n tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces

R E S E A R C H

Open Access

n-tuplet fixed point theorems for contractive

type mappings in partially ordered metric

spaces

Müzeyyen Ertürk

_{and Vatan Karakaya}

A correction to this article has been published:

http://www.journalofinequalitiesandapplications.com/content/2013/1/368.

*_{Correspondence:} merturk3263@gmail.com; merturk@yildiz.edu.tr

1_{Department of Mathematics, The} Faculty of Arts and Sciences, Yildiz Technical University, Davutpasa Campus, Esenler, Istanbul, 34210, Turkey

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

Abstract

In this paper, we study existence and uniquennes of fixed points of operator

F:Xn_{→Xwherenis an arbitrary positive integer andXis partially ordered complete}

metric space.

MSC: Primary 47H10; 54H25; secondary 54E50

Keywords: fixed point theorems; nonlinear contraction; partially ordered metric space;n-tuplet fixed point; mixedg-monotone

1 Introduction

As it is known, fixed point theory is one of the oldest and most famous theory in math-ematics, and it has become an important tool for other areas of science such as approxi-mation theory, statistics, engineering and economics.

Among hundreds of fixed point theorems, the Banach contraction theorem [] is par-ticularly well known due to its simplicity and usefulness. It states that any contraction mapping of a complete metric space has a unique fixed point.

In , the Banach contraction principle were extended to metric space endowed with partial order by Ran and Reuring []. They pointed out that the contractivity condition on the nonlinear and monotone map is only assumed to hold on elements which are com-parable in the partial order. Afterward, Nieto and Rodriguez-Lopez [] extended results of Ran and Reuring for non-decreasing mapping and studied existence and uniqueness of first-order differential equations.

In , by following the above mentioned trend, Bhaskar and Lakshmikantham [] in-troduced mixed monotone property and gave their coupled fixed point theorem for map-pings with mixed monotone property. Also, they produced some applications related with the existence and uniqueness of solution for a periodic boundary value problem. This work of Bhaskar and Lakshmikantham has attracted the attention of many researchers. The con-cept of coupled fixed point for various contractive type mappings was studied by several authors [–]. Lakshmikantham and Ciric [] extended the results of [] for monotone non-linear contractive mapping and generalized mixed monotone concept. Berinde and Borcut [] introduced tripled fixed point theorem for non-linear mapping in partially

dered complete metric space as a generalization and extension of the coupled fixed point theorem.

Motivated by these studies, the quadruple fixed point theorem was given for different contractive type mappings [–].

In this paper, we generalize mentioned trend in the above for an arbitrary positive num-bern, that is, we introduce the concept ofn-tuplet fixed point theorem and prove some results.

2 Main results

Let us give new definitions for our aim.

Definition  Let (X,≤) be partially ordered set andF :Xn_{→X. We say that F has}

the mixed monotone property if F(x,x,x, . . . ,xn) is monotone non-decreasing in its

odd argument and it is monotone non-increasing in its even argument. That is, for any

x,x,x, . . . ,xn∈X y,z∈X, y≤z

⇒ F(y,x,x, . . . ,xn)≤F(z,x,x, . . . ,xn), y,z∈X, y≤z

⇒ F(x,y,x, . . . ,xn)≥F(x,z,x, . . . ,xn),

.. .

yn,zn∈X, yn≤zn

⇒ F(x,x,x, . . . ,yn)≤F(x,x,x, . . . ,zn) (ifnis odd), yn,zn∈X, yn≤zn

⇒ F(x,x,x, . . . ,yn)≥F(x,x,x, . . . ,zn) (ifnis even).

(.)

Definition  LetXbe a nonempty set and F:Xn_{→Xa given mapping. An element}

(x,x,x, . . . ,xn)∈Xnis called an-tuplet fixed point ofFif F(x,x,x, . . . ,xn) =x,

F(x,x, . . . ,xn,x) =x,

.. .

F(xn,x,x, . . . ,xn–) =xn.

(.)

Definition  Let (X,≤) be partially ordered set andF:Xn_{→Xandg:X→X. We say that} Fhas the mixedg-monotone property ifF(x,x,x, . . . ,xn) is monotoneg-non-decreasing

in its odd argument and it is monotoneg-non-increasing in its even argument. That is, for anyx,x,x, . . . ,xn∈X

y,z∈X, g(y)≤g(z)

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

..

. (.)

yn,zn∈X, g(yn)≤g(zn)

⇒ F(x,x,x, . . . ,yn)≤F(x,x,x, . . . ,zn) (ifnis odd), yn,zn∈X, g(xn)≤g(yn)

⇒ F(x,x,x, . . . ,yn)≥F(x,x,x, . . . ,zn) (ifnis even).

Note that ifgis the identity mapping, this definition reduces to Definition .

Definition  LetX be a nonempty set andF:Xn_{→Xa given mapping. An element}

(x,x,x, . . . ,xn)∈Xnis called an-tuplet coincidence point ofF:Xn→Xandg:X→Xif F(x,x,x, . . . ,xn) =g(x),

F(x,x, . . . ,xn,x) =g(x),

.. .

F(xn,x,x, . . . ,xn–) =g(xn).

(.)

Note that ifgis the identity mapping, this definition reduces to Definition .

Definition  Let (X,≤) be partially ordered set andF:Xn_{→Xandg:X→X.Fandg}

called commutative if

gF(x,x,x, . . . ,xn)

=Fg(x),g(x),g(x), . . . ,g(xn)

(.)

for allx,x,x, . . . ,xn∈X.

Letdenote the all functionsφ : [,∞)→[,∞), which are continuous and satisfy that

(i) φ(t) <t,

(ii) lim_r→t+φ(r) <tfor eachr> .

Since we want to shorten expressions in the following theorem, consider Condition  in the following forXanF.

Condition  Suppose either

(i) Fis continuous, or

(ii) Xhas the following property:

(a) if non-decreasing sequencexk→x, thenxk≤xfor allk, (b) if non-increasing sequenceyk→y, thenyk≥yfor allk.

Theorem  Let(X,≤)be partially ordered set and suppose that(X,d)is complete met-ric space.Assume F:Xn_{→X and g:X→X are such that F has the mixed g-monotone} property and

dF(x,x,x, . . . ,xn),F(y,y,y, . . . ,yn)

≤φ

d(g(x),g(y)) +d(g(x),g(y)) +· · ·+d(g(xn),g(yn)) n

for all x,x,x, . . . ,xn∈X for which g(x)≤g(y),g(x)≥g(y), . . . ,g(xn)≤g(yn) (if n is odd),g(xn)≥g(yn) (if n is even).Assume that F(Xn)⊂g(X)and g commutes with F.Also, suppose that Conditionis satisfied.If there exist x

,x,x, . . . ,xn∈X such that

gx_≤Fx_,x_,x_, . . . ,xn_,

gx_≥Fx_,x_, . . . ,xn_,x_, ..

.

gxn_≤Fxn_,x_,x_, . . . ,xn_– (if n is odd),

gxn_≥Fxn_,x_,x_, . . . ,xn_– (if n is even)

(.)

then there exist x,x,x, . . . ,xn∈X such that F(x,x,x, . . . ,xn) =g(x),

F(x,x, . . . ,xn,x) =g(x),

.. .

F(xn,x,x, . . . ,xn–) =g(xn),

that is,F and g have a n-tuplet coincidence point.

Proof Letx

,x,x, . . . ,xn∈Xbe such that (.). SinceF(Xn)⊂g(X), we construct the

sequence (x

k), (xk), . . . , (xnk) as follows:

gx

k

=Fx

k–,xk–, . . . ,xnk–

,

gx_k=Fx_k–, . . . ,xn_k–,x_k–,

.. .

gxn_k=Fxn_k–,x_k–, . . . ,xn_k–_–

(.)

fork= , , , . . . . We claim that

gx_k–≤gx_k,

gx_k–≥gx_k, ..

.

gxn_k–≤gxn_k (ifnis odd),

gxn_k–≥gxn_k (ifnis even)

(.)

for allk≥. For this, we will use the mathematical induction. The inequalities in (.) holdk=  because of (.), that is, we have

gx_≤Fx_,x_, . . . ,xn_=gx_,

.. .

gxn_≤Fxn_,x_,x_, . . . ,xn_–=gxn_ (ifnis odd),

gxn_≥Fxn_,x_,x_, . . . ,xn_–=gxn_ (ifnis even).

Thus, our claim is true fork= . Now, suppose that the inequalities in (.) holdk=m. In this case,

gx_m–≤gx_m,

gx_m–≥gx_m, ..

.

gxn_m–≤gxn_m (ifnis odd),

gxn_m–≥gxn_m (ifnis even).

(.)

Now, we must show that the inequalities in (.) holdk=m+ . If we consider (.) and mixedg-monotone property ofFtogether with (.), we have

gx_m=Fx_m–,x_m–, . . . ,xn_m–

≤Fx_m,x_m–, . . . ,xn_m–

≤Fx_m,x_m,x_m_–, . . . ,xn_m–_–,xn_m–

.. .

≤Fx_m,x_m, . . . ,xn_m–,xn_m–

≤Fx_m,x_m, . . . ,xn_m=gx_m+,

gx_m=Fx_m–, . . . ,xn_m–,x_m–

≥Fx_m,x_m–, . . . ,xn_m–,x_m–

≥Fx_m,x_m, . . . ,xn_m–,x_m–

.. .

≥Fx_m,x_m, . . . ,xn_m,x_m–

≥Fx_m,x_m, . . . ,xn_m,x_m=gx_m+, ..

.

gxn_m=Fxn_m–,x_m–, . . . ,xn_m–_–

≤Fxn_m,x_m_–, . . . ,xn_m–_–

≤Fxn_m,x_m,x_m–, . . . ,xn_m–_–

.. .

≤Fxn_m,x_m, . . . ,x_mn–=gxn_m+ (ifnis odd),

gxn_m=Fxn_m–,x_m–, . . . ,xn_m–_–

≥Fxn_m,x_m_–, . . . ,xn_m–_–

≥Fxn_m,x_m,x_m–, . . . ,xn_m–_–

.. .

≥Fxn_m,x_m,x_m, . . . ,xn_m–,xn_m–_–

≥Fxn_m,x_m, . . . ,x_mn–=gxn_m+ (ifnis even).

Thus, (.) is satisfied for allk≥. So, we have,

· · · ≥gx_k≥gx_k–≥ · · · ≥gx_≥gx_,

· · · ≤gx_k≤gx_k–≤ · · · ≤gx_≤gx_, ..

.

· · · ≥gxn_k≥gxn_k–≥ · · · ≥gxn_≥gxn_ (ifnis odd),

· · · ≤gxn_k≤gxn_k–≤ · · · ≤gxn_≤gxn_ (ifnis even).

(.)

For the simplicity, we define

δk=d

gx_k,gx_k++dgx_k,gx_k++· · ·+dgxn_k,gxn_k+. We will show that

δk+≤n·φ

δk n

. (.)

By (.), (.) and (.), we get

dgx_k+,gx_k+

=dFx_k,x_k, . . . ,xn_k,Fx_k+,x_k+, . . . ,xn_k+

≤φ

d(g((x_k),g(x_k+))) +d(g(x_k),g(x_k+)) +· · ·+d(g(xn_k),g(xn_k+))

n

=φ

δk n

, (.)

dgx_k+,gx_k+

=dFx_k, . . . ,xn_k,x_k,Fx_k+, . . . ,xn_k+,x_k+

≤φ

d(g(x

k),g(xk+)) +· · ·+d(g(xnk),g(xnk+)) +d(g(xk),g(xk+))

n

=φ

δk n

, (.)

dgxn_k,gxn_k+

=dFxn_k,x_k, . . . ,xn_k–,Fxn_k+,x_k_+, . . . ,xn_k–_+

≤φ

d(g((xn_k),g(x_kn_+))) +d(g(x_k),g(x_k_+)) +· · ·+d(g(x_kn–),g(xn_k–_+))

n

=φ

δk n

. (.)

Due to (.)-(.), we conclude that

dgx_k+,gx_k++dgx_k+,gx_k++· · · +dgxn_k+,gxn_k+≤nφ

δk n

. (.)

Hence, we get (.).

Sinceφ(t) <tfor allt> , thenδk+≤nφ(δ_nk) <n·δ_nk =δkfor allk∈N. So, (δk) is

mono-tone decreasing. Since it is bounded below, there is someδ≥ such that

lim

k→∞δk=δ+. (.)

We want to show thatδ= . Suppose thatδ> . Then taking the limit asδk→δ+ of both

sides of (.) and keeping in mind that we assume thatlimr→t+φ(r) <tfor allt> , we

have

δ= lim

k→∞δk+≤klim→∞n·φ

δk n

=n lim

δk→δ+

φ

δk n

<nδ

n=δ (.)

which is a contradiction. Thus,δ= , that is

lim

k→∞

dgx_k,gx_k_++dgx_k,gx_k++· · ·+dgxn_k,gxn_k+= . (.)

Now we prove thatg(x

k),g(xk), . . . ,g(xnk) are Cauchy sequences. Suppose that at least one

ofg(x

k),g(xk), . . . ,g(xnk) is not Cauchy. So, there exists anε>  for which we can find

sub-sequence of integerj(k),l(k) withj(k) >l(k)≥ksuch that

tk=d

gx_j(k),gx_l(k)+dgx_j(k),gx_l(k)+· · ·+dgxn_j(k),gxn_l(k)≥ε. (.)

Additionally, corresponding toj(k), we can choosel(k) such that it is the smallest integer satisfying (.) andj(k) >l(k)≥k. Thus,

dgx_j(k)–,gx_l(k)+dgx_j(k)–,gx_l(k)+· · ·+dgxn_j(k)–,gx_ln_(k)<ε. (.)

By using triangle inequality and having (.) and (.) in mind

ε≤tk

≤dgx_j(k),gx_j(k)–+dgx_j(k)–,gx_l(k)

+dgx_j(k),gx_j(k)–+dgx_j(k)–,gx_l(k)

+· · ·+dgxn_j(k),gxn_j(k)–+dgxn_j(k)–,gxn_l(k)

< dgx_j(k),gx_j(k)–+dgx_j(k),gx_j(k)–

+· · ·+dgxn_j(k),gxn_j(k)–+ε

< δj(k)–+ε. (.)

Lettingk→ ∞in (.) and using (.)

lim

k→∞tk=klim→∞

dgx_j(k),gx_l_(k)+dgx_j(k),gx_l(k)+· · ·+dgxn_j(k),gxn_l(k)

=ε+. (.)

We apply triangle inequality to (.) as the following.

tk =d

gx_j(k),gx_l(k)+dgx_j(k),gx_l(k)

+· · ·+dgxn_j(k),gxn_l(k)

≤dgx_j(k),gx_j(k)++dgx_j(k)+,gx_l(k)++dgx_l(k)+,gx_l(k)

+dgx_j(k),gx_j(k)++dgx_j(k)+,gx_l(k)++dgx_l(k)+,gx_l(k)

+· · ·+dgxn_j(k),gxn_j(k)++dgxn_j(k)+,gxn_l(k)++dgxn_l(k)+,gxn_l(k)

≤δj(k)++δl(k)++d

gx_j(k)+,gx_l(k)++dgx_j(k)+,gx_l(k)+

+· · ·+dgxn_j(k)+,gxn_l(k)+. (.) Sincej(k) >l(k), then

gx_j(k)≥gx_l(k),

gx_j(k)≤gx_l(k), ..

.

gxn_j(k)≥gxn_l(k) (ifnis odd),

gxn_j(k)≤gxn_l(k) (ifnis even).

(.)

So, from (.), (.) and (.), we get

dgx_j(k)+,gx_l(k)+

=dFx_j(k),x_j(k), . . . ,xn_j(k),Fx_l(k),x_l_(k), . . . ,xn_l(k)

≤φ



n

dgx_j(k),gx_l_(k)+dgx_j(k),gx_l(k)+· · ·+dgxn_j(k),gxn_l(k)

=φ

tk n

dgx_j(k)+,gx_l(k)+

=dFx_j(k), . . . ,xn_j(k),x_j(k),Fx_l(k), . . . ,x_ln_(k),x_l(k)

≤φ



n

dgx_j(k),gx_l_(k)+· · ·+dgxn_j(k),gxn_l(k)+dgx_j(k),gx_l(k)

=φ tk n , (.) .. .

dgxn_j(k)+,gxn_l(k)+

=dFxn_j(k),x_j(k), . . . ,xn_j(–_k),Fxn_l(k),x_l_(k), . . . ,xn_l(–_k)

≤φ



n

dgxn_j(k),gxn_l(k)+dgx_j(k),gx_l(k)+· · ·+dgxn_j(–_k),gxn_l(–_k)

=φ tk n . (.)

Combining (.) with (.)-(.), we get

tk≤δj(k)++δl(k)+

+dgx

j(k)+

,gx

l(k)+

+dgx

k(k)+

,gx

l(k)+

+· · ·+dgxn_j(k)+,gxn_l(k)+

≤δj(k)++δl(k)++tk+n·φ

tk n

<δj(k)++δl(k)++n·

tk

n. (.)

Lettingk→ ∞, we obtain a contradiction. This show thatg(x

k),g(xk), . . . ,g(xnk) are Cauchy

sequences. SinceXis complete metric space, there existx_,x_{, . . . ,x}n_{∈Xsuch that}

lim

k→∞g

x_k=x, lim

k→∞g

x_k=x, . . . , lim

k→∞g

xn_k=xn. (.)

Sincegis continuous, (.) implies that

lim

k→∞g

gx_k=gx, lim

k→∞g

gx_k=gx, . . . ,

lim

k→∞g

gxn_k=gxn.

(.)

From (.) and by regarding commutativity ofFandg

ggx_k+=gFx_k,x_k, . . . ,xn_k=Fgx_k,gx_k, . . . ,gxn_k,

ggx_k+=gFx_k, . . . ,xn_k,x_k=Fgx_k, . . . ,gxn_k,gx_k,

.. .

ggxn_k+=gFxn_k,x_k, . . . ,xn_k–=Fgxn_k,gx_k, . . . ,gxn_k–.

We shall show that

Fx,x, . . . ,xn=gx,

Fx, . . . ,xn,x=gx,

.. .

Fxn,x, . . . ,xn–=gxn.

Suppose now, (i) holds. Then by (.), (.) and (.), we have

gx= lim

k→∞g

gx_k+= lim

k→∞g

Fx_k,x_k, . . . ,xn_k

= lim

k→∞F

gx_k,gx_k, . . . ,gxn_k

=F

lim

k→∞g

x_k, lim

k→∞g

x_k, . . . , lim

k→∞g

xn_k

=Fx,x, . . . ,xn. (.)

Analogously,

gx=ggx_k+= lim

k→∞g

Fx_k, . . . ,xn_k,x_k

= lim

k→∞F

gx_k, . . . ,gxn_k,gx_k

=F

lim

k→∞g

x_k, . . . ,lim

k→∞g

xn_k, lim

k→∞g

x_k

=Fx, . . . ,xn,x, (.)

.. .

gxn=ggxn_k+= lim

k→∞g

Fxn,x, . . . ,xn–

= lim

k→∞F

gxn_k,gx_k, . . . ,gxn_k–

=Flim

k→∞g

xn_k, lim

k→∞g

x_k, . . . ,lim

k→∞g

xn_k–

=Fxn,x, . . . ,xn–. (.)

Thus, we have

gx=Fx,x, . . . ,xn,

gx=Fx, . . . ,xn,x,

.. .

gxn=Fxn,x, . . . ,xn–.

g(xn_k–) are non-decreasing andg(x

k),g(xk), . . . ,g(xnk) are non-increasing and by considering g(x

k)→x,g(xk)→x, . . . ,g(xnk)→xnwe have

gx_k≥x, gx_k≤x, . . . , gxn_k≤xn (ifnis odd),

gxn_k≥xn (ifnis even)

(.)

for allk. Thus, by triangle inequality and (.)

dgx,Fx,x, . . . ,xn

≤dgx,ggx_k+

+dggx_k+,Fx,x, . . . ,xn

≤dgx,ggx_k+

+φ



n

dggx_k,gx+dggx_k,gx

+· · ·+dggxn_k,gxn. (.)

Lettingk→ ∞implies thatd(g(x),F(x,x, . . . ,xn))≤. Hence,g(x) =F(x,x, . . . ,xn). Analogously, we can get that

gx=Fx, . . . ,xn,x, . . . , gxn=Fxn,x, . . . ,xn–.

Thus, we proved thatFandghave an-tuplet coincidence point.

Corollary  The above theorem reduces to Theorem.of[]for n= and g(x) =x if(i)is

satisfied andφ(t) =ct where c∈(, ).

The following corollary is a generalization of Corollary . in [] and Theorem . in [].

Corollary  Let(X,≤)be a partially ordered set and suppose that(X,d)is complete metric space.Suppose F:Xn_{→X and there existφ∈such that F has the mixed g-monotone} property and there exist a m∈[, )with

φFx,x, . . . ,xn,Fy,y, . . . ,yn

≤m

n

dgx,gy+dgx,gy+· · ·+dgxn,gyn (.)

for all x_,x_{, . . . ,x}n_,y_,y_{, . . . ,y}n_{∈X for which g(x}_)≤g(y_),g(x_)≥g(y_{), . . . ,g(x}n_)≤g(yn₎

that

gz≤Fz,z, . . . ,zn,

gz≥Fz, . . . ,zn,z, ..

.

gzn≤Fzn,z, . . . ,zn– (if n is odd),

gzn≥Fzn,z, . . . ,zn– (if n is even)

(.)

then there exist x,x, . . . ,xn_{∈X such that}

gx=Fx,x, . . . ,xn,

gx=Fx, . . . ,xn,x,

.. .

gxn=Fxn,x, . . . ,xn–,

that is,F and g have a n-tuplet coincidence point.

Proof It is sufficient to takeφ=mtwithm∈[, ) in previous theorem.

3 Uniqueness ofn-tuplet fixed point

For all (x_,x_{, . . . ,x}n_{), (y}_,y_{, . . . ,y}n_)∈Xn_,

x,x, . . . ,xn≤y,y, . . . ,yn

⇐⇒ x≤y, x≥y, . . . , xn≤yn (ifnis odd),

xn≥yn (ifnis even). (.)

We say that (x_,x_{, . . . ,x}n_{) is equal to (y}_,y_{, . . . ,y}n_{) if and only ifx}_=y_,x_=y_{, . . . ,x}n_=yn_.

Theorem  In addition to hypothesis Theorem , assume that for all (x_,x_{, . . . ,x}n_),

(y,y, . . . ,yn)∈Xnthere exist(z,z, . . . ,zn)∈Xnsuch that

Fz,z, . . . ,zn,Fz, . . . ,zn,z, . . . ,Fzn,z, . . . ,zn–

is comparable to

Fx,x, . . . ,xn,Fx, . . . ,xn,x, . . . ,Fxn,x, . . . ,xn–

and

Then F and g have a unique n-tuplet common fixed point,that is,there exist(x_,x_{, . . . ,x}n_)∈ Xn_{such that}

x=gx=Fx,x, . . . ,xn,

x=gx=Fx, . . . ,xn,x,

.. .

xn=gxn=Fxn,x, . . . ,xn–.

Proof From the Theorem , the set ofn-tuplet coincidences is non-empty. We will show that if (x,x, . . . ,xn) and (y,y, . . . ,yn) aren-tuplet coincidence points, that is, if

gx=Fx,x, . . . ,xn,

gx=Fx, . . . ,xn,x,

.. .

gxn=Fxn,x, . . . ,xn–

and

gy=Fy,y, . . . ,yn,

gy=Fy, . . . ,yn,y,

.. .

gyn=Fyn,y, . . . ,yn–,

then

gx=gy,

gx=gy,

.. .

gxn=gyn.

(.)

By assumption there is (z_,z_{, . . . ,z}n_)∈Xn_{such that}

Fz,z, . . . ,zn,Fz, . . . ,zn,z, . . . ,Fzn,z, . . . ,zn– (.)

is comparable with

Fx,x, . . . ,xn,Fx, . . . ,xn,x, . . . ,Fxn,x, . . . ,xn– (.)

and

Define sequences (g(z

k)), (g(zk)), . . . , (g(znk)) such thatz=z,z=z, . . . ,zn=znand

gz_k=Fz_k–,z_k–, . . . ,zn_k–,

gz_k=Fz_k–, . . . ,zn_k–,z_k–,

.. .

gz_kn=Fzn_k–,z_k–, . . . ,z_kn–_–.

(.)

Since (.) and (.) comparable with (.), we may assume that

gx,gx, . . . ,gxn≥gz,gz, . . . ,gzn=gz_,gz_, . . . ,gz_n.

By using (.), we get that

gx,gx, . . . ,gxn≥gz_k,gz_k, . . . ,gzn_k

for allk. From (.), we have

gx≤gz_k,

gx≥gz_k, ..

.

gxn≤gzn_k (ifnis odd),

gxn≥gzn_k (ifnis even).

(.)

By (.) and (.), we have

dgx,gz_k+

=dFx,x, . . . ,xn,Fz_k,z_k, . . . ,zn_k

≤φ



n

dgx,gz_k+dgx,gz_k+· · ·+dgxn_,gzn k

, (.)

dgx,gz_k+

=dFx, . . . ,xn,x,Fz_k, . . . ,zn_k,z_k

≤φ



n

dgx,gz_k+· · ·+dgxn,gzn_k+dgx,gz_k, (.) ..

.

dgxn,gzn_k+

=dFxn,x, . . . ,xn–,Fzn_k,z_k, . . . ,zn_k–

≤φ



n

Adding (.)-(.), we get

d(g(x_),g(z

k+)) +d(g(x),g(zk+)) +· · ·+d(g(xn),g(znk+))

n

≤φ



n

dgx,gz_k+· · ·+dgxn,gzn_k+dgx,gz_k. (.)

Hence, it follows

d(g(x_),g(z

k+)) +d(g(x),g(zk+)) +· · ·+d(g(xn),g(znk+))

n

≤φk



n

dgx,gz_k+· · ·+dgxn,gzn_k+dgx,gz_k (.)

for eachk≥. It is known thatφ(t) <tandlimr→t+φ(r) <timplylim_k→∞φk(t) =  for

eacht> . Thus, from (.)

lim

k→∞d

gx,gz_k+= ,

lim

k→∞d

gx,gz_k+= ,

.. .

lim

k→∞d

gx,gz_k+= .

(.)

Analogously, we can show that

lim

k→∞d

gy,gz_k_+= ,

lim

k→∞d

gy,gz_k+= ,

.. .

lim

k→∞d

gy,gz_k+= .

(.)

Combining (.) and (.) and by using the triangle inequality

dgx,gy≤dgx,gz_k++dgy,gz_k+→ ask→ ∞,

dgx,gy≤dgx,gz_k++dgy,gz_k+→ ask→ ∞, ..

.

dgxn,gyn≤dgxn,gzn_k++dgyn,gzn_k+→ ask→ ∞.

(.)

By commutativity ofFandg,

ggx=gFx,x, . . . ,xn=Fgx,gx, . . . ,gxn,

ggx=gFx, . . . ,xn,x=Fgx, . . . ,gxn,gx,

.. .

ggxn=gFxn,x, . . . ,xn–=Fgxn,gx, . . . ,gxn–.

(.)

Denoteg(x_{) =w}_,g(x_{) =w}_{, . . . ,g(x}n_{) =w}n_{. Since (.), we get}

gw=Fw,w, . . . ,wn,

gw=Fw, . . . ,wn,w,

.. .

gwn=Fwn,w, . . . ,wn–.

(.)

Thus, (w_,w_{, . . . ,w}n_{) is an-tuplet coincidence point. Then from assumption in theorem}

withy_=w_{, . . . ,y}n_=wn_{it followsg(w}_{) =g(x}_),g(w_{) =g(x}_{), . . . ,g(w}n_{) =g(x}n_{), that is}

gw=w, . . . , gwn=wn. (.)

From (.) and (.),

w=gw=Fw,w, . . . ,wn,

w=gw=Fw, . . . ,wn,w,

.. .

wn=gwn=Fwn,w, . . . ,wn–.

Therefore, (w_,w_{, . . . ,w}n_{) isn-tuplet common fixed point ofFandg. To prove the}

unique-ness, assume that (u_,u_{, . . . ,u}n_{) is anothern-tuplet common fixed point. Then by}

assump-tion in theorem we have

u=gu=gw,

u=gu=gw,

.. .

un=gun=gwn.

property and

dFx,x,x, . . . ,xn,Fy,y,y, . . . ,yn

≤φ

d(g(x_),g(y_{)) +d(g(x}_),g(y_{)) +· · ·+d(g(x}n_),g(yn₎₎ n

(.)

for all x,x,x, . . . ,xn,y,y,y, . . . ,yn∈_{X for which x}≤_y_,x≥_y_{, . . . ,x}n≤_yn_{(if n is odd),}

xn_≥yn_{(if n is even).Suppose there exist x}

,x,x, . . . ,xn∈X such that

x_≤Fx_,x_,x_, . . . ,xn_,

x_≥Fx_,x_, . . . ,xn_,x_,

xn_≤Fxn_,x_,x_, . . . ,xn_– (if n is odd), ..

.

xn_≥Fxn_,x_,x_, . . . ,xn_– (if n is even).

(.)

Assume also that Conditionholds.Then there exist x_,x_,x_{, . . . ,x}n_{such that}

x=Fx,x,x, . . . ,xn,

x=Fx,x, . . . ,xn,x,

.. .

xn=Fxn,x,x, . . . ,xn–.

(.)

That is F has a n-tuplet fixed point.

Proof Takeg(x) =x, then the assumption in Theorem  are satisfied. Thus, we get the

result.

Corollary  Let(X,≤)be partially ordered set and suppose that(X,d)is complete metric space.Suppose F:Xn_{→X and there existφ∈such that F has the mixed g-monotone} property and there exist m∈[, )with

φFx,x, . . . ,xn,Fy,y, . . . ,yn

≤m

n

dgx,gy+dgx,gy+· · ·+dgxn,gyn (.)

for all x_,x_{, . . . ,x}n_,y_,y_{, . . . ,y}n_{∈X for which x}_≤y_{, x}_≥y_{, . . . , x}n_≤yn_{(if n is odd),} xn≥yn(if n is even).Suppose there exist x,x,x, . . . ,xn∈X such that

x_≤Fx_,x_,x_, . . . ,xn_,

x_≥Fx_,x_, . . . ,xn_,x_, ..

xn_≤Fxn_,x_,x_, . . . ,xn_– (if n is odd),

xn_≥Fxn_,x_,x_, . . . ,xn_– (if n is even).

Assume also that Conditionholds.Then there exist x_,x_,x_{, . . . ,x}n_{such that}

x=Fx,x,x, . . . ,xn,

x=Fx_,x_, . . . ,xn_,x_,

.. .

xn=Fxn,x,x, . . . ,xn–.

(.)

That is F and g have n-tuplet coincidence point.

Proof Takingφ(t) =m·twithm∈[, ) in above corollary we obtain this corollary.

Competing interests

The authors declare that they have no competing interests.

Author details

1_{Department of Mathematics, The Faculty of Arts and Sciences, Yildiz Technical University, Davutpasa Campus, Esenler,} Istanbul, 34210, Turkey.2_{Current address: Department of Mathematical Engineering, Yildiz Technical University,} Davutpasa Campus, Esenler, Istanbul, 34220, Turkey.

Acknowledgements

This work is supported by Yildiz Technical University Scientific Research Projects Coordination Unit under the project number BAPK 2012-07-03-DOP03.

Received: 4 January 2013 Accepted: 5 April 2013 Published: 22 April 2013

References

1. Banach, S: Sur les operations dans les ensembles abstraits et leur applications aux equations integrales. Fundam. Math.3, 133-181 (1922)

2. 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). doi:10.1090/s0002-9939-03-07220-4

3. Nieto, J, Rodríguez-López, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order22(3), 223-239 (2005). doi:10.1007/s11083-005-9018-5

4. Bhaskar, TG, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal., Theory Methods Appl.65(7), 1379-1393 (2006). doi:10.1016/j.na.2005.10.017

5. Abbas, M, Khan, MA, Radenovic, S: Common coupled fixed point theorems in cone metric spaces forw-compatible mappings. Appl. Math. Comput.217(1), 195-202 (2010). doi:10.1016/j.amc.2010.05.042

6. Abbas, M, Damjanovic, B, Lazovic, R: Fuzzy common fixed point theorems for generalized contractive mappings. Appl. Math. Lett.23(11), 1326-1330 (2010). doi:10.1016/j.aml.2010.06.023

7. Aydi, H, Karapinar, E, Shatanawi, W: Coupled fixed point results for (ψ,φ)-weakly contractive condition in ordered partial metric spaces. Comput. Math. Appl.62(12), 4449-4460 (2011). doi:10.1016/j.camwa.2011.10.021 8. Berinde, V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric

spaces. Nonlinear Anal., Theory Methods Appl.74(18), 7347-7355 (2011). doi:10.1016/j.na.2011.07.053 9. Berinde, V: Coupled coincidence point theorems for mixed monotone nonlinear operators. Comput. Math. Appl.

64(6), 1770-1777 (2012). doi:10.1016/j.camwa.2012.02.012

10. Gordji, ME, Ghods, S, Ghods, V, Hadian, M: Coupled fixed point theorem for generalized fuzzy Meir-Keeler contraction in fuzzy metric spaces. J. Comput. Anal. Appl.14(2), 271-277 (2012)

11. Lakshmikantham, V, Ciric, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal., Theory Methods Appl.70(12), 4341-4349 (2009). doi:10.1016/j.na.2008.09.020

12. Berinde, V, Borcut, M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal., Theory Methods Appl.74(15), 4889-4897 (2011). doi:10.1016/j.na.2011.03.032

13. Aydi, H, Karapınar, E, Yüce, IS: Quadruple fixed point theorems in partially ordered metric spaces depending on another function. ISRN Appl. Math. (2012). doi:10.5402/2012/539125

14. Karapinar, E, Shatanawi, W, Mustafa, Z: Quadruple fixed point theorems under nonlinear contractive conditions in partially ordered metric spaces. J. Appl. Math. (2012). doi:10.1155/2012/951912

15. Mustafa, Z, Aydi, H, Karapinar, E: Mixedg-monotone property and quadruple fixed point theorems in partially ordered metric spaces. Fixed Point Theory Appl.20121-19 (2012). doi:10.1186/1687-1812-2012-71

doi:10.1186/1029-242X-2013-196