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# Fixed point theory of cyclical generalized contractive conditions in partial metric spaces

- Chi-Ming Chen
^{1}Email author

**2013**:17

https://doi.org/10.1186/1687-1812-2013-17

© Chen; licensee Springer. 2013

**Received:**6 November 2012**Accepted:**13 January 2013**Published:**28 January 2013

## Abstract

The purpose of this paper is to study fixed point theorems for a mapping satisfying the cyclical generalized contractive conditions in complete partial metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.

**MSC:**47H10, 54C60, 54H25, 55M20.

## Keywords

- fixed point
- cyclic $\mathcal{CW}$-contraction
- cyclic $\mathcal{MK}$-contraction
- partial metric space

## 1 Introduction and preliminaries

*D*be a subset of

*X*and $f:D\to X$ be a map. We say

*f*is contractive if there exists $\alpha \in [0,1)$ such that for all $x,y\in D$,

*f*is contractive and $(X,d)$ is complete, then

*f*has a unique fixed point in

*X*. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, in 1969, Boyd and Wong [2] introduced the notion of Φ-contraction. A mapping $f:X\to X$ on a metric space is called Φ-contraction if there exists an upper semi-continuous function $\mathrm{\Phi}:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ such that

In 1994, Mattews [3] introduced the following notion of partial metric spaces.

**Definition 1** [3] A partial metric on a nonempty set *X* is a function $p:X\times X\to {\mathbb{R}}^{+}$ such that for all $x,y,z\in X$,

(p_{1}) $x=y$ if and only if $p(x,x)=p(x,y)=p(y,y)$;

(p_{2}) $p(x,x)\le p(x,y)$;

(p_{3}) $p(x,y)=p(y,x)$;

(p_{4}) $p(x,y)\le p(x,z)+p(z,y)-p(z,z)$.

A partial metric space is a pair $(X,p)$ such that *X* is a nonempty set and *p* is a partial metric on *X*.

**Remark 1** It is clear that if $p(x,y)=0$, then from (p_{1}) and (p_{2}), $x=y$. But if $x=y$, $p(x,y)$ may not be 0.

*p*on

*X*generates a ${\mathcal{T}}_{0}$ topology ${\tau}_{p}$ on

*X*which has as a base the family of open

*p*-balls $\{{B}_{p}(x,\gamma ):x\in X,\gamma >0\}$, where ${B}_{p}(x,\gamma )=\{y\in X:p(x,y)<p(x,x)+\gamma \}$ for all $x\in X$ and $\gamma >0$. If

*p*is a partial metric on

*X*, then the function ${d}_{p}:X\times X\to {\mathbb{R}}^{+}$ given by

is a metric on *X*.

We recall some definitions of a partial metric space as follows.

**Definition 2** [3]

- (1)
a sequence $\{{x}_{n}\}$ in a partial metric space $(X,p)$ converges to $x\in X$ if and only if $p(x,x)={lim}_{n\to \mathrm{\infty}}p(x,{x}_{n})$;

- (2)
a sequence $\{{x}_{n}\}$ in a partial metric space $(X,p)$ is called a Cauchy sequence if and only if ${lim}_{m,n\to \mathrm{\infty}}p({x}_{m},{x}_{n})$ exists (and is finite);

- (3)
a partial metric space $(X,p)$ is said to be complete if every Cauchy sequence $\{{x}_{n}\}$ in

*X*converges, with respect to ${\tau}_{p}$, to a point $x\in X$ such that $p(x,x)={lim}_{m,n\to \mathrm{\infty}}p({x}_{m},{x}_{n})$; - (4)
a subset

*A*of a partial metric space $(X,p)$ is closed if whenever $\{{x}_{n}\}$ is a sequence in*A*such that $\{{x}_{n}\}$ converges to some $x\in X$, then $x\in A$.

**Remark 2** The limit in a partial metric space is not unique.

- (a)
$\{{x}_{n}\}$

*is a Cauchy sequence in a partial metric space*$(X,p)$*if and only if it is a Cauchy sequence in the metric space*$(x,{d}_{p})$; - (b)
*a partial metric space*$(X,p)$*is complete if and only if the metric space*$(X,{d}_{p})$*is complete*.*Furthermore*, ${lim}_{n\to \mathrm{\infty}}{d}_{p}({x}_{n},x)=0$*if and only if*$p(x,x)={lim}_{n\to \mathrm{\infty}}p({x}_{n},x)={lim}_{n\to \mathrm{\infty}}p({x}_{n},{x}_{m})$.

In 2003, Kirk, Srinivasan and Veeramani [5] introduced the following notion of the cyclic representation.

**Definition 3** [5]

*X*be a nonempty set, $m\in \mathbb{N}$ and $f:X\to X$ be an operator. Then $X={\bigcup}_{i=1}^{m}{A}_{i}$ is called a cyclic representation of

*X*with respect to

*f*if

- (1)
${A}_{i}$, $i=1,2,\dots ,m$ are nonempty subsets of

*X*; - (2)
$f({A}_{1})\subset {A}_{2},f({A}_{2})\subset {A}_{3},\dots ,f({A}_{m-1})\subset {A}_{m},f({A}_{m})\subset {A}_{1}$.

Kirk, Srinivasan and Veeramani [5] also proved the following theorem.

**Theorem 1** [5]

*Let*$(X,d)$

*be a complete metric space*, $m\in \mathbb{N}$, ${A}_{1},{A}_{2},\dots ,{A}_{m}$,

*be closed nonempty subsets of*

*X*

*and*$X={\bigcup}_{i=1}^{m}{A}_{i}$.

*Suppose that*

*f*

*satisfies the following condition*:

*where* $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ *is upper semi*-*continuous from the right and* $0\le \psi (t)<t$ *for* $t>0$. *Then* *f* *has a fixed point* $z\in {\bigcap}_{i=1}^{n}{A}_{i}$.

Recently, the fixed theorems for an operator $f:X\to X$ defined on a metric space *X* with a cyclic representation of *X* with respect to *f* have appeared in the literature (see, *e.g.*, [6–8]). In 2010, Pǎcurar and Rus [7] introduced the following notion of a cyclic weaker *φ*-contraction.

**Definition 4** [7]

*X*and $X={\bigcup}_{i=1}^{m}{A}_{i}$. An operator $f:X\to X$ is called a cyclic weaker

*φ*-contraction if

- (1)
$X={\bigcup}_{i=1}^{m}{A}_{i}$ is a cyclic representation of

*X*with respect to*f*; - (2)there exists a continuous, non-decreasing function $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $\phi (t)>0$ for $t\in (0,\mathrm{\infty})$ and $\phi (0)=0$ such that$d(fx,fy)\le d(x,y)-\phi (d(x,y))$

for any $x\in {A}_{i}$, $y\in {A}_{i+1}$, $i=1,2,\dots ,m$, where ${A}_{m+1}={A}_{1}$.

And Pǎcurar and Rus [7] proved the following main theorem.

**Theorem 2** [7]

*Let* $(X,d)$ *be a complete metric space*, $m\in \mathbb{N}$, ${A}_{1},{A}_{2},\dots ,{A}_{m}$ *be closed nonempty subsets of* *X* *and* $X={\bigcup}_{i=1}^{m}{A}_{i}$. *Suppose that* *f* *is a cyclic weaker* *φ*-*contraction*. *Then* *f* *has a fixed point* $z\in {\bigcap}_{i=1}^{n}{A}_{i}$.

In the recent years, fixed point theory has developed rapidly on cyclic contraction mappings, see [9–15].

The purpose of this paper is to study fixed point theorems for a mapping satisfying the cyclical generalized contractive conditions in complete partial metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.

## 2 Fixed point theorems (I)

In the section, we denote by Ψ the class of functions $\psi :{{\mathbb{R}}^{+}}^{3}\to {\mathbb{R}}^{+}$ satisfying the following conditions:

(${\psi}_{1}$) *ψ* is an increasing and continuous function in each coordinate;

(${\psi}_{2}$) for $t\in {\mathbb{R}}^{+}$, $\psi (t,t,t)\le t$, $\psi (t,0,0)\le t$ and $\psi (0,0,t)\le t$.

Next, we denote by Θ the class of functions $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ satisfying the following conditions:

(${\phi}_{1}$) *φ* is continuous and non-decreasing;

(${\phi}_{2}$) for $t>0$, $\phi (t)>0$ and $\phi (0)=0$.

And we denote by Φ the class of functions $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ satisfying the following conditions:

(${\varphi}_{1}$) *ϕ* is continuous;

(${\varphi}_{2}$) for $t>0$, $\varphi (t)>0$ and $\varphi (0)=0$.

We now state a new notion of cyclic $\mathcal{CW}$-contractions in partial metric spaces as follows.

**Definition 5**Let $(X,p)$ be a partial metric space, $m\in \mathbb{N}$, ${A}_{1},{A}_{2},\dots ,{A}_{m}$ be nonempty subsets of

*X*and $Y={\bigcup}_{i=1}^{m}{A}_{i}$. An operator $f:Y\to Y$ is called a cyclic $\mathcal{CW}$-contraction if

- (1)
${\bigcup}_{i=1}^{m}{A}_{i}$ is a cyclic representation of

*Y*with respect to*f*; - (2)for any $x\in {A}_{i}$, $y\in {A}_{i+1}$, $i=1,2,\dots ,m$,$\phi (p(fx,fy))\le \psi (\phi (p(x,y)),\phi (p(x,fx)),\phi (p(y,fy)))-\varphi (M(x,y)),$(2.1)

where $\psi \in \mathrm{\Psi}$, $\phi \in \mathrm{\Theta}$, $\varphi \in \mathrm{\Phi}$, and $M(x,y)=max\{p(x,y),p(x,fx),p(y,fy)\}$.

**Theorem 3** *Let* $(X,p)$ *be a complete partial metric space*, $m\in \mathbb{N}$, ${A}_{1},{A}_{2},\dots ,{A}_{m}$ *be nonempty closed subsets of* *X* *and* $Y={\bigcup}_{i=1}^{m}{A}_{i}$. *Let* $f:Y\to Y$ *be a cyclic* $\mathcal{CW}$-*contraction*. *Then* *f* *has a unique fixed point* $z\in {\bigcap}_{i=1}^{m}{A}_{i}$.

*Proof* Given ${x}_{0}$ and let ${x}_{n+1}=f{x}_{n}={f}^{n}{x}_{0}$ for $n=0,1,2,\dots $ . If there exists ${n}_{0}\in \mathbb{N}$ such that ${x}_{{n}_{0}+1}={x}_{{n}_{0}}$, then we finished the proof. Suppose that ${x}_{n+1}\ne {x}_{n}$ for any $n=0,1,2,\dots $ . Notice that for any $n\ge 0$, there exists ${i}_{n}\in \{1,2,\dots ,m\}$ such that ${x}_{n}\in {A}_{{i}_{n}}$ and ${x}_{n+1}\in {A}_{{i}_{n}+1}$.

which implies that $\varphi (p({x}_{n},{x}_{n+1}))=0$, and hence $p({x}_{n},{x}_{n+1})=0$. This contradicts our initial assumption.

*φ*and

*ϕ*, we get

_{2}), we also have

Step 2. We show that $\{{x}_{n}\}$ is a Cauchy sequence in the metric space $(Y,{d}_{p})$. We claim that the following result holds.

**Claim** For every $\epsilon >0$, there exists $n\in \mathbb{N}$ such that if $r,q\ge n$ with $r-q=1modm$, then ${d}_{p}({x}_{r},{x}_{q})<\epsilon $.

*f*is a cyclic $\mathcal{CW}$-contraction, we have

which implies $\varphi (\frac{\u03f5}{2})=0$, that is, $\u03f5=0$. So, we get a contradiction. Therefore, our claim is proved.

for any $n\ge {n}_{2}$.

Thus, $\{{x}_{n}\}$ is a Cauchy sequence in the metric space $(Y,{d}_{p})$.

Step 3. We show that *f* has a fixed point *ν* in ${\bigcap}_{i=1}^{m}{A}_{i}$.

*Y*is closed, the subspace $(Y,p)$ is complete. Then from Lemma 1, we have that $(Y,{d}_{p})$ is complete. Thus, there exists $\nu \in X$ such that

*X*with respect to

*f*, the sequence $\{{x}_{n}\}$ has infinite terms in each ${A}_{i}$ for $i\in \{1,2,\dots ,m\}$. Now, for all $i=1,2,\dots ,m$, we may take a subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$ with ${x}_{{n}_{k}}\in {A}_{i-1}$ and also all converge to

*ν*. Using (2.10) and (2.11), we have

which implies $\varphi (p(\nu ,f\nu ))=0$, that is, $p(\nu ,f\nu )=0$. So, $\nu =f\nu $.

*μ*,

*ν*are fixed points of

*f*. Then using the inequality (2.1), we obtain that

which implies that $\varphi (p(\mu ,\nu ))=0$, and hence $p(\mu ,\nu )=0$, that is, $\mu =\nu $. So, we complete the proof. □

The following provides an example for Theorem 3.

**Example 1**Let $X=[0,1]$ and $A=[0,1]$, $B=[0,\frac{1}{2}]$, $C=[0,\frac{1}{4}]$. We define the partial metric

*p*on

*X*by

Then *f* is a cyclic $\mathcal{CW}$-contraction and 0 is the unique fixed point.

*Proof*We claim that

*f*is a cyclic $\mathcal{CW}$-contraction.

- (1)
Note that $f(A)=[0,\frac{1}{2}]\subset B$, $f(B)=[0,\frac{1}{6}]\subset C$ and $f(C)=[0,\frac{1}{20}]\subset A$. Thus, $A\cup B\cup C$ is a cyclic representation of

*X*with respect to*f*; - (2)

On the other hand, for $x\in C$ and $y\in A$, without loss of generality, we may assume that $x\le y$, then it is easy to get the above inequality.

Note that Example 1 satisfies all of the hypotheses of Theorem 3, and we get that 0 is the unique fixed point. □

## 3 Fixed point theorems (II)

In this article, we also recall the notion of a Meir-Keeler function (see [16]). A function $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is said to be a Meir-Keeler function if for each $\eta >0$, there exists $\delta >0$ such that for $t\in [0,\mathrm{\infty})$ with $\eta \le t<\eta +\delta $, we have $\varphi (t)<\eta $. We now introduce a new notion of a weaker Meir-Keeler function $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ in a partial metric space $(X,p)$ as follows.

**Definition 6** Let $(X,p)$ be a partial metric space. We call $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ a weaker Meir-Keeler function in *X* if for each $\eta >0$, there exists $\delta >0$ such that for $x,y\in X$ with $\eta \le p(x,y)<\eta +\delta $, there exists ${n}_{0}\in \mathbb{N}$ such that ${\varphi}^{{n}_{0}}(p(x,y))<\eta $.

In the section, we denote by Φ the class of weaker Meir-Keeler functions $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ in a partial metric space in $(X,p)$ satisfying the following conditions:

(${\varphi}_{1}$) $\varphi (t)>0$ for $t>0$, $\varphi (0)=0$;

(${\varphi}_{2}$) ${\{{\varphi}^{n}(t)\}}_{n\in \mathbb{N}}$ is decreasing;

- (a)
if ${lim}_{n\to \mathrm{\infty}}{t}_{n}=\gamma >0$, then ${lim}_{n\to \mathrm{\infty}}\varphi ({t}_{n})<\gamma $ and

- (b)
if ${lim}_{n\to \mathrm{\infty}}{t}_{n}=0$, then ${lim}_{n\to \mathrm{\infty}}\varphi ({t}_{n})=0$.

And we denote by the class Ψ of functions $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ a continuous function satisfying $\psi (t)>0$ for $t>0$, $\psi (0)=0$.

First, we state a new notion of cyclic $\mathcal{MK}$-contractions in partial metric spaces as follows.

**Definition 7**Let $(X,p)$ be a partial metric space, $m\in \mathbb{N}$, ${A}_{1},{A}_{2},\dots ,{A}_{m}$ be nonempty subsets of

*X*and $Y={\bigcup}_{i=1}^{m}{A}_{i}$. An operator $f:Y\to Y$ is called a cyclic $\mathcal{MK}$-contraction if

- (1)
${\bigcup}_{i=1}^{m}{A}_{i}$ is a cyclic representation of

*Y*with respect to*f*; - (2)for any $x\in {A}_{i}$, $y\in {A}_{i+1}$, $i=1,2,\dots ,m$,$p(fx,fy)\le \varphi (p(x,y))-\psi (p(x,y)),$(3.1)

where ${A}_{m+1}={A}_{1}$, $\varphi \in \mathrm{\Phi}$ and $\psi \in \mathrm{\Psi}$.

**Theorem 4** *Let* $(X,p)$ *be a complete partial metric space*, $m\in \mathbb{N}$, ${A}_{1},{A}_{2},\dots ,{A}_{m}$ *be nonempty closed subsets of* *X* *and* $Y={\bigcup}_{i=1}^{m}{A}_{i}$. *Let* $f:Y\to Y$ *be a cyclic* $\mathcal{MK}$-*contraction*. *Then* *f* *has a unique fixed point* $z\in {\bigcap}_{i=1}^{m}{A}_{i}$.

*Proof*Given ${x}_{0}$ and let ${x}_{n+1}=f{x}_{n}={f}^{n}{x}_{0}$, for $n=0,1,2,\dots $ . If there exists ${n}_{0}\in \mathbb{N}$ such that ${x}_{{n}_{0}+1}={x}_{{n}_{0}}$, then we finished the proof. Suppose that ${x}_{n+1}\ne {x}_{n}$ for any $n=0,1,2,\dots $ . Notice that for any $n\ge 0$, there exists ${i}_{n}\in \{1,2,\dots ,m\}$ such that ${x}_{n}\in {A}_{{i}_{n}}$ and ${x}_{n+1}\in {A}_{{i}_{n}+1}$. Then by (3.1), we have

*f*is a cyclic $\mathcal{MK}$-contraction, we can conclude that

*ϕ*, there exists $\delta >0$ such that for ${x}_{0},{x}_{1}\in X$ with $\eta \le p({x}_{0},{x}_{1})<\delta +\eta $, there exists ${n}_{0}\in \mathbb{N}$ such that ${\varphi}^{{n}_{0}}(p({x}_{0},{x}_{1}))<\eta $. Since ${lim}_{n\to \mathrm{\infty}}{\varphi}^{n}(p({x}_{0},{x}_{1}))=\eta $, there exists ${k}_{0}\in \mathbb{N}$ such that $\eta \le {\varphi}^{k}(p({x}_{0},{x}_{1}))<\delta +\eta $, for all $k\ge {k}_{0}$. Thus, we conclude that ${\varphi}^{{k}_{0}+{n}_{0}}(p({x}_{0},{x}_{1}))<\eta $. So, we get a contradiction. Therefore, ${lim}_{n\to \mathrm{\infty}}{\varphi}^{n}(p({x}_{0},{x}_{1}))=0$, and so we have

_{2}), we also have

Step 2. We show that $\{{x}_{n}\}$ is a Cauchy sequence in the metric space $(Y,{d}_{p})$. We claim that the following result holds.

**Claim** For every $\epsilon >0$, there exists $n\in \mathbb{N}$ such that if $r,q\ge n$ with $r-q=1modm$, then ${d}_{p}({x}_{r},{x}_{q})<\epsilon $.

*f*is a cyclic $\mathcal{MK}$-contraction, we have

*ϕ*, we obtain that

and consequently, $\psi (\frac{\u03f5}{2})=0$. By the definition of a function *ψ*, we get $\u03f5=0$ which is a contraction. Therefore, our claim is proved.

for any $n\ge {n}_{2}$.

Thus, $\{{x}_{n}\}$ is a Cauchy sequence in the metric space $(Y,{d}_{p})$.

Step 3. We show that *f* has a fixed point *ν* in ${\bigcap}_{i=1}^{m}{A}_{i}$.

*Y*is closed, the subspace $(Y,p)$ is complete. Then from Lemma 1, we have that $(Y,{d}_{p})$ is complete. Thus, there exists $\nu \in X$ such that

*X*with respect to

*f*, the sequence $\{{x}_{n}\}$ has infinite terms in each ${A}_{i}$ for $i\in \{1,2,\dots ,m\}$. Now, for all $i=1,2,\dots ,m$, we may take a subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$ with ${x}_{{n}_{k}}\in {A}_{i-1}$ and also all converge to

*ν*. Using (3.9) and (3.10), we have

and so $\nu =f\nu $.

*μ*be another fixed point of

*f*in ${\bigcap}_{i=1}^{m}{A}_{i}$. By the cyclic character of

*f*, we have $\mu ,\nu \in {\bigcap}_{i=1}^{n}{A}_{i}$. Since

*f*is a cyclic weaker $\mathcal{MK}$-contraction, we have

which implies $p(\nu ,\mu )=0$. So, we have $\mu =\nu $. We complete the proof. □

The following provides an example for Theorem 4.

**Example 2**Let $X=[0,1]$ and $A=[0,1]$, $B=[0,\frac{1}{2}]$, $C=[0,\frac{1}{4}]$. We define the partial metric

*p*on

*X*by

Then *f* is a cyclic $\mathcal{MK}$-contraction and 0 is the unique fixed point.

By Theorem 4, it is easy to get the following corollary.

**Corollary 1**

*Let*$(X,p)$

*be a complete partial metric space*, $m\in \mathbb{N}$, ${A}_{1},{A}_{2},\dots ,{A}_{m}$

*be nonempty closed subsets of*

*X*, $Y={\bigcup}_{i=1}^{m}{A}_{i}$

*and let*$f:Y\to Y$.

*Assume that*

- (1)
${\bigcup}_{i=1}^{m}{A}_{i}$

*is a cyclic representation of**Y**with respect to**f*; - (2)
*for any*$x\in {A}_{i}$, $y\in {A}_{i+1}$, $i=1,2,\dots ,m$,$p(fx,fy)\le \varphi (p(x,y)),$

*where* ${A}_{m+1}={A}_{1}$ *and* $\varphi \in \mathrm{\Phi}$.

*Then* *f* *has a unique fixed point* $z\in {\bigcap}_{i=1}^{m}{A}_{i}$.

## Declarations

### Acknowledgements

The authors would like to thank referee(s) for many useful comments and suggestions for the improvement of the paper.

## Authors’ Affiliations

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