun=F(un–,vn–),gvn= F(vn–,un–) forn∈N. Taking into account the properties of coincidence points, it is easy

to see that it can be done so thatxn=x,yn=yandun=u,vn=v,i.e.,

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Since (F(x,y),F(y,x)) = (gx,gy) and (F(w,z),F(z,v)) = (gw,gz) are comparable, then, for

example,gxgw,gygzand in a similar way,gxgwn,gygzn. Thus, we can apply

the contractive condition (.) to obtain

d(gx,gwn+) =d

F(x,y),F(wn,zn)

≤θd(gx,gwn),d(gy,gzn)maxd(gx,gwn),d(gy,gzn)

and

d(gy,gzn+) =d

F(zn,wn),F(y,x)

≤θd(gy,gzn),d(gx,gwn)maxd(gy,gzn),d(gx,gwn)

=θd(gx,gwn),d(gy,gzn)maxd(gx,gwn),d(gy,gzn).

This implies that

maxd(gx,gwn+),d(gy,gzn+)

≤θd(gx,gwn),d(gy,gzn)maxd(gx,gwn),d(gy,gzn)

<maxd(gx,gwn),d(gy,gzn). (.)

Therefore, we get that the sequence dn:=max{d(gx,gwn),d(gy,gzn)} is decreasing and hencedn→dasn→ ∞for somed≥. Now, we prove thatd= . Assume to the contrary thatd> ; then from (.) we have

max{d(gx,gwn+),d(gy,gzn+)}

max{d(gx,gwn),d(gy,gzn)} ≤θ

d(gx,gwn),d(gy,gzn)< .

Taking the limit asn→ ∞in the above inequality, we have

θd(gx,gwn),d(gy,gzn)

→ asn→ ∞.

By the property (θ) ofθ∈, we get

d(gx,gwn)→ and d(gy,gzn)→

asn→ ∞. Now we have

dn:=maxd(gx,gwn),d(gy,gzn)→ asn→ ∞,

which contradicts with d> . Therefore, we conclude that dn=max{d(gx,gwn),d(gy,

gzn)} → asn→ ∞and then

lim

n→∞d(gx,gwn) =  and nlim→∞d(gy,gzn) = .

In a similar way, we have

lim

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By the triangle inequality, we have

d(gx,gu)≤d(gx,gwn) +d(gwn,gu) and d(gy,gv)≤d(gy,gzn) +d(gzn,gv)

for alln∈N. Takingn→ ∞in the above two inequalities, we get thatd(gx,gu) =  and

d(gy,gv) = . Therefore, we havegx=guandgy=gv. Now we letp:=gxandq:=gy. Hence we have

gp=g(gx) =gF(x,y) and gq=g(gy) =gF(y,x).

By the deﬁnition of sequences{xn}and{yn}we have

gxn=F(x,y) =F(xn–,yn–) and gyn=F(y,x) =F(yn–,xn–)

for alln∈N. So we have

lim

n→∞F(xn,yn) =nlim→∞gxn=F(x,y) and nlim→∞F(yn,xn) =nlim→∞gyn=F(y,x).

SincegandFare compatible, we have

lim n→∞d

gF(xn,yn),F(gxn,gyn)= ,

that is,gF(x,y) =F(gx,gy). Therefore, we get

gp=gF(x,y) =F(gx,gy) =F(p,q)

and in a similar way

gq=gF(y,x) =F(gy,gx) =F(q,p).

This implies that (p,q) is another coincidence point. By the property we have just proved, it follows thatgp=gx=pandgq=gy=q. So,

p=gp=F(p,q) and q=gq=F(q,p),

and (p,q) is a common coupled ﬁxed point ofgandF. For the uniqueness of the common

coupled ﬁxed point, we can proceed easily.

Corollary . In addition to the hypotheses of Corollary.assume that

(vi) for any two elements(x,y), (u,v)∈X×X,there exists(w,z)∈X×Xsuch that (F(w,z),F(z,w))is comparable to both(F(x,y),F(y,x))and(F(u,v),F(v,u)). Then g and F have a unique common coupled ﬁxed point.

Theorem . In addition to the hypotheses of Corollary.,let the condition(vi)of Theo-rem.be satisfied.Then the coupled fixed point of F is unique.Moreover,if for the terms of sequences{xn},{yn}defined by xn=F(xn–,yn–)and yn=F(xn–,yn–),xnynholds for

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Proof We have only to prove the last statement. Suppose that forn suﬃciently large,

xnyn. Then, by (.) (withg=IX), we get

d(xn+,yn+) =d

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

≤θd(xn,yn),d(xn,yn)d(xn,yn). (.)

This implies that

d(xn+,yn+) <d(xn,yn),

that is, the sequencedn:=d(xn,yn) is decreasing. Hencedn→dasn→ ∞for somed≥.

Next, we prove thatd= . Assume to the contrary thatd> ; then from (.) we have

d(xn+,yn+)

d(xn,yn)

≤θd(xn,yn),d(xn,yn)< .

Lettingn→ ∞, we getθ(d(xn,yn),d(xn,yn))→. Sinceθ∈, we haved(xn,yn)→ as n→ ∞, which contradicts withd> . Therefore, we haved(xn,yn)→ asn→ ∞.

By the triangle inequality, we have

d(x,y)≤d(x,xn+) +d(xn+,yn+) +d(yn+,y)

=d(x,xn+) +d

F(xn,yn),F(yn,xn)+d(yn+,y)

≤d(x,xn+) +θ

d(xn,yn),d(xn,yn)d(xn,yn) +d(yn+,y)

<d(x,xn+) +d(xn,yn) +d(yn+,y).

Passing to the limit asn→ ∞, sincexn→x,yn→yandd(xn,yn)→, we get thatd(x,y)≤

 and thusx=y. This completes the proof.

Corollary . In addition to the hypotheses of Corollary.,let the condition(vi)of The-orem.be satisfied.Then the coupled fixed point of F is unique.Moreover,if for the terms of sequences{xn},{yn}defined by xn=F(xn–,yn–)and yn=F(xn–,yn–),xnynholds for

all n,then the coupled ﬁxed point of F has the form(x,x).

Finally, we give an example showing that our theorem can be used when many results in this ﬁeld cannot.

Example . LetX= [, ]⊆Rwith the usual metric and order. Consider the mappings

g:X→XandF:X×X→Xdeﬁned by

gx=x, F(x,y) =ln

 +x



 +

y

 .

Note thatFdoes not satisfy the mixedg-monotone property of [] and []. Indeed, for

y=_∈Xandy=_∈X, we havegygy. But forx= ∈X, we get

F(x,y) =ln



ln



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Hence a lot of results in literature that used theg-mixed monotone property cannot be applied in this example. Moreover,gandFdo not commute as in [].

Next, we show that Theorem . can be used in this example.

Letθ∈be deﬁned by

θ(s,t) = ⎧ ⎨ ⎩

ln(+_s+_t)

s

+t , s>  ort> ,

, s=  andt= .

We will show that condition (.) holds with the functionθ. Letx,y,u,vbe arbitrary points inX. We obtain that

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

ln

 +x



 +

y

 ,ln

 +u



 +

v



=ln

 +x



 +

y

 –ln

 +u



 +

v



=ln + x

 +

y



 +u

 +

v



=ln

 +( x  + y ) – ( u  + v )

 +u_ +v_ ≤ln  + x  + y  – u  + v  ≤ln  +

x

_–_u₊

y

_–_v

=ln( +



|x–u|+ 

|y–v|)

(_|x_–_u|₊

|y–v|)

 x

_–_u₊

y

_–_v

=ln( +



d(gx,gu) + 

d(gy,gv))

(_d(gx,gu) +_d(gy,gv))



d(gx,gu) + 

d(gy,gv)

=θd(gx,gu),d(gy,gv)

d(gx,gu) + 

d(gy,gv)

≤θd(gx,gu),d(gy,gv)maxd(gx,gu),d(gy,gv).

This shows that condition (.) holds with the functionθ.

Next, we check thatg andFare compatible. Let{xn}and{yn}be two sequences inX

such that

lim

n→∞gxn=nlim→∞F(xn,yn) =a and nlim→∞gyn=nlim→∞F(yn,xn) =b.

Thenln( +a_+b_) =aandln( +a_+b_) =b, wherefrom it follows thata=b= . Then

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

ln

 +x

 n  + y n   –ln  +x



n

 +

y

n

 → (n→ ∞),

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We can easily check that all other conditions of Theorem . hold. Therefore,g and

F have a unique common coupled ﬁxed point. In this example, we can see that a point (, )∈X×Xis a unique common coupled ﬁxed point ofgandF.

Remark . We conclude remarking that the approach used in this paper (i.e., using

g-monotone instead ofg-mixed monotone mappings) has the advantage when considering

n-tupled common ﬁxed points for oddn. For example, one can consider mappings F:

X_→_X_and_g_:_X_→_X_{and say that the mapping}_F_is_g_{-monotone if}_F₍_x_,_y_,_z_{) is}_g

-non-decreasing in all three variables, that is, for allx,y,z∈X,

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

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

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

In particular, ifg=IX,Fis said to be monotone. An element (x,y,z)∈X_{is called a tripled}

coincidence point ofFandg ifF(x,y,z) =gx,F(y,z,x) =gyandF(z,x,y) =gz. Moreover, (x,y,z) is called a tripled common ﬁxed point ofFandgif

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

In particular, forg=IX, the element (x,y,z) is called a tripled ﬁxed point ofF.

Results concerning these notions can be found,e.g., in []. Such an approach is not possible for mixed monotone mappings whenn=  (see []).

Open problems

• In Theorem ., can the set{d(gx,gu),d(gy,gv)}be replaced by other sets?

• Can Theorem . be improved by replacing the control functionθby other functions? • Can Theorem . be extended and generalized replacing condition(.)by other

conditions?

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The main idea of this paper was proposed by ZK, PK, SR and WS prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the ﬁnal manuscript.

Author details

1_{Faculty of Mathematics, University of Belgrade, Beograd, 11000, Serbia.}2_{Department of Mathematics, Faculty of Science,}

King Mongkut’s University of Technology Thonburi, 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok, 10140, Thailand.3_{Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King}

Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok, 10140, Thailand.4_{Faculty of Mathematics and Information Technology, Teacher Education, Dong Thap University, Cao Lanh City,}

Dong Thap Province, Vietnam. 5_{Department of Mathematics and Statistics, Faculty of Science and Technology,}

Thammasat University, Rangsit Center, Pathumthani, 12121, Thailand.

Acknowledgements

The first author is thankful to the Ministry of Education, Science and Technological Development of Serbia, Project No. 174002. The second author was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU) and Theoretical and Computational Science Center (TaCS). The fourth author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financial support during the preparation of this manuscript.

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