Open Access

A new type of multivalued contraction in partial Hausdorff metric spaces endowed with a graph

Journal of Inequalities and Applications20152015:205

https://doi.org/10.1186/s13660-015-0728-y

Received: 26 February 2015

Accepted: 3 June 2015

Published: 18 June 2015

Abstract

A generalization, namely the K-comparison function, of a comparison function is introduced. Using K-comparison functions we introduce \(K_{G}\)-contractive mappings. We obtain a fixed point theorem for such mappings on partial Hausdorff metric spaces endowed with a graph. We also construct examples in support of our results.

Keywords

K-comparison function \(K_{G}\)-contractive mapping partial metric space partial Hausdorff metric space

MSC

47H10 54H25

1 Introduction

Development in metric fixed point theory is based on two things: the first is to modify contraction condition and the second is to modify the structure of a metric space. Matthews [1] introduced the notion of a partial metric space and extended Banach contraction principle in the setting of partial metric space. The work of Matthews [1] has been extended by many authors; see for example [217]. Using the notion of a partial metric on a set X, Aydi et al. [18] defined a partial Hausdorff metric on the set of closed and bounded subsets of the set X. Moreover, they [18] extended Nadler’s fixed point theorem in the setting of a partial Hausdorff metric spaces. Jachymaski [19] generalized the Banach fixed point theorem for mappings of a complete metric space endowed with a graph. He introduced the notion of Banach G-contractions. Here G stands for a directed graph in a metric space whose vertex set coincides with the metric space. Many authors extended the Banach G-contraction in different ways; see, for example, [1928].

In this paper, we introduce the notions of a K-comparison function and a \(K_{G}\)-contractive mapping. We establish a fixed point theorem for \(K_{G}\)-contractive mappings, in the setting of partial Hausdorff metric spaces endowed with a graph.

2 Preliminaries

In this section we recollect some definitions from partial (Hausdorff) metric spaces and comparison functions. We also present some results from partial (Hausdorff) metric spaces for ready reference. Throughout this paper, \(\mathbb{R} ^{+}=[0,\infty)\).

Definition 2.1

[1]

Let X be a nonempty set. A mapping \(p:X\times X\rightarrow\mathbb{R} ^{+}\) is a partial metric on X, if for all \(x,y,z\in X\). We have
  1. (P1)

    \(p ( x,x ) =p ( y,y ) =p ( x,y ) \) if and only if \(x=y\);

     
  2. (P2)

    \(p ( x,x ) \leq p ( x,y )\);

     
  3. (P3)

    \(p ( x,y ) =p ( y,x )\);

     
  4. (P4)

    \(p ( x,z ) \leq p ( x,y ) +p ( y,z ) -p ( y,y ) \).

     

Remark 2.2

[18]

If \(p ( x,y ) =0\), then (P1) and (P2) implies \(x=y\) but the converse is not true in general.

Example 2.3

[1]

Let X be the set of all closed intervals of real line \(\mathbb{R} \), that is, \(X= \{ [ a,b ] :a,b\in \mathbb{R} ,a\leq b \} \) and define a function \(p:X\times X\rightarrow\mathbb{R} ^{+}\) by \(p ( [ a,b ] , [ c,d ] ) =\max \{ b,d \} -\min \{ a,c \} \), then \(( X,p ) \) is a partial metric space.

Lemma 2.4

[1]

Every metric space is a partial metric space.

Remark 2.5

[1]

Every partial metric p on X generates a \(T_{0}\) topology \(\tau_{p}\) on X with as a base the family of the open balls (p balls) \(\{ B_{p} ( x,\epsilon ) :x\in X,\epsilon>0 \} \), where
$$ B_{p} ( x,\epsilon ) = \bigl\{ y\in X:p ( x,y ) < p ( x,x ) +\epsilon \bigr\} . $$

Definition 2.6

[1]

Let \(( X,p ) \) be a partial metric space. Then:
  1. (a)

    A sequence \(\{ x_{n} \} \) in \(( X,p ) \) is said to be convergent to a point \(x\in X\) with respect to \(\tau_{p}\) if and only if \(p ( x,x ) =\lim_{n\rightarrow\infty}p ( x,x_{n} ) \).

     
  2. (b)

    A sequence \(\{ x_{n} \} \) in X will be a Cauchy sequence if and only if \(\lim_{n,m\rightarrow\infty} p ( x_{n},x_{m} ) \) exists and is finite.

     
  3. (c)

    A partial metric space \(( X,p ) \) is called a complete partial metric space if every Cauchy sequence \(\{ x_{n} \} \) in X converges with respect to \(\tau_{p}\) to a point \(x\in X\).

     

Remark 2.7

[1]

Let \(( X,p ) \) be a partial metric space, then the function \(d_{p}:X\times X\to[0,\infty)\) defined by
$$ d_{p} ( x,y ) =2p ( x,y ) -p ( x,x ) -p ( y,y ) $$
is a metric on X.

Lemma 2.8

[1]

Let \(( X,p ) \) be a partial metric space, a sequence \(\{ x_{n} \} \) in \(( X,d_{p} ) \) is said to be convergent to a point \(x\in X\) if and only if
$$ p ( x,x ) =\lim_{n\rightarrow\infty}p ( x,x_{n} ) =\lim _{n,m\rightarrow\infty}p ( x_{n},x_{m} ). $$

Lemma 2.9

[18]

Let \(( X,p ) \) be a partial metric space. Then:
  1. (a)

    A sequence \(\{ x_{n} \} \) in X is Cauchy with respect to p if and only if it is Cauchy with respect to \(d_{p}\).

     
  2. (b)

    A partial metric space \(( X,p ) \) is complete if and only if the metric space \(( X,d_{p} ) \) is complete.

     

A subset A of a partial metric space \((X,p)\) is a bounded [18], if there exists \(x_{0}\in A\) such that \(p(x_{0},a)< p(x_{0},x_{0})+M\). A subset A of a partial metric space is closed if it is closed with respect to the topology \(\tau_{p}\) on X. Let \(\operatorname{CB}_{p}(X)\) be the family of all nonempty closed and bounded subsets of a partial metric space \((X,p)\). We use the following notions and terminologies [18]. For \(A,B\in \operatorname{CB}_{p}(X)\), \(p ( x,A ) =\inf \{ p ( x,a ) :a\in A \} \), \(p ( A,B ) =\inf \{ p ( x,y ) :x\in A,y\in B \} \). The functions \(\delta _{p}:\operatorname{CB}_{p} ( X ) \times \operatorname{CB}_{p} ( X ) \rightarrow \mathbb{R} ^{+}\) and \(H_{p}:\operatorname{CB}_{p} ( X ) \times \operatorname{CB}_{p} ( X ) \rightarrow\mathbb{R} ^{+}\) are defined by \(\delta_{p} ( A,B ) =\sup \{ p ( a,B ) :a\in A \} \) and \(H_{p} ( A,B ) =\max \{ \delta_{p} ( A,B ) ,\delta_{p} ( B,A ) \}\), respectively.

Remark 2.10

[18]

If \(d_{p} ( x,A ) =\inf \{ d_{p} ( x,a ) :a\in A \} \), then it is easy to prove that \(p ( x,A ) =0\) implies that \(d_{p} ( x,A ) =0\).

Lemma 2.11

[18]

Let \(( X,p ) \) be a partial metric space and A be any nonempty subset of X, then \(a\in\overline{A}\) if and only if \(p ( a,A ) =p ( a,a ) \).

Lemma 2.12

[18]

Let \(( X,p ) \) be a partial metric space and A be any nonempty subset of X. If A is closed in \(( X,p ) \), then A is closed in \(( X,d_{p} )\).

Proposition 2.13

[18]

Let \((X,p)\) be a partial metric space. For \(A,B\in \operatorname{CB}_{p}(X)\), the following properties hold:
  1. (1)

    \(H_{p}(A,A)\leq H_{p}(A,B)\);

     
  2. (2)

    \(H_{p}(A,B)=H_{p}(B,A)\);

     
  3. (3)

    \(H_{p}(A,C)\leq H_{p}(A,B)+H_{p}(B,C)-\inf_{c\in C}p(c,c)\);

     
  4. (4)

    \(H_{p}(A,B)=0\) implies that \(A=B\).

     

Lemma 2.14

[18]

Let \((X,p)\) be a partial metric space, let \(A,B\in \operatorname{CB}_{p}(X)\) and \(h>1\). For any \(a\in A\), there exists \(b\in B\) such that \(p(a,b)\leq hH_{p}(A,B)\).

Let \(\xi:[0,\infty)\to[0,\infty)\) be a function. Consider the following conditions:
  1. (i)

    ξ is an increasing function;

     
  2. (ii)

    \(\xi(t)< t\) for each \(t>0\);

     
  3. (iii)

    \(\xi(0)=0\);

     
  4. (iv)

    \(\{\xi^{n}(t)\}\) converges to 0 for each \(t\geq0\);

     
  5. (v)

    \(\sum_{n=0}^{\infty} {\xi^{n}(t)}\) converges for each \(t> 0\).

     
The function ξ satisfying (i) and (iv) is said to be a comparison function [29]. The function ξ satisfying (i) and (v) is known as a \((c)\)-comparison function [29]. It is easily seen that (i) and (iv) imply (ii); and (i) and (ii) imply (iii) [29].

Property (A):

([19], Remark 3.1)

For any sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in X, if \(x_{n}\rightarrow x\) and \((x_{n}, x_{n+1})\in E\) for \(n\in\mathbb{N}\), then \((x_{n}, x)\in E\).

3 Main results

We begin this section by introducing the notion of a K-comparison function.

Definition 3.1

A mapping \(\zeta:[0,\infty)\to[0,\infty)\) is said to be a K-comparison if the following conditions hold:
  1. (i)

    for each \(t>0\), we have \(\zeta(t)< t\);

     
  2. (ii)

    \(\zeta(0)=0\).

     

Note that any comparison or \((c)\)-comparison function is K-comparison function but converse is not true in general.

Example 3.2

Let \(\zeta:[0,\infty)\to[0,\infty)\) be a mapping such that
$$\zeta(t)= \textstyle\begin{cases} \frac{t}{2} &\mbox{if } 0\leq t\leq2, \\ \sqrt{t} & \mbox{otherwise}. \end{cases} $$
Thus, ζ is a K-comparison function but neither a comparison nor a \((c)\)-comparison function.

We denote the class of K-comparison functions by \(\mathfrak{K}\). Throughout this section, \((X,p)\) is a partial metric space, \(G=(V,E)\) is a directed graph without parallel edges such that \(V=X\) and \(\triangle=\{(x,x):x\in X\} \subset E\). For basic terminologies of graph theory we refer the reader to the excellent text by Chartrand et al. [30].

Definition 3.3

Let \((X,p)\) be a partial metric space. A mapping \(T:X\to \operatorname{CB}_{p}(X)\) is said to be \(K_{G}\)-contractive, if there exists \(\zeta\in \mathfrak{K}\) with \(\sup_{t>0}\frac{\zeta(t)}{t}<1\) such that
  1. (i)
    for each \((x,y)\in E\) with \(x\neq y\), we have
    $$ H_{p}(Tx,Ty)\leq \zeta \biggl(\max \biggl\{ p(x,y), \frac{p(x,Tx)+p(y,Ty)}{2},\frac {p(x,Ty)+p(y,Tx)}{2} \biggr\} \biggr); $$
    (3.1)
     
  2. (ii)
    if \(s\in Tx\) and \(t\in Ty\) are such that
    $$ p(s,t)< p(x,y), $$
    (3.2)
    then we have \((s,t)\in E\), whenever \((x,y)\in E\) with \(x\neq y\).
     

Theorem 3.4

Let \((X,p)\) be a complete partial metric space endowed with the graph G and Property (A). Let \(T:X\to \operatorname{CB}_{p}(X)\) be a \(K_{G}\)-contractive mapping. Assume that there exist \(x_{0}\in X\) and \(x_{1}\in Tx_{0}\) such that \((x_{0},x_{1})\in E\). Then T has a fixed point.

Proof

By hypothesis, we have \(x_{0}\in X\) and \(x_{1}\in Tx_{0}\) such that \((x_{0},x_{1})\in E\). If \(x_{0}=x_{1}\), then \(x_{0}\) is a fixed point. Suppose that \(x_{0}\neq x_{1}\). Since T is a \(K_{G}\)-contractive mapping, from (3.1), we have
$$\begin{aligned} H_{p}(Tx_{0},Tx_{1}) \leq&\zeta \biggl(\max \biggl\{ p(x_{0},x_{1}),\frac {p(x_{0},Tx_{0})+p(x_{1},Tx_{1})}{2}, \frac{p(x_{0},Tx_{1})+p(x_{1},Tx_{0})}{2} \biggr\} \biggr) \\ < &\max \biggl\{ p(x_{0},x_{1}),\frac{p(x_{0},Tx_{0})+p(x_{1},Tx_{1})}{2}, \frac{p(x_{0},Tx_{1})+p(x_{1},Tx_{0})}{2} \biggr\} . \end{aligned}$$
(3.3)
Then there exists \(a_{1}\in(0,l]\), where \(l=\sup_{t>0}\frac{\zeta(t)}{t}\), and obviously \(a_{1}\) depends on \(x_{0}\) and \(x_{1}\), such that
$$\begin{aligned}& H_{p}(Tx_{0},Tx_{1}) \\& \quad \leq a_{1}\max \biggl\{ p(x_{0},x_{1}), \frac{p(x_{0},Tx_{0})+p(x_{1},Tx_{1})}{2}, \frac{p(x_{0},Tx_{1})+p(x_{1},Tx_{0})}{2} \biggr\} . \end{aligned}$$
(3.4)
Since \(a_{1}<1\), then \(1/\sqrt{a_{1}}>1\). Thus, by using Lemma 2.14, we have \(x_{2}\in Tx_{1}\) such that
$$ p(x_{1},x_{2})\leq\frac{1}{\sqrt{a_{1}}} H_{p}(Tx_{0},Tx_{1}). $$
(3.5)
From (3.4) and (3.5), we get
$$\begin{aligned} p(x_{1},x_{2}) \leq&\sqrt{a_{1}} \max \biggl\{ p(x_{0},x_{1}),\frac {p(x_{0},x_{1})+p(x_{1},x_{2})}{2}, \frac{p(x_{0},x_{2})+p(x_{1},x_{1})}{2} \biggr\} \\ \leq&\sqrt{a_{1}}\max\bigl\{ p(x_{0},x_{1}),p(x_{1},x_{2}) \bigr\} . \end{aligned}$$
(3.6)
If we assume that \(\max\{p(x_{0},x_{1}),p(x_{1},x_{2})\}=p(x_{1},x_{2})\), then we get a contradiction to (3.6). Thus, \(\max\{p(x_{0},x_{1}),p(x_{1},x_{2})\}=p(x_{0},x_{1})\). From (3.6), we have
$$ p(x_{1},x_{2})\leq\sqrt{a_{1}}p(x_{0},x_{1})< p(x_{0},x_{1}). $$
(3.7)
From (3.2) and (3.7), we have \((x_{1},x_{2})\in E\). If \(x_{1}=x_{2}\), then \(x_{1}\) is a fixed point. Suppose that \(x_{1}\neq x_{2}\). Again, from (3.1), we have
$$\begin{aligned} H_{p}(Tx_{1},Tx_{2}) \leq&\zeta \biggl(\max \biggl\{ p(x_{1},x_{2}),\frac {p(x_{1},Tx_{1})+p(x_{2},Tx_{2})}{2}, \frac{p(x_{1},Tx_{2})+p(x_{2},Tx_{1})}{2} \biggr\} \biggr) \\ < &\max \biggl\{ p(x_{1},x_{2}),\frac{p(x_{1},Tx_{1})+p(x_{2},Tx_{2})}{2}, \frac{p(x_{1},Tx_{2})+p(x_{2},Tx_{1})}{2} \biggr\} . \end{aligned}$$
Then there exists \(a_{2}\in(0,l]\), and obviously \(a_{2}\) depends on \(x_{1}\) and \(x_{2}\), such that
$$\begin{aligned}& H_{p}(Tx_{1},Tx_{2}) \\& \quad \leq a_{2}\max \biggl\{ p(x_{1},x_{2}), \frac{p(x_{1},Tx_{1})+p(x_{2},Tx_{2})}{2}, \frac{p(x_{1},Tx_{2})+p(x_{2},Tx_{1})}{2} \biggr\} . \end{aligned}$$
(3.8)
Since \(a_{2}<1\), \(1/\sqrt{a_{2}}>1\). Again by using Lemma 2.14, we have \(x_{3}\in Tx_{2}\) such that
$$ p(x_{2},x_{3})\leq\frac{1}{\sqrt{a_{2}}} H_{p}(Tx_{1},Tx_{2}). $$
(3.9)
From (3.8) and (3.9), we get
$$\begin{aligned} p(x_{2},x_{3}) \leq&\sqrt{a_{1}} \max \biggl\{ p(x_{1},x_{2}),\frac {p(x_{1},x_{2})+p(x_{2},x_{3})}{2}, \frac{p(x_{1},x_{3})+p(x_{2},x_{2})}{2} \biggr\} \\ \leq&\sqrt{a_{1}}\max\bigl\{ p(x_{1},x_{2}),p(x_{2},x_{3}) \bigr\} . \end{aligned}$$
(3.10)
If we assume that \(\max\{p(x_{1},x_{2}),p(x_{2},x_{3})\}=p(x_{2},x_{3})\), then we get a contradiction to (3.10). Thus, \(\max\{p(x_{1},x_{2}),p(x_{2},x_{3})\}=p(x_{1},x_{2})\). From (3.10), we have
$$ p(x_{2},x_{3})\leq\sqrt{a_{2}}p(x_{1},x_{2})< p(x_{1},x_{2}). $$
(3.11)
Also, we have
$$p(x_{2},x_{3})\leq\sqrt{a_{2}}p(x_{1},x_{2}) \leq\sqrt{a_{2}}\sqrt{a_{1}}p(x_{0},x_{1}) . $$
Continuing the same way we get sequences \(\{a_{n}\}\subset(0,l]\) and \(\{x_{n}\}\subset X\) such that \(x_{n-1}\in Tx_{n}\), \(x_{n-1}\neq x_{n}\), and \((x_{n-1},x_{n})\in E\), with
$$p(x_{n},x_{n+1})\leq\sqrt{a_{n}} \sqrt{a_{n-1}}\cdots\sqrt{a_{1}}p(x_{0},x_{1}) \quad \text{for each }n\in\mathbb{N}. $$
Let \(n,m\in\mathbb{N}\), by using the triangular inequality, we have
$$\begin{aligned} p(x_{n},x_{n+m}) \leq& p(x_{n},x_{n+1})+p(x_{n+1},x_{n+2})+ \cdots+p(x_{n+m-1},x_{n+m})-\sum_{i=n+1}^{n+m-1}p(x_{i},x_{i}) \\ \leq& p(x_{n},x_{n+1})+p(x_{n+1},x_{n+2})+ \cdots+p(x_{n+m-1},x_{n+m}) \\ \leq& \sqrt{a_{n}}\sqrt{a_{n-1}}\cdots\sqrt{a_{1}}p(x_{0},x_{1})+ \sqrt{a_{n+1}}\sqrt{a_{n}}\cdots\sqrt{a_{1}}p(x_{0},x_{1}) \\ &{}+\cdots+ \sqrt{a_{n+m-1}}\sqrt{a_{n+m-2}}\cdots \sqrt{a_{1}}p(x_{0},x_{1}). \end{aligned}$$
(3.12)
Let \(b=\sup\{\sqrt{a_{i}}: i\in\mathbb{N}\}\), clearly, \(b<1\). Then from (3.12), we get
$$\begin{aligned} p(x_{n},x_{n+m}) \leq& \sqrt{a_{n}} \sqrt{a_{n-1}}\cdots\sqrt{a_{1}}p(x_{0},x_{1})+ \sqrt{a_{n+1}}\sqrt{a_{n}}\cdots\sqrt{a_{1}}p(x_{0},x_{1}) \\ &{}+\cdots+ \sqrt{a_{n+m-1}}\sqrt{a_{n+m-2}}\cdots \sqrt{a_{1}}p(x_{0},x_{1}) \\ \leq& \bigl[b^{n} +b^{n+1}+\cdots+ b^{n+m-1} \bigr]p(x_{0},x_{1}) \\ < &\frac{b^{n}}{1-b}p(x_{0},x_{1}). \end{aligned}$$
(3.13)
Consequently, we have
$$d_{p}(x_{n},x_{n+m})\leq2p(x_{n},x_{n+m})< \frac{2b^{n}}{1-b}p(x_{0},x_{1}). $$
Thus, we conclude that \(\{x_{n}\}\) is a Cauchy sequence in \((X,d_{p})\). Since \((X,p)\) is a complete partial metric space, by Lemma 2.9(b), \((X,d_{p})\) is a complete metric space. Then there exists \(x^{\ast}\in X\) such that \(x_{n}\to x^{\ast}\in X\) with respect to \(d_{p}\), as \(n\to\infty\). By Lemma 2.8, we have
$$ p\bigl(x^{\ast},x^{\ast}\bigr)=\lim _{n\to\infty}p\bigl(x_{n},x^{\ast}\bigr)=\lim _{n\to \infty}p(x_{n},x_{n+m})=0. $$
(3.14)
By Property (A), we have \((x_{n},x^{\ast})\in E\) for each \(n\in \mathbb{N}\). Now, we claim that \(p(x^{\ast},Tx^{\ast})=0\). On the contrary suppose that \(p(x^{\ast},Tx^{\ast})>0\). By using the triangular inequality and (3.1), we have
$$\begin{aligned} p\bigl(x^{\ast},Tx^{\ast}\bigr) \leq& p \bigl(x^{\ast},x_{n+1}\bigr)+p\bigl(x_{n+1},Tx^{\ast} \bigr)-p(x_{n+1},x_{n+1}) \\ \leq& p\bigl(x^{\ast},x_{n+1}\bigr)+H_{p} \bigl(Tx_{n},Tx^{\ast}\bigr) \\ \leq&p\bigl(x^{\ast},x_{n+1}\bigr)+\zeta \biggl(\max \biggl\{ p \bigl(x_{n},x^{\ast}\bigr),\frac{p(x_{n},Tx_{n})+p(x^{\ast },Tx^{\ast})}{2}, \\ &\frac{p(x_{n},Tx^{\ast})+p(x^{\ast},Tx_{n})}{2} \biggr\} \biggr) \\ < &p\bigl(x^{\ast},x_{n+1}\bigr)+\max \biggl\{ p \bigl(x_{n},x^{\ast}\bigr),\frac {p(x_{n},Tx_{n})+p(x^{\ast},Tx^{\ast})}{2}, \\ &\frac{p(x_{n},Tx^{\ast})+p(x^{\ast},Tx_{n})}{2} \biggr\} \\ \leq&p\bigl(x^{\ast},x_{n+1}\bigr)+\max \biggl\{ p \bigl(x_{n},x^{\ast}\bigr),\frac {p(x_{n},x_{n+1})+p(x^{\ast},Tx^{\ast})}{2}, \\ &\frac{p(x_{n},x^{\ast})+p(x^{\ast},Tx^{\ast})-p(x^{\ast},x^{\ast })+p(x^{\ast},x_{n+1})}{2} \biggr\} . \end{aligned}$$
Letting \(n\to\infty\) in the above inequality,
$$p\bigl(x^{\ast},Tx^{\ast}\bigr)\leq\frac{p(x^{\ast},Tx^{\ast})}{2}, $$
but this is impossible for \(p(x^{\ast},Tx^{\ast})>0\). Thus, \(p(x^{\ast},Tx^{\ast})=0\). Therefore, we have
$$p\bigl(x^{\ast},Tx^{\ast}\bigr)=0=p\bigl(x^{\ast},x^{\ast} \bigr). $$
This implies that \(x^{\ast}\in Tx^{\ast}\). □

Example 3.5

Let \(X=\mathbb{N}\cup\{0\}\) endowed with the partial metric \(p(x,y)=\max\{x,y\}\) and a graph \(G=(V,E)\) be defined as \(V=X\) and \(E=\{(x,y):x,y\in\{0,2,4,6,8,10,12,14,16\}\}\cup\{(x,x):x\in \mathbb{N}\}\). Let \(T:X\to \operatorname{CB}_{p}(X)\) be defined by
$$Tx= \textstyle\begin{cases} \{0\} & \mbox{if } x\in\{0,2,6,10,12,14\}, \\ \{0,2\}& \mbox{if } x=4, \\ \{0,4\}& \mbox{if } x=8, \\ \{0,8\}& \mbox{if } x=16, \\ \{x+1,x+2\}& \mbox{otherwise}, \end{cases} $$
and \(\zeta:[0,\infty)\to[0,\infty)\) by
$$\zeta(t)= \textstyle\begin{cases} \frac{t}{2} & \mbox{if } 0\leq t\leq4, \\ \frac{2t}{3} & \mbox{if } 4< t\leq8, \\ \frac{t}{2} & \mbox{if } t> 8. \end{cases} $$
To see that (3.1) holds it is sufficient to consider the following cases:
  1. (i)

    If \((x,y)\in E\) with \(x\neq y\in\{0,2,6,10,12,14\}\), then (3.1) trivially holds.

     
  2. (ii)
    If \((x,y)\in E\) with \(x\in\{0,2,6,10,12,14\}\) and \(y=4\), then
    $$H_{p}(Tx,Ty)=2\leq\zeta\bigl(M_{p}(x,y)\bigr). $$
     
  3. (iii)
    If \((x,y)\in E\) with \(x\in\{0,2,6,10,12,14\}\) and \(y=8\), then
    $$H_{p}(Tx,Ty)=4\leq\zeta\bigl(M_{p}(x,y)\bigr). $$
     
  4. (iv)
    If \((x,y)\in E\) with \(x\in\{0,2,6,10,12,14\}\) and \(y=16\), then
    $$H_{p}(Tx,Ty)=8\leq\zeta\bigl(M_{p}(x,y)\bigr). $$
     
  5. (v)
    If \((x,y)\in E\) with \(x=4\) and \(y=8\), then
    $$H_{p}(Tx,Ty)=4\leq\zeta\bigl(M_{p}(x,y)\bigr). $$
     
  6. (vi)
    If \((x,y)\in E\) with \(x=4\) and \(y=16\), then
    $$H_{p}(Tx,Ty)=8\leq\zeta\bigl(M_{p}(x,y)\bigr). $$
     
  7. (vii)
    If \((x,y)\in E\) with \(x=8\) and \(y=16\), then
    $$H_{p}(Tx,Ty)=8\leq\zeta\bigl(M_{p}(x,y)\bigr), $$
     
where \(M_{p}(x,y)=\max \{p(x,y),\frac{p(x,Tx)+p(y,Ty)}{2},\frac {p(x,Ty)+p(x,Ty)}{2} \}\). Thus, (3.1) holds. Further it can be observed that for \((x,y)\in E\) with \(x\neq y\), if \(s\in Tx\) and \(t\in Ty\) are such that \(p(s,t)< p(x,y)\), then we have \((s,t)\in E\). For \(x_{0}=8\), we have \(x_{1}=4\in Tx_{0}\) such that \((8,4)\in E\). Moreover, Property (A) holds. Therefore, Theorem 3.4 guarantees the existence of a fixed point of T.

Example 3.6

Let \(X=[0,\infty)\times[0,\infty)\) be endowed with the partial metric \(p(x,y)=p((x_{1},x_{2}),(y_{1},y_{2}))=\max\{x_{1},y_{1}\}+\max\{x_{2},y_{2}\}\) and a graph \(G=(V,E)\) be defined as \(V=X\) and \(E=\{(r,s):r=(r_{1},0),s =(s_{1},0) \text{ with } r_{1},s_{1}\geq0 \} \cup\{(x,x):x\in X\}\). Let \(T:X\to \operatorname{CB}_{p}(X)\) be defined by
$$T(b,a)= \bigl\{ (0,0), (b/2,a)\bigr\} \quad \text{for each }(b,a)\in X $$
and \(\zeta :[0,\infty)\to[0,\infty)\) be defined as
$$\zeta(t)= \textstyle\begin{cases} \frac{2t}{3} &\mbox{if } 0\leq t \leq10, \\ \frac{t}{2} & \mbox{otherwise}. \end{cases} $$
To see that (3.1) holds, we consider the following cases:
  1. (i)
    If \(((u,0),(v,0))\in E\) with \(0 \leq u< v\), then
    $$H_{p}\bigl(T(u,0),T(v,0)\bigr)=\frac{v}{2}\leq\zeta \bigl(M_{p}(x,y)\bigr). $$
     
  2. (ii)
    If \(((u,0),(v,0))\in E\) with \(0 \leq v< u\), then
    $$H_{p}\bigl(T(u,0),T(v,0)\bigr)=\frac{u}{2}\leq\zeta \bigl(M_{p}(x,y)\bigr), $$
     
where \(M_{p}(x,y)=\max \{p(x,y),\frac{p(x,Tx)+p(y,Ty)}{2},\frac {p(x,Ty)+p(x,Ty)}{2} \}\). Thus, (3.1) holds. Further it can be observed that for \((x,y)\in E\) with \(x\neq y\), if \(s\in Tx\) and \(t\in Ty\) are such that \(p(s,t)< p(x,y)\), then we have \((s,t)\in E\). For \(x_{0}=(1,0)\), we have \(x_{1}=(0.5,0)\in Tx_{0}\) such that \(((1,0),(0.5,0))\in E\). Moreover, Property (A) holds. Therefore, Theorem 3.4 guarantees the existence of a fixed point of T.

Note that the following results are direct consequences of our result.

Corollary 3.7

Let \((X,p)\) be a complete partial metric space endowed with a graph G and Property (A). Let \(T:X\to \operatorname{CB}_{p}(X)\) be a mapping such that
  1. (i)
    for each \((x,y)\in E\) with \(x\neq y\), we have
    $$ H_{p}(Tx,Ty)\leq\phi\bigl(M_{p}(x,y)\bigr)M_{p}(x,y), $$
    where \(M_{p}(x,y)=\max \{p(x,y),\frac{p(x,Tx)+p(y,Ty)}{2}, \frac{p(x,Ty)+p(x,Ty)}{2} \}\), and \(\phi:[0,\infty)\to[0,1)\) is such that \(\limsup_{t\to r^{+}}\phi(t)<1\) for each \(r\in[0,\infty)\);
     
  2. (ii)

    if \(s\in Tx\) and \(t\in Ty\) are such that \(p(s,t)< p(x,y)\), then we have \((s,t)\in E\), whenever \((x,y)\in E\) with \(x\neq y\).

     
Further, assume that there exist \(x_{0}\in X\) and \(x_{1}\in Tx_{0}\) such that \((x_{0},x_{1})\in E\). Then T has a fixed point.

This result can be obtained from Theorem 3.4, by considering \(\zeta(t)=\phi(t)t\) for each \(t\geq0\).

Corollary 3.8

Let \((X,d)\) be a complete metric space endowed with a graph G and Property (A). Let \(T:X\to \operatorname{CB}(X)\) be a mapping such that
  1. (i)
    for each \((x,y)\in E\) with \(x\neq y\), we have
    $$ H(Tx,Ty)\leq \zeta \biggl(\max \biggl\{ d(x,y),\frac{d(x,Tx)+d(y,Ty)}{2}, \frac {d(x,Ty)+d(x,Ty)}{2} \biggr\} \biggr), $$
    where ζ is a K-comparison function with \(\sup_{t>0}\frac{\zeta(t)}{t}<1\);
     
  2. (ii)

    if \(s\in Tx\) and \(t\in Ty\) are such that \(d(s,t)< d(x,y)\), then we have \((s,t)\in E\), whenever \((x,y)\in E\) with \(x\neq y\).

     
Further, assume that there exist \(x_{0}\in X\) and \(x_{1}\in Tx_{0}\) such that \((x_{0},x_{1})\in E\). Then T has a fixed point.

Corollary 3.9

Let \((X,p)\) be a complete partial metric space endowed with a graph G and Property (A). Let \(T:X\to \operatorname{CB}_{p}(X)\) be a mapping such that
  1. (i)
    for each \((x,y)\in E\) with \(x\neq y\), we have
    $$ H_{p}(Tx,Ty)\leq \zeta \biggl(\max \biggl\{ p(x,y),\frac{p(x,Tx)+p(y,Ty)}{2}, \frac {p(x,Ty)+p(x,Ty)}{2} \biggr\} \biggr), $$
    where ζ is comparison function with \(\sup_{t>0}\frac{\zeta(t)}{t}<1\);
     
  2. (ii)

    if \(s\in Tx\) and \(t\in Ty\) are such that \(p(s,t)< p(x,y)\), then we have \((s,t)\in E\), whenever \((x,y)\in E\) with \(x\neq y\).

     
Further, assume that there exist \(x_{0}\in X\) and \(x_{1}\in Tx_{0}\) such that \((x_{0},x_{1})\in E\). Then T has a fixed point.

Corollary 3.10

Let \((X,p)\) be a complete partial metric space endowed with a graph G and Property (A). Let \(T:X\to \operatorname{CB}_{p}(X)\) be a mapping such that
  1. (i)
    for each \((x,y)\in E\) with \(x\neq y\), we have
    $$ H_{p}(Tx,Ty)\leq \zeta \biggl(\max \biggl\{ p(x,y),\frac{p(x,Tx)+p(y,Ty)}{2}, \frac {p(x,Ty)+p(x,Ty)}{2} \biggr\} \biggr), $$
    where ζ is a \((c)\)-comparison function with \(\sup_{t>0}\frac{\zeta(t)}{t}<1\);
     
  2. (ii)

    if \(s\in Tx\) and \(t\in Ty\) are such that \(p(s,t)< p(x,y)\), then we have \((s,t)\in E\), whenever \((x,y)\in E\) with \(x\neq y\).

     
Further, assume that there exist \(x_{0}\in X\) and \(x_{1}\in Tx_{0}\) such that \((x_{0},x_{1})\in E\). Then T has a fixed point.

Corollary 3.11

Let \((X,p)\) be a complete partial metric space endowed with a graph G and Property (A). Let \(T:X\to \operatorname{CB}_{p}(X)\) be a mapping such that
  1. (i)
    for each \((x,y)\in E\) with \(x\neq y\), we have
    $$ H_{p}(Tx,Ty)\leq a \max \biggl\{ p(x,y),\frac{p(x,Tx)+p(y,Ty)}{2}, \frac {p(x,Ty)+p(x,Ty)}{2} \biggr\} , $$
    where \(a\in[0,1)\);
     
  2. (ii)

    if \(s\in Tx\) and \(t\in Ty\) are such that \(p(s,t)< p(x,y)\), then we have \((s,t)\in E\), whenever \((x,y)\in E\) with \(x\neq y\).

     
Further, assume that there exist \(x_{0}\in X\) and \(x_{1}\in Tx_{0}\) such that \((x_{0},x_{1})\in E\). Then T has a fixed point.

Declarations

Acknowledgements

The authors are grateful to the reviewers for their valuable suggestions. One of the authors (LA Khan) wishes to thank King Abdulaziz City of Science and Technology for their financial support under the National Science, Technology and Innovation Plan, Project ID 14-MAT1272-03 and also the Deanship of Scientific Research, King Abdulaziz University, Jeddah, for their technical support.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology
(2)
Department of Mathematics, Quaid-i-Azam University
(3)
Department of Mathematics, Faculty of Science, King Abdulaziz University

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© Ali et al. 2015