# Some common fixed point theorems for a family of non-self mappings in cone metric spaces

- Xianjiu Huang
^{1}, - Chuanxi Zhu
^{1}Email author, - Xi Wen
^{2}and - Ljubica Lalović
^{3}

**2013**:144

https://doi.org/10.1186/1687-1812-2013-144

© Huang et al.; licensee Springer. 2013

**Received: **17 December 2012

**Accepted: **15 April 2013

**Published: **4 June 2013

## Abstract

Some common fixed point theorems for a family of non-self mappings defined on a closed subset of a metrically convex cone metric space (over the cone which is not necessarily normal) are obtained which generalize earlier results due to Imdad *et al.* and Janković *et al.*

**MSC:**47H10, 54H25.

## Keywords

## 1 Introduction and preliminaries

The existing literature of fixed point theory contains many results enunciating fixed point theorems for self-mappings in metric and Banach spaces. Recently, Huang and Zhang [1] have replaced the real numbers by ordering Banach space and defining cone metric space. They have proved some fixed point theorems of contractive mappings on cone metric spaces. The study of fixed point theorems in such spaces is followed by some other mathematicians; see [2–17]. However, fixed point theorems for non-self mappings are not frequently discussed and so they form a natural subject for further investigation. The study of fixed point theorems for non-self mappings in metrically convex metric spaces was initiated by Assad and Kirk [18]. Recently, Janković *et al.* [10] obtained a fixed point theorem for two non-self mappings in cone metric spaces. Motivated by Janković *et al.* [10], we prove some common fixed point theorems for a family of non-self mappings on cone metric spaces in which the cone need not be normal.

Consistent with Huang and Zhang [1], the following definitions and results will be needed in the sequel.

*E*be a real Banach space. A subset

*P*of

*E*is called a cone if and only if:

- (a)
*P*is closed, nonempty and $P\ne \{\theta \}$; - (b)
$a,b\in R$, $a,b\ge 0$, $x,y\in P$ implies $ax+by\in P$;

- (c)
$P\cap (-P)=\{\theta \}$.

*P*by $x\u2aafy$ if and only if $y-x\in P$. A cone

*P*is called normal if there is a number $K>0$ such that for all $x,y\in E$,

The least positive number *K* satisfying the above inequality is called the normal constant of *P*, while $x\ll y$ stands for $y-x\in intP$ (interior of *P*).

**Definition 1.1** [1]

*X*be a nonempty set. Suppose that the mapping $d:X\times X\to E$ satisfies:

- (d1)
$\theta \u2aafd(x,y)$ for all $x,y\in X$ and $d(x,y)=\theta $ if and only if $x=y$;

- (d2)
$d(x,y)=d(y,x)$ for all $x,y\in X$;

- (d3)
$d(x,y)\u2aafd(x,z)+d(z,y)$ for all $x,y,z\in X$.

Then *d* is called a cone metric on *X* and $(X,d)$ is called a cone metric space.

The concept of a cone metric space is more general than that of a metric space.

**Definition 1.2** [1]

- (e)
a Cauchy sequence if for every $c\in E$ with $\theta \ll c$, there is an

*N*such that for all $n,m>N$, $d({x}_{n},{x}_{m})\ll c$; - (f)
a convergent sequence if for every $c\in E$ with $\theta \ll c$, there is an

*N*such that for all $n>N$, $d({x}_{n},x)\ll c$ for some fixed $x\in X$.

A cone metric space *X* is said to be complete if for every Cauchy sequence in *X*, it is convergent in *X*. It is known that $\{{x}_{n}\}$ converges to $x\in X$ if and only if $d({x}_{n},x)\to \theta $ as $n\to \mathrm{\infty}$. It is a Cauchy sequence if and only if $d({x}_{n},{x}_{m})\to \theta $ ($n,m\to \mathrm{\infty}$).

**Remark 1.1** [19]

Let *E* be an ordered Banach (normed) space. Then *c* is an interior point of *P*, if and only if $[-c,c]$ is a neighborhood of *θ*.

**Corollary 1.1** [9]

(1) *If* $a\u2aafb$ *and* $b\ll c$, *then* $a\ll c$.

*Indeed*, $c-a=(c-b)+(b-a)\u2ab0c-b$ *implies* $[-(c-a),c-a]\supseteq [-(c-b),c-b]$.

(2) *If* $a\ll b$ *and* $b\ll c$, *then* $a\ll c$.

*Indeed*, $c-a=(c-b)+(b-a)\u2ab0c-b$ *implies* $[-(c-a),c-a]\supseteq [-(c-b),c-b]$.

(3) *If* $\theta \u2aafu\ll c$ *for each* $c\in intP$, *then* $u=\theta $.

**Remark 1.2** [11]

If $c\in intP$, $\theta \u2aaf{a}_{n}$ and ${a}_{n}\to \theta $, then there exists an ${n}_{0}$ such that for all $n>{n}_{0}$, we have ${a}_{n}\ll c$.

**Remark 1.3** [11]

If *E* is a real Banach space with cone *P* and if $a\u2aafka$, where $a\in P$ and $0<k<1$, then $a=\theta $.

We find it convenient to introduce the following definition.

**Definition 1.3** [11]

*C*be a nonempty closed subset of

*X*, and let $f,g:C\to X$ be non-self mappings. Denote, for $x,y\in C$,

*f*is called a generalized ${g}_{{M}_{1}}$-contractive mapping of

*C*into

*X*if, for some $\lambda \in (0,\sqrt{2}-1)$, there exists $u(x,y)\in {M}_{1}^{f,g}$ such that for all $x,y\in C$,

**Definition 1.4** [2]

Let *f* and *g* be self-maps of a set *X* (*i.e.*, $f,g:X\to X$). If $w=fx=gx$ for some *x* in *X*, then *x* is called a coincidence point of *f* and *g*, and *w* is called a point of coincidence of *f* and *g*. Self-maps *f* and *g* are said to be coincidentally commuting if they commute at their coincidence point; *i.e.*, if $fx=gx$ for some $x\in X$, then $fgx=gfx$.

## 2 Main results

Recently, Janković *et al.* [10] proved some fixed point theorems for a pair of non-self mappings defined on a nonempty closed subset of complete metrically convex cone metric spaces with new contractive conditions.

**Theorem 2.1** [10]

*Let*$(X,d)$

*be a complete cone metric space*,

*let*

*C*

*be a nonempty closed subset of*

*X*

*such that for each*$x\in C$

*and*$y\notin C$

*there exists a point*$z\in \partial C$ (

*the boundary of*

*C*)

*such that*

*Suppose that*$f,g:C\to X$

*are such that*

*f*

*is a generalized*${g}_{{M}_{1}}$-

*contractive mapping of*

*C*

*into*

*X*,

*and*

- (i)
$\partial C\subseteq gC$, $fC\cap C\subseteq gC$,

- (ii)
$gx\in \partial C\Rightarrow fx\in C$,

- (iii)
*gC**is closed in**X*.

*Then the pair* $(f,g)$ *has a coincidence point*. *Moreover*, *if* $(f,g)$ *are coincidentally commuting*, *then* *f* *and* *g* *have a unique common fixed point*.

The purpose of this paper is to extend the above theorem for a family of non-self mappings in cone metric spaces. We begin with the following definition.

**Definition 2.1**Let $(X,d)$ be a complete cone metric space, let

*C*be a nonempty closed subset of

*X*, and let ${\{{F}_{n}\}}_{n=1}^{\mathrm{\infty}},S,T:C\to X$ be non-self mappings. Denote, for $x,y\in C$,

*C*into

*X*if for some $\lambda \in (0,1)$ there exists $u(x,y)\in {M}_{1}^{{F}_{n},S,T}$ such that for all $x,y\in C$ with $x\ne y$,

Notice that by setting ${F}_{i}={F}_{j}=f$, $T=S=g$ and $\lambda \in (0,\sqrt{2}-1)$ in (2.1), one deduces a slightly generalized form of (1.1).

We state and prove our main result as follows.

**Theorem 2.2**

*Let*$(X,d)$

*be a complete cone metric space*,

*let*

*C*

*be a nonempty closed subset of*

*X*

*such that for each*$x\in C$

*and*$y\notin C$

*there exists a point*$z\in \partial C$

*such that*

*Suppose that*${F}_{n},S,T:C\to X$

*are such that*$({F}_{i},{F}_{j})$

*is a pair of generalized*${(T,S)}_{{M}_{1}}$-

*contractive mappings of*

*C*

*into*

*X*

*for all*$i=2n-1$, $j=2n$ ($n\in N$),

*and*

- (I)
$\partial C\subseteq SC\cap TC$, ${F}_{i}C\cap C\subseteq SC$, ${F}_{j}C\cap C\subseteq TC$,

- (II)
$Tx\in \partial C$

*implies that*${F}_{i}x\in C$, $Sx\in \partial C$*implies that*${F}_{j}x\in C$, - (III)
*SC**and**TC*(*or*${F}_{i}C$*and*${F}_{j}C$)*are closed in**X*.

*Then*

- (IV)
$({F}_{i},T)$

*has a point of coincidence*, - (V)
$({F}_{j},S)$

*has a point of coincidence*.

*Moreover*, *if* $({F}_{i},T)$ *and* $({F}_{j},S)$ *are coincidentally commuting pairs*, *then* ${\{{F}_{n}\}}_{n=1}^{\mathrm{\infty}}$, *S* *and* *T* *have a unique common fixed point*.

*Proof*Let $x\in \partial C$ be arbitrary. Then (due to $\partial C\subseteq TC$) there exists a point ${x}_{0}\in C$ such that $x=T{x}_{0}$. Since $T{x}_{0}\in \partial C$, from (I) and (II), we have ${F}_{1}{x}_{0}\in {F}_{1}C\cap C\subseteq SC$. Thus, there exists ${x}_{1}\in C$ such that ${y}_{1}=S{x}_{1}={F}_{1}{x}_{0}\in C$. Since ${y}_{1}={F}_{1}{x}_{0}$, there exists a point ${y}_{2}={F}_{2}{x}_{1}$ such that

- (a)
${y}_{2n}={F}_{2n}{x}_{2n-1}$, ${y}_{2n+1}={F}_{2n+1}{x}_{2n}$,

- (b)${y}_{2n}\in C$ implies that ${y}_{2n}=T{x}_{2n}$ or ${y}_{2n}\notin C$ implies that $T{x}_{2n}\in \partial C$ and$d(S{x}_{2n-1},T{x}_{2n})+d(T{x}_{2n},{y}_{2n})=d(S{x}_{2n-1},{y}_{2n}),$
- (c)${y}_{2n+1}\in C$ implies that ${y}_{2n+1}=S{x}_{2n+1}$ or ${y}_{2n+1}\notin C$ implies that $S{x}_{2n+1}\in \partial C$ and$d(T{x}_{2n},S{x}_{2n+1})+d(S{x}_{2n+1},{y}_{2n+1})=d(T{x}_{2n},{y}_{2n+1}).$

Note that $(T{x}_{2n},S{x}_{2n+1})\notin {P}_{1}\times {Q}_{1}$, as if $T{x}_{2n}\in {P}_{1}$, then ${y}_{2n}\ne T{x}_{2n}$ and one infers that $T{x}_{2n}\in \partial C$, which implies that ${y}_{2n+1}={F}_{2n+1}{x}_{2n}\in C$. Hence ${y}_{2n+1}=S{x}_{2n+1}\in {Q}_{0}$. Similarly, one can argue that $(S{x}_{2n-1},T{x}_{2n})\notin {Q}_{1}\times {P}_{1}$.

Now, we distinguish the following three cases.

*n*such that at least one of the following three cases holds:

- (1)
$d(T{x}_{2n},S{x}_{2n+1})\u2aaf\lambda d({y}_{2n-1},{y}_{2n})=\lambda d(S{x}_{2n-1},T{x}_{2n})$;

- (2)
$d(T{x}_{2n},S{x}_{2n+1})\u2aaf\lambda d({y}_{2n},{y}_{2n+1})$ implies that $d(T{x}_{2n},S{x}_{2n+1})=\theta \u2aaf\lambda d(S{x}_{2n-1},T{x}_{2n})$;

- (3)
$d(T{x}_{2n},S{x}_{2n+1})\u2aaf\lambda \frac{d({y}_{2n-1},{y}_{2n+1})}{2}\u2aaf\frac{\lambda}{2}d({y}_{2n-1},{y}_{2n})+\frac{1}{2}d({y}_{2n},{y}_{2n+1})$ implies that $d(T{x}_{2n},S{x}_{2n+1})\u2aaf\lambda d(S{x}_{2n-1},T{x}_{2n})$.

Now, proceeding as in Case 1, we have that (2.3) holds.

in view of Case 1.

and we proved (2.9).

in view of Case 1.

and we proved (2.13).

This implies that $d(S{x}_{2n+1},T{x}_{2n+2})\u2aafd(S{x}_{2n+1},{y}_{2n+2})$.

in view of Case 1.

From Remark 1.2 and Corollary 1.1(1), $d(T{x}_{2n},S{x}_{2m+1})\ll c$.

*z*. Furthermore, subsequences $\{T{x}_{2{n}_{k}}\}$ and $\{S{x}_{2{n}_{k}+1}\}$ both converge to $z\in C$ as

*C*is a closed subset of a complete cone metric space $(X,d)$. We assume that there exists a subsequence $\{T{x}_{2{n}_{k}}\}\subseteq {P}_{0}$ for each $k\in N$, and

*TC*as well as

*SC*are closed in

*X*. Since $\{T{x}_{2{n}_{k}}\}$ is a Cauchy sequence in

*TC*, it converges to a point $z\in TC$. Let $w\in {T}^{-1}z$, then $Tw=z$. Similarly, $\{S{x}_{2{n}_{k}+1}\}$ being a subsequence of the Cauchy sequence $\{T{x}_{0},S{x}_{1},T{x}_{2},S{x}_{3},\dots ,S{x}_{2n-1},T{x}_{2n},S{x}_{2n-1},\dots \}$ also converges to

*z*as

*SC*is closed. Using (2.2), one can write

for any odd integer $i\in N$ and even integer $j\in N$.

*n*.

- (1)
$d({F}_{i}w,z)\u2aaf\lambda d(z,S{x}_{2{n}_{k}-1})+d({F}_{j}{x}_{2{n}_{k}-1},z)\ll \lambda \frac{c}{2\lambda}+\frac{c}{2}=c$;

- (2)
$d({F}_{i}w,z)\u2aaf\lambda d(z,{F}_{i}w)+d({F}_{j}{x}_{2{n}_{k}-1},z)\Rightarrow d({F}_{i}w,z)\u2aaf\frac{1}{1-\lambda}d({F}_{j}{x}_{2{n}_{k}-1},z)\ll \frac{1}{1-\lambda}(1-\lambda )c=c$;

- (3)$\begin{array}{rcl}d({F}_{i}w,z)& \u2aaf& \lambda d(S{x}_{2{n}_{k}-1},{F}_{j}{x}_{2{n}_{k}-1})+d({F}_{j}{x}_{2{n}_{k}-1},z)\\ \u2aaf& \lambda (d(S{x}_{2{n}_{k}-1},z)+d(z,{F}_{j}{x}_{2{n}_{k}-1}))+d({F}_{j}{x}_{2{n}_{k}-1},z)\\ \u2aaf& (\lambda +1)d({F}_{j}{x}_{2{n}_{k}-1},z)+\lambda d(S{x}_{2{n}_{k}-1},z)\ll (\lambda +1)\frac{c}{2(\lambda +1)}+\lambda \frac{c}{2\lambda}=c;\end{array}$
- (4)$\begin{array}{c}\begin{array}{rl}d({F}_{i}w,z)& \u2aaf\lambda \frac{d(z,{F}_{j}{x}_{2{n}_{k}-1})+d({F}_{i}w,S{x}_{2{n}_{k}-1})}{2}+d({F}_{j}{x}_{2{n}_{k}-1},z)\\ \u2aaf\lambda \frac{d(z,{F}_{j}{x}_{2{n}_{k}-1})+d(z,S{x}_{2{n}_{k}-1})}{2}+\frac{1}{2}d({F}_{i}w,z)+d({F}_{j}{x}_{2{n}_{k}-1},z)\end{array}\hfill \\ \begin{array}{rl}\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d({F}_{i}w,z)& \u2aaf(2+\lambda )d({F}_{j}{x}_{2{n}_{k}-1},z)+\lambda d(z,S{x}_{2{n}_{k}-1})\\ \ll (2+\lambda )\frac{c}{2(2+\lambda )}+\lambda \frac{c}{2\lambda}=c.\end{array}\hfill \end{array}$

In all cases, we obtain $d({F}_{i}w,z)\ll c$ for each $c\in intP$. Using Corollary 1.1(3), it follows that $d({F}_{i}w,z)=\theta $ or ${F}_{i}w=z$. Thus, ${F}_{i}w=z=Tw$, that is, *z* is a coincidence point of ${F}_{i}$, *T* for any odd integer $i\in N$.

for any odd integer $i\in N$ and even integer $j\in N$.

Using Remark 1.3 and Corollary 1.1(3), it follows that $Sv={F}_{j}v$; therefore, $Sv=z={F}_{j}v$, that is, *z* is a coincidence point of $({F}_{j},S)$ for any even integer $j\in N$.

In case ${F}_{i}C$ and ${F}_{j}C$ are closed in *X*, then $z\in {F}_{i}C\cap C\subseteq SC$ or $z\in {F}_{j}C\cap C\subseteq TC$. The analogous arguments establish (IV) and (V). If we assume that there exists a subsequence $\{S{x}_{2{n}_{k}+1}\}\subseteq {Q}_{0}$ with *TC* as well as *SC* closed in *X*, then noting that $\{S{x}_{2{n}_{k}+1}\}$ is a Cauchy sequence in *SC*, foregoing arguments establish (IV) and (V).

Using Remark 1.3 and Corollary 1.1(3), it follows that ${F}_{i}z=z$. Thus, ${F}_{i}z=z=Tz$.

Similarly, we can prove ${F}_{j}z=z=Sz$. Therefore $z={F}_{i}z={F}_{j}z=Sz=Tz$, that is, *z* is a common fixed point of ${F}_{n}$, *S* and *T*.

The uniqueness of the common fixed point follows easily from (2.2). □

**Example 2.1**Let $E={C}^{1}([0,1],R)$, $P=\{\phi \in E:\phi (t)\ge 0,t\in [0,1]\}$, $X=[0,+\mathrm{\infty})$, $C=[0,2]$ and $d:X\times X\to E$ defined by $d(x,y)=|x-y|\phi $, where $\phi \in P$ is a fixed function,

*e.g.*, $\phi (t)={e}^{t}$. Then $(X,d)$ is a complete cone metric space with a non-normal cone having the nonempty interior. Define ${F}_{i}$, ${F}_{j}$,

*S*and $T:C\to X$ as

Since $\partial C=\{0,2\}$. Clearly, for each $x\in C$ and $y\notin C$, there exists a point $z=2\in \partial C$ such that $d(x,z)+d(z,y)=d(x,y)$. Further, $SC\cap TC=[0,20]\cap [0,10]=[0,10]\supset \{0,2\}=\partial C$, ${F}_{i}C\cap C=[\frac{4}{5},\frac{14}{5}]\cap [0,2]=[\frac{4}{5},2]\subset SC$, ${F}_{j}C\cap C=[\frac{4}{5},\frac{24}{5}]\cap [0,2]=[\frac{4}{5},2]\subset TC$, and *SC*, *TC*, ${F}_{i}C$ and ${F}_{j}C$ are closed in *X*.

that is, (2.2) is satisfied with $\lambda =\frac{1}{5}$.

Evidentially, $1=T(\frac{1}{5})={F}_{i}(\frac{1}{5})\ne \frac{1}{5}$ and $1=S(\frac{1}{\sqrt{5}})={F}_{j}(\frac{1}{\sqrt{5}})\ne \frac{1}{\sqrt{5}}$. Notice that two separate coincidence points are not common fixed points as ${F}_{i}T(\frac{1}{5})\ne T{F}_{i}(\frac{1}{5})$ and $S{F}_{j}(\frac{1}{\sqrt{5}})\ne {F}_{j}S(\frac{1}{\sqrt{5}})$, which shows the necessity of coincidentally commuting property in Theorem 2.2.

Next, we furnish an illustrative example in support of our result. In doing so, we are essentially inspired by Imdad and Kumar [21].

**Example 2.2**Let $E={C}^{1}([0,1],R)$, $P=\{\phi \in E:\phi (t)\ge 0,t\in [0,1]\}$, $X=[1,+\mathrm{\infty})$, $C=[1,3]$ and $d:X\times X\to E$ defined by $d(x,y)=|x-y|\phi $, where $\phi \in P$ is a fixed function,

*e.g.*, $\phi (t)={e}^{t}$. Then $(X,d)$ is a complete cone metric space with a non-normal cone having the nonempty interior. Define ${F}_{i}$, ${F}_{j}$,

*S*and $T:C\to X$ as

Note that $\partial C=\{1,3\}$. Clearly, for each $x\in C$ and $y\notin C$, there exists a point $z=3\in \partial C$ such that $d(x,z)+d(z,y)=d(x,y)$. Further, $SC\cap TC=[1,253]\cap [1,61]=[1,61]\supset \{1,3\}=\partial C$, ${F}_{i}C\cap C=[1,\frac{n+3}{n}]\cap [1,3]\subset SC$ and ${F}_{j}C\cap C=[1,\frac{n+7}{n}]\cap [1,3]\subset TC$.

Therefore, condition (2.2) is satisfied if we choose $\lambda =max\{\frac{1}{4n({x}^{2}+2)},\frac{1}{4n({x}^{2}+{y}^{3})}\}\in (0,1)$. Moreover, 1 is a point of coincidence as $T1={F}_{i}1$ as well as $S1={F}_{j}1$, whereas both the pairs $({F}_{i},T)$ and $({F}_{j},S)$ are weakly compatible as $T{F}_{i}1=1={F}_{i}T1$ and $S{F}_{j}1=1={F}_{j}S1$. Also, *SC*, *TC*, ${F}_{i}C$ and ${F}_{j}C$ are closed in *X*. Thus, all the conditions of Theorem 2.2 are satisfied and 1 is the unique common fixed point of ${F}_{i}$, ${F}_{j}$, *S* and *T*. One may note that 1 is also a point of coincidence for both the pairs $({F}_{i},T)$ and $({F}_{j},S)$.

**Remark 2.1** Setting ${F}_{i}=F$ and ${F}_{j}=G$ in Theorem 2.2, we obtain the following result.

**Corollary 2.1**

*Let*$(X,d)$

*be a complete cone metric space*,

*let*

*C*

*be a nonempty closed subset of*

*X*

*such that for each*$x\in C$

*and*$y\notin C$

*there exists a point*$z\in \partial C$

*such that*

*Suppose that*$F,G,S,T:C\to X$

*are such that*$(F,G)$

*is a pair of generalized*${(T,S)}_{{M}_{1}}$-

*contractive mappings of*

*C*

*into*

*X*,

*and*

- (I)
$\partial C\subseteq SC\cap TC$, $FC\cap C\subseteq SC$, $GC\cap C\subseteq TC$,

- (II)
$Tx\in \partial C$

*implies that*$Fx\in C$, $Sx\in \partial C$*implies that*$Gx\in C$, - (III)
*SC**and**TC*(*or**FC**and**GC*)*are closed in**X*.

*Then*

- (IV)
$(F,T)$

*has a point of coincidence*, - (V)
$(G,S)$

*has a point of coincidence*.

*Moreover*, *if* $(F,T)$ *and* $(G,S)$ *are coincidentally commuting pairs*, *then* *F*, *G*, *S* *and* *T* *have a unique common fixed point*.

**Remark 2.2** 1. Theorem 2.2 in [10] is a special case of Theorem 2.2 with ${F}_{i}={F}_{j}=f$, $T=S=g$ and $\lambda \in (0,\sqrt{2}-1)$.

2. Setting ${F}_{i}={F}_{j}=f$ and $T=S={I}_{X}$ (the identity mapping on *X*) in Theorem 2.2, we obtain the following result.

**Corollary 2.2**

*Let*$(X,d)$

*be a complete cone metric space*,

*and let*

*C*

*be a nonempty closed subset of*

*X*

*such that for each*$x\in C$

*and*$y\notin C$

*there exists a point*$z\in \partial C$

*such that*

*Suppose that*$f:C\to X$

*satisfies the condition*

*where*

*for all* $x,y\in C$, $\lambda \in [0,1)$ *and* *f* *has the additional property that for each* $x\in \partial C$, $fx\in C$. *Then* *f* *has a unique fixed point*.

**Remark 2.3** The following definition is a special case of Definition 2.1 when $(X,d)$ is a metric space. But when $(X,d)$ is a cone metric space, which is not a metric space, this is not true. Indeed, there may exist $x,y\in X$ such that the vectors $d(Tx,{F}_{i}x)$, $d(Sy,{F}_{j}y)$ and $\frac{d(Tx,{F}_{i}x)+d(Sy,{F}_{j}y)}{2}$ are incomparable. For the same reason Theorems 2.2 and 2.3 (given below) are incomparable.

**Definition 2.2**Let $(X,d)$ be a complete cone metric space, let

*C*be a nonempty closed subset of

*X*, and let ${\{{F}_{n}\}}_{n=1}^{\mathrm{\infty}},S,T:C\to X$ be non-self mappings. Denote, for $x,y\in C$,

*C*into

*X*if for some $\lambda \in [0,1)$ there exists $u(x,y)\in {M}_{2}^{{F}_{n},S,T}$ such that for all $x,y\in C$ with $x\ne y$,

Our next result is the following.

**Theorem 2.3**

*Let*$(X,d)$

*be a complete cone metric space*,

*let*

*C*

*be a nonempty closed subset of*

*X*

*such that for each*$x\in C$

*and*$y\notin C$

*there exists a point*$z\in \partial C$

*such that*

*Suppose that*${F}_{n},S,T:C\to X$

*are such that*$({F}_{i},{F}_{j})$

*is a pair of generalized*${(T,S)}_{{M}_{2}}$-

*contractive mappings of*

*C*

*into*

*X*

*for all*$i=2n-1$, $j=2n$ ($n\in N$),

*and*

- (I)
$\partial C\subseteq SC\cap TC$, ${F}_{i}C\cap C\subseteq SC$, ${F}_{j}C\cap C\subseteq TC$,

- (II)
$Tx\in \partial C$

*implies that*${F}_{i}x\in C$, $Sx\in \partial C$*implies that*${F}_{j}x\in C$, - (III)
*SC**and**TC*(*or*${F}_{i}C$*and*${F}_{j}C$)*are closed in**X*.

*Then*

- (IV)
$({F}_{i},T)$

*has a point of coincidence*, - (V)
$({F}_{j},S)$

*has a point of coincidence*.

*Moreover*, *if* $({F}_{i},T)$ *and* $({F}_{j},S)$ *are coincidentally commuting pairs*, *then* ${F}_{n}$, *S* *and* *T* *have a unique common fixed point*.

The proof of this theorem is very similar to the proof of Theorem 2.2 and it is omitted.

**Remark 2.4** Setting ${F}_{i}=F$ and ${F}_{j}=G$ in Theorem 2.3, we obtain the following result.

**Corollary 2.3**

*Let*$(X,d)$

*be a complete cone metric space*,

*let*

*C*

*be a nonempty closed subset of*

*X*

*such that for each*$x\in C$

*and*$y\notin C$

*there exists a point*$z\in \partial C$

*such that*

*Suppose that*$F,G,S,T:C\to X$

*are such that*$(F,G)$

*is a pair of generalized*${(T,S)}_{{M}_{2}}$-

*contractive mappings of*

*C*

*into*

*X*,

*and*

- (I)
$\partial C\subseteq SC\cap TC$, $FC\cap C\subseteq SC$, $GC\cap C\subseteq TC$,

- (II)
$Tx\in \partial C$

*implies that*$Fx\in C$, $Sx\in \partial C$*implies that*$Gx\in C$, - (III)
*SC**and**TC*(*or**FC**and**GC*)*are closed in**X*.

*Then*

- (IV)
$(F,T)$

*has a point of coincidence*, - (V)
$(G,S)$

*has a point of coincidence*.

*Moreover*, *if* $(F,T)$ *and* $(G,S)$ *are coincidentally commuting pairs*, *then* *F*, *G*, *S* *and* *T* *have a unique common fixed point*.

We now list some corollaries of Theorems 2.2 and 2.3.

**Corollary 2.4**

*Let*$(X,d)$

*be a complete cone metric space*,

*let*

*C*

*be a nonempty closed subset of*

*X*

*such that for each*$x\in C$

*and*$y\notin C$

*there exists a point*$z\in \partial C$

*such that*

*Let*${F}_{n},S,T:C\to X$

*be such that*

*for some* $\lambda \in [0,1)$ *and for all* $i=2n-1$, $j=2n$ ($n\in N$), $x,y\in C$ *with* $x\ne y$.

*Suppose*,

*further*,

*that*${F}_{n}$,

*S*,

*T*

*and*

*C*

*satisfy the following conditions*:

- (I)
$\partial C\subseteq SC\cap TC$, ${F}_{i}C\cap C\subseteq SC$, ${F}_{j}C\cap C\subseteq TC$,

- (II)
$Tx\in \partial C$

*implies that*${F}_{i}x\in C$, $Sx\in \partial C$*implies that*${F}_{j}x\in C$, - (III)
*SC**and**TC*(*or*${F}_{i}C$*and*${F}_{j}C$)*are closed in**X*.

*Then*

- (IV)
$({F}_{i},T)$

*has a point of coincidence*, - (V)
$({F}_{j},S)$

*has a point of coincidence*.

*Moreover*, *if* $({F}_{i},T)$ *and* $({F}_{j},S)$ *are coincidentally commuting pairs*, *then* ${\{{F}_{n}\}}_{n=1}^{\mathrm{\infty}}$, *S* *and* *T* *have a unique common fixed point*.

**Corollary 2.5**

*Let*$(X,d)$

*be a complete cone metric space*,

*let*

*C*

*be a nonempty closed subset of*

*X*

*such that for each*$x\in C$

*and*$y\notin C$

*there exists a point*$z\in \partial C$

*such that*

*Let*${F}_{n},S,T:C\to X$

*be such that*

*for some* $\lambda \in [0,1/2)$ *and for all* $i=2n-1$, $j=2n$ ($n\in N$), $x,y\in C$ *with* $x\ne y$.

*Suppose*,

*further*,

*that*${F}_{n}$,

*S*,

*T*

*and*

*C*

*satisfy the following conditions*:

- (I)
$\partial C\subseteq SC\cap TC$, ${F}_{i}C\cap C\subseteq SC$, ${F}_{j}C\cap C\subseteq TC$,

- (II)
$Tx\in \partial C$

*implies that*${F}_{i}x\in C$, $Sx\in \partial C$*implies that*${F}_{j}x\in C$, - (III)
*SC**and**TC*(*or*${F}_{i}C$*and*${F}_{j}C$)*are closed in**X*.

*Then*

- (IV)
$({F}_{i},T)$

*has a point of coincidence*, - (V)
$({F}_{j},S)$

*has a point of coincidence*.

*Moreover*, *if* $({F}_{i},T)$ *and* $({F}_{j},S)$ *are coincidentally commuting pairs*, *then* ${\{{F}_{n}\}}_{n=1}^{\mathrm{\infty}}$, *S* *and* *T* *have a unique common fixed point*.

**Corollary 2.6**

*Let*$(X,d)$

*be a complete cone metric space*,

*let*

*C*

*be a nonempty closed subset of*

*X*

*such that for each*$x\in C$

*and*$y\notin C$

*there exists a point*$z\in \partial C$

*such that*

*Let*${F}_{n},S,T:C\to X$

*be such that*

*for some* $\lambda \in [0,1/2)$ *and for all* $i=2n-1$, $j=2n$ ($n\in N$), $x,y\in C$ *with* $x\ne y$.

*Suppose*,

*further*,

*that*${F}_{n}$,

*S*,

*T*

*and*

*C*

*satisfy the following conditions*:

- (I)
$\partial C\subseteq SC\cap TC$, ${F}_{i}C\cap C\subseteq SC$, ${F}_{j}C\cap C\subseteq TC$,

- (II)
$Tx\in \partial C$

*implies that*${F}_{i}x\in C$, $Sx\in \partial C$*implies that*${F}_{j}x\in C$, - (III)
*SC**and**TC*(*or*${F}_{i}C$*and*${F}_{j}C$)*are closed in**X*.

*Then*

- (IV)
$({F}_{i},T)$

*has a point of coincidence*, - (V)
$({F}_{j},S)$

*has a point of coincidence*.

*Moreover*, *if* $({F}_{i},T)$ *and* $({F}_{j},S)$ *are coincidentally commuting pairs*, *then* ${\{{F}_{n}\}}_{n=1}^{\mathrm{\infty}}$, *S* *and* *T* *have a unique common fixed point*.

**Remark 2.5** Setting ${F}_{i}={F}_{j}=f$ and $T=S=g$ in Corollaries 2.4-2.6, we obtain the following result.

**Corollary 2.7**

*Let*$(X,d)$

*be a complete cone metric space*,

*let*

*C*

*be a nonempty closed subset of*

*X*

*such that for each*$x\in C$

*and*$y\notin C$

*there exists a point*$z\in \partial C$

*such that*

*Let*$f,g:C\to X$

*be such that*

*for some*$\lambda \in [0,1)$

*and for all*$x,y\in C$.

*Suppose*,

*further*,

*that*

*f*,

*g*

*and*

*C*

*satisfy the following conditions*:

- (I)
$\partial C\subseteq gC$, $fC\cap C\subseteq gC$,

- (II)
$gx\in \partial C$

*implies that*$fx\in C$, - (III)
*gC**is closed in**X*.

*Then there exists a coincidence point* *z* *of* *f*, *g* *in* *C*. *Moreover*, *if* $(f,g)$ *are coincidentally commuting*, *then* *z* *is the unique common fixed point of* *f* *and* *g*.

**Corollary 2.8**

*Let*$(X,d)$

*be a complete cone metric space*,

*let*

*C*

*be a nonempty closed subset of*

*X*

*such that for each*$x\in C$

*and*$y\notin C$

*there exists a point*$z\in \partial C$

*such that*

*Let*$f,g:C\to X$

*be such that*

*for some*$\lambda \in [0,1/2)$

*and for all*$x,y\in C$.

*Suppose*,

*further*,

*that*

*f*,

*g*

*and*

*C*

*satisfy the following conditions*:

- (I)
$\partial C\subseteq gC$, $fC\cap C\subseteq gC$,

- (II)
$gx\in \partial C$

*implies that*$fx\in C$, - (III)
*gC**is closed in**X*.

*Then there exists a coincidence point* *z* *of* *f*, *g* *in* *C*. *Moreover*, *if* $(f,g)$ *are coincidentally commuting*, *then* *z* *is the unique common fixed point of* *f* *and* *g*.

**Corollary 2.9**

*Let*$(X,d)$

*be a complete cone metric space*,

*let*

*C*

*be a nonempty closed subset of*

*X*

*such that for each*$x\in C$

*and*$y\notin C$

*there exists a point*$z\in \partial C$

*such that*

*Let*$f,g:C\to X$

*be such that*

*for some*$\lambda \in [0,1/2)$

*and for all*$x,y\in C$.

*Suppose*,

*further*,

*that*

*f*,

*g*

*and*

*C*

*satisfy the following conditions*:

- (I)
$\partial C\subseteq gC$, $fC\cap C\subseteq gC$,

- (II)
$gx\in \partial C$

*implies that*$fx\in C$, - (III)
*gC**is closed in**X*.

*Then there exists a coincidence point* *z* *of* *f*, *g* *in* *C*. *Moreover*, *if* $(f,g)$ *are coincidentally commuting*, *then* *z* *is the unique common fixed point of* *f* *and* *g*.

**Remark 2.6** Corollaries 2.7-2.9 are the corresponding theorems of Abbas and Jungck from [2] in the case that *f*, *g* are non-self mappings.

## Declarations

### Acknowledgements

The authors would like to express their sincere appreciation to the referees for their very helpful suggestions and kind comments. Project is supported by the National Natural Science Foundation of China (11071108) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20114BAB201003) and the Science and Technology Project of Educational Commission of Jiangxi Province, China (GJJ11346).

## Authors’ Affiliations

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