Skip to main content

A new extension to the controlled metric type spaces endowed with a graph

Abstract

In this paper, we initiate a new extension of b-metric spaces, called controlled metric-like spaces, by changing the condition

$$ \bigl[\wp (s,r)=0 \Leftrightarrow s=r\bigr]\quad \text{by } \bigl[\wp (s,r)=0 \Rightarrow s=r\bigr] $$

and that means basically we may have a non-zero self-distance. We prove some fixed point theorems which generalize many results in the literature. Also, we present an interesting application to illustrate our results by considering controlled metric-like spaces endowed with a graph.

Introduction

Banach [1] in 1922 started a new field in mathematics, so called fixed point theory. That was the starting point for researchers around the globe to start generalize his result whether by changing the contractions or by generalizing the type of metric spaces, so it can cover a larger class of metrics; see [223]. Lately, in [24], an extension of b-metric spaces to extended b-metric spaces was given by Kamran et al. For related work, see [2530]. Not much later, Mlaiki et al. in [31], introduced another generalization to the b-metric spaces, so called controlled metric type spaces.

In this manuscript, we introduce the concept of controlled metric-like spaces, which involves a generalization of the controlled metric type spaces, that is, by assuming that the “self-distance” of an element is not necessary zero. However, if the value of the metric between two elements is zero, then these two elements are equal. One may ask: what is the point of all these generalizations? In fact the answer to that is quite simple: the larger the class of functions or metrics, the more fields that results can be applied to, such as computer science and engineering. Inspired by the work of Jachymski in [32], where he introduced the concept of metric spaces endowed with a graph, we present in the last section some fixed point results on a controlled metric-like space endowed with a graph.

Preliminaries

In 2017, Kamran et al. [24] introduced the notion of extended b-metric spaces.

Definition 2.1

([24])

Given \(F\neq \emptyset \) and \(\theta : F\times F \rightarrow [1,\infty )\). The function \(\Omega :F\times F\rightarrow [0,\infty )\) is called an extended b-metric if

  1. (1)

    \(\Omega (s,r)=0 \Longleftrightarrow s=r\);

  2. (2)

    \(\Omega (s,r) = \Omega (r,s)\);

  3. (3)

    \(\Omega (s,r) \leq \theta (s,r) [\Omega (s,z) + \Omega (z,r)]\),

for all \(s,r,z \in F\). In 2018, Mlaiki et al. in [31] introduced the following generalization of extended b-metric spaces, called controlled metric type spaces.

Definition 2.2

([31])

Given \(F\neq \emptyset \) and \(\eta : F\times F\rightarrow [1,\infty )\). The function \(\Omega : F\times F\rightarrow [0,\infty )\) is said to be of controlled metric type if

  1. (1)

    \(\Omega (s,r)=0\) \(s=r\);

  2. (2)

    \(\Omega (s,r)=\Omega (r,s)\);

  3. (3)

    \(\Omega (s,r)\leq \eta (s,z) \Omega (s,z)+\eta (z,r) \Omega (z,r)\),

for all \(s,r,z\in F\). The pair \((F,\Omega )\) is called a controlled metric type space.

Now, we present the definition of controlled metric-like spaces.

Definition 2.3

Given \(F\neq \emptyset \) and \(\eta : F\times F\rightarrow [1,\infty )\). The function \(\Omega : F\times F\rightarrow [0,\infty )\) verifying:

  1. (CL1)

    \(\wp (s,r)=0\) \(s=r\);

  2. (CL2)

    \(\wp (s,r)=\wp (r,s)\);

  3. (CL3)

    \(\wp (s,r)\leq \eta (s,z) \wp (s,z)+\eta (z,r) \wp (z,r)\),

for all \(s,r,z\in F\), is called a controlled metric-like space on F.

Notice that every space called a controlled metric type is a controlled metric-like space, but the converse is not always true, and to prove it we present the following example of a controlled metric-like space, that is not a controlled metric type space.

Example 2.4

Choose \(F=\{1,2,\ldots \}\). Take \(\wp :F \times F \to [0,\infty ) \) as

$$ \wp (s,r)= \textstyle\begin{cases} 0, &\text{if $r=s$ and both even,} \\ \frac{1}{s+r}, &\text{if $s=r$ and both odd,} \\ \frac{1}{s}, &\text{if $s$ is even and $r$ is odd,} \\ \frac{1}{r}, &\text{if $s$ is odd and $r$ is even,} \\ 1, &\text{otherwise}. \end{cases} $$

Given \(\eta : F\times F\rightarrow [1,\infty )\) as

$$ \eta (s,r)= \textstyle\begin{cases} s, &\text{if $s$ is even and $r$ is odd,} \\ r, &\text{if $s$ is odd and $r$ is even,} \\ 1, &\text{otherwise}. \end{cases} $$

\((CL1)\) and \((CL2)\) are obvious. We claim that (\(CL3\)) is satisfied.

Case 1: If \(z=s\) or \(z=r\), (\(CL3\)) is satisfied.

Case 2: If \(z\neq s\) and \(z\neq r\), (\(CL3\)) holds if \(s=r\). Now, assume that \(s\neq r\), so \(s\neq r\neq z\). Again, (\(CL3\)) is verified in the following subcases:

  1. (s1):

    s, z are even and r is odd;

  2. (s2):

    s is even and r, z are odd;

  3. (s3):

    s, z are odd and r is even;

  4. (s4):

    s, z are even and r is odd;

  5. (s5):

    s, r, z are even;

  6. (s6):

    s, r are even and z is odd;

  7. (s7):

    s, r are odd and z is even;

  8. (s8):

    s, r, z are odd.

Thus, \((F,\wp )\) is a controlled metric-like space. It is not difficult to see that \((F,\wp )\) is not a controlled metric type space.

Example 2.5

Take \(F=\{0,1,2\}\). Define the function by

$$ \wp (0,0)=\wp (1,1)=0,\quad\quad \wp (2,2)=\frac{1}{10} $$

and

$$ \wp (0,1)=\wp (1,0)=1,\quad\quad \wp (0,2)=\wp (2,0)=\frac{1}{2},\quad \quad \wp (1,2)=\wp (2,1)=\frac{2}{5}. $$

Take \(\eta : F\times F\rightarrow [1,\infty )\) to be symmetric and to be defined by

$$ \eta (0,0)=\eta (1,1)=\eta (2,2)=\eta (0,2)=1, \quad\quad \eta (1,2)= \frac{5}{4}, \quad\quad \eta (0,1)=\frac{11}{10}. $$

Here, is controlled metric-like on F.

We have \(\wp (2,2)=\frac{1}{10}\neq 0\), which implies that \((F,\wp )\) is not a controlled metric type space.

For work in the same direction, see [33]. The definition of being of Cauchy type and of convergence in controlled metric-like spaces is given as follows.

Definition 2.6

Let \((F,\wp )\) be a controlled metric-like space and \(\{s_{n}\}_{n\ge 0}\) be a sequence in F.

  1. (1)

    \(\{s_{n}\}\) is convergent to s in F, if and only if

    $$ \lim_{n\rightarrow \infty }\wp (s_{n},s)=\wp (s,s). $$

    Here, one writes \(\lim_{n \rightarrow \infty }{s_{n}=s}\).

  2. (2)

    \(\{s_{n}\}\) is Cauchy, if and only if \(\lim_{n,m\rightarrow \infty }\wp (s_{n},s_{m})\) exists and is finite.

  3. (3)

    \((F,\wp )\) is called complete if, for each Cauchy sequence \(\{s_{n}\}\), there is some \(s\in F\) so that

    $$ \lim_{n\rightarrow \infty }\wp (s_{n},s)=\wp (s,s)=\lim _{n,m \rightarrow \infty }\wp (s_{n},s_{m}). $$

Definition 2.7

Let \((F,\wp )\) be a controlled metric-like space. Let \(s\in F\) and \(\tau >0\).

  1. (i)

    The open ball \(B(s,\tau )\) is

    $$ B(s,\tau )=\bigl\{ y\in F, \bigl\vert \wp (s,r)-\wp (s,s) \bigr\vert < \tau \bigr\} . $$
  2. (ii)

    The mapping \(\zeta :F\rightarrow F\) is called continuous at \(s\in F\) if for \(\upsilon >0\), there is \(\nu >0\) so that \(\zeta (B(s,\nu ))\subseteq B(\zeta (s),\upsilon )\). Thus, if ζ is continuous at s, then, for any sequence \(\{s_{n}\}\) converging to s, we have \(\lim_{n \rightarrow \infty }\zeta s_{n}=\zeta s\), that is,

    $$ \lim_{n\rightarrow \infty }\wp (\zeta s_{n},\zeta s)=\wp (\zeta s, \zeta s). $$

Notice that each controlled metric-like space is a controlled metric space. But the converse is not always true (see Example 2.4 and Example 2.5).

Main results

The following result corresponds to the Banach contraction principle on controlled metric-like spaces.

Theorem 3.1

Let ζ be a self-mapping on a complete controlled metric-like space \((F,\wp )\) so that

$$ \wp \bigl(\zeta (s),\zeta (r)\bigr)\leq k\wp (s,r), $$
(3.1)

for all \(s,r\in F\), where \(k\in (0,1)\). For \(s_{0}\in F\), take \(s_{n}=\zeta ^{n}(s_{0})\). Suppose that

$$ \sup_{m\geq 1}\lim_{i\rightarrow \infty } \frac{\eta (s_{i+1},s_{i+2})}{\eta (s_{i},s_{i+1})}\eta (s_{i+1},s_{m})< \frac{1}{k}. $$
(3.2)

Also, assume for each \(s\in F\),

$$ \lim_{n\rightarrow \infty }\eta (s_{n},s) \quad \textit{and}\quad \lim_{n \rightarrow \infty }\eta (s,s_{n}) \quad \textit{exist and are finite}. $$
(3.3)

Then ζ possesses a unique fixed point, say \(\tau \in F\). We have \(\wp (\tau ,\tau )=0\).

Proof

Take the sequence \(\{s_{n}=\zeta ^{n}(s_{0})\}\). By using (3.1), we get \(\wp (s_{n},s_{n+1})\le k^{n}\wp (s_{0},s_{1})\) for all \(n\geq 0\). For all integers \(n< m\), one writes

$$\begin{aligned} \wp (s_{n},s_{m})&\le \eta (s_{n},s_{n+1}) \wp (s_{n},s_{n+1})+\eta (s_{n+1},s_{m}) \wp (s_{n+1},s_{m}) \\ &\le \eta (s_{n},s_{n+1})\wp (s_{n},s_{n+1})+ \eta (s_{n+1},s_{m}) \eta (s_{n+1},s_{n+2}) \wp (s_{n+1},s_{n+2}) \\ &\quad {} +\eta (s_{n+1},s_{m})\eta (s_{n+2},s_{m}) \wp (s_{n+2},s_{m}) \\ &\le \eta (s_{n},s_{n+1})\wp (s_{n},s_{n+1})+ \eta (s_{n+1},s_{m}) \eta (s_{n+1},s_{n+2}) \wp (s_{n+1},s_{n+2}) \\ &\quad {} +\eta (s_{n+1},s_{m})\eta (s_{n+2},s_{m}) \eta (s_{n+2},s_{n+3})\wp (s_{n+2},s_{n+3}) \\ &\quad {} +\eta (s_{n+1},s_{m})\eta (s_{n+2},s_{m}) \eta (s_{n+3},s_{m})\wp (s_{n+3},s_{m}) \\ &\le \cdots \\ &\le \eta (s_{n},s_{n+1})\wp (s_{n},s_{n+1})+ \sum_{i=n+1}^{m-2} \Biggl(\prod _{j=n+1}^{i}\eta (s_{j},s_{m}) \Biggr)\eta (s_{i},s_{i+1}) \wp (s_{i},s_{i+1}) \\ &\quad {} +\prod_{k=n+1}^{m-1} \eta (s_{k},s_{m})\wp (s_{m-1},s_{m}) \\ &\le \eta (s_{n},s_{n+1})k^{n}\wp (s_{0},s_{1})+\sum_{i=n+1}^{m-2} \Biggl(\prod_{j=n+1}^{i}\eta (s_{j},s_{m}) \Biggr)\eta (s_{i},s_{i+1}) k^{i}\wp (s_{0},s_{1}) \\ &\quad {} +\prod_{i=n+1}^{m-1} \eta (s_{i},s_{m})k^{m-1}\wp (s_{0},s_{1}) \\ &\le \eta (s_{n},s_{n+1})k^{n}\wp (s_{0},s_{1})+\sum_{i=n+1}^{m-2} \Biggl(\prod_{j=n+1}^{i}\eta (s_{j},s_{m}) \Biggr)\eta (s_{i},s_{i+1}) k^{i}\wp (s_{0},s_{1}) \\ &\quad {} + \Biggl(\prod_{i=n+1}^{m-1} \eta (s_{i},s_{m}) \Biggr)k^{m-1}\eta (s_{m-1},s_{m}) \wp (s_{0},s_{1}) \\ &= \eta (s_{n},s_{n+1})k^{n}\wp (s_{0},s_{1})+\sum_{i=n+1}^{m-1} \Biggl(\prod_{j=n+1}^{i}\eta (s_{j},s_{m}) \Biggr)\eta (s_{i},s_{i+1}) k^{i}\wp (s_{0},s_{1}). \end{aligned}$$

Thus, due to \(\eta (s,r) \geq 1\),

$$ \wp (s_{n},s_{m})\leq \eta (s_{n},s_{n+1})k^{n} \wp (s_{0},s_{1})+ \sum_{i=n+1}^{m-1} \Biggl(\prod_{j=0}^{i}\eta (s_{j},s_{m}) \Biggr) \eta (s_{i},s_{i+1}) k^{i}\wp (s_{0},s_{1}). $$

Let

$$ \Upsilon _{p}= \sum_{i=0}^{p} \Biggl(\prod_{j=0}^{i}\eta (s_{j},s_{m}) \Biggr)\eta (s_{i},s_{i+1}) k^{i}. $$

Hence, we have

$$ \wp (s_{n},s_{m})\leq \wp (s_{0},s_{1}) \bigl[ k^{n} \eta (s_{n},s_{n+1})+( \Upsilon _{m-1}-\Upsilon _{n}) \bigr]. $$
(3.4)

Condition (3.2), by using the ratio test, implies that \(\lim_{n\rightarrow \infty }\Upsilon _{n}\) exists. Hence, \(\{\Upsilon _{n}\}\) is a real Cauchy sequence. Letting \(n,m\rightarrow \infty \) in the inequality (3.4), we get

$$ \lim_{n,m\rightarrow \infty }\wp (s_{n},s_{m})=0, $$
(3.5)

that is, the sequence \(\{s_{n}\}\) is Cauchy in \((F,\wp )\), which is a complete controlled metric-like space, so \(\{s_{n}\}\) converges to some \(s\in F\). We have

$$ \lim_{n\rightarrow \infty }\wp (s_{n},s)=\wp (s,s)= \lim_{n,m \rightarrow \infty }\wp (s_{n},s_{m})=0. $$
(3.6)

Then \(\wp (s,s)=0\). We claim that \(\zeta s=s\). Applying \((CL3)\), one writes

$$ \wp (s,s_{n+1})\leq \eta (s,s_{n})\wp (u, s_{n})+ \eta (s_{n},s_{n+1}) \wp (s_{n},s_{n+1}). $$

Using (3.2), (3.3), (3.5) and (3.6),

$$ \lim_{n\rightarrow \infty }\wp (s,s_{n+1})=0. $$
(3.7)

Again, in view of \((CL3)\) and using (3.1),

$$\begin{aligned} \wp \bigl(s,\zeta (s)\bigr)&\leq \eta (s,s_{n+1})\wp (s, s_{n+1})+ \eta \bigl(s_{n+1}, \zeta (s)\bigr)\wp \bigl(s_{n+1}, \zeta (s)\bigr) \\ &\le \eta (s,s_{n+1})\wp (s, s_{n+1})+ k\eta \bigl(s_{n+1},\zeta (s)\bigr)\wp (s_{n}, s). \end{aligned}$$

Letting \(n\rightarrow \infty \) and inserting (3.3) and (3.7), we conclude that \(\wp (s,\zeta (s))=0\), i.e., \(\zeta (s)=s\).

Assume that \(\zeta \sigma =\sigma \) and \(\zeta \varsigma =\varsigma \). Here,

$$ \wp (\sigma ,\varsigma )=\wp \bigl(\zeta (\sigma ),\zeta (\varsigma )\bigr)\le k \wp (\sigma ,\varsigma ). $$

It holds unless \(\wp (\sigma ,\varsigma )=0\), so \(\sigma =\varsigma \). □

Theorem 3.1 is supported by the two following examples.

Example 3.2

Choose \(F=\{0,1,2\}\). Take to be symmetric and to be defined by

$$ \wp (0,0)=0,\quad\quad \wp (1,1)=0,\quad\quad \wp (2,2)=\frac{1}{100}, $$

and

$$ \wp (0,1)=1,\quad\quad \wp (0,2)=\frac{11}{20},\quad\quad \wp (1,2)= \frac{2}{5}. $$

Given \(\eta : F\times F\rightarrow [1,\infty )\) as

$$ \eta (0,0)=\eta (1,1)=\eta (2,2)=\eta (0,2)=\eta (2,0)=1, \quad\quad \eta (1,2)= \eta (2,1)=\frac{49}{40}, $$

and

$$ \eta (0,1)=\eta (1,0)=\frac{11}{10}. $$

Clearly, is controlled metric-like ( is not a controlled metric type on F). Define the self-mapping ζ on F by

$$ \zeta (0)=2\quad \text{and}\quad \zeta (1)=\zeta (2)=1. $$

Set \(k=\frac{4}{5}\). It is clear that (3.1) is verified. For each \(s_{0}\in F\), (3.2) holds. All hypotheses of Theorem 3.1 are fulfilled, and hence ζ possesses a unique fixed point, which is \(s=1\).

Example 3.3

Let \(F =[0,\infty )\). Define \(\wp :F ^{2}\rightarrow {}[ 0,\infty )\) by

$$ \wp (x,y)= \textstyle\begin{cases} 0&\text{if } x=y=0, \\ \frac{y}{1+y}&\text{if } x=0,y\neq 0, \\ \frac{x}{1+x}&\text{if } y=0,x\neq 0, \\ x+y&\text{if } x\neq 0,y\neq 0. \end{cases} $$

Consider \(\eta :F^{2}\rightarrow {}[ 1,\infty )\) as \(\eta (x,y )=2+2x +2y\). Note that is controlled metric-like on F.

First, \((CL1)\) and \((CL2)\) are obvious. We need to prove \((CL3)\). For this, let x, y and z in F. We state the following cases:

Case 1: \(x=z=0\). Here, \((CL3)\) holds.

Case 2: \(x=0\) and \(z\neq 0\).

Subcase 1: \(y=0\). We have

$$ \begin{aligned} \wp (x,z)&=\frac{z}{1+z} \\ &\leq (2) (0)+(2+2z)\frac{z}{1+z} \\ &= \eta (x,y)\wp (x,y)+\eta (y,z)\wp (y,z). \end{aligned} $$

Subcase 2: \(y\neq 0\). We have

$$ \begin{aligned} \wp (x,z)&=\frac{z}{1+z} \\ &\leq (2+2y)\frac{y}{1+y}+(2+2z) (y+z) \\ &= \eta (x,y)\wp (x,y)+\eta (y,z)\wp (y,z). \end{aligned} $$

Case 3: \(z=0\) and \(x\neq 0\). Proceeding similarly to Case 2, \((CL3)\) holds.

Case 4: \(x\neq 0\) and \(z\neq 0\).

Subcase 1: \(y=0\). We have

$$ \begin{aligned} \wp (x,z)&=x+z \\ &\leq (2+2x)\frac{x}{1+x}+(2+2z)\frac{z}{1+z} \\ &= \eta (x,y)\wp (x,y)+\eta (y,z)\wp (y,z). \end{aligned} $$

Subcase 2: \(y\neq 0\). We have

$$ \begin{aligned} \wp (x,z)&=x+z \\ &\leq (2+2x+2y) (x+y)+(2+2y+2z) (y+z) \\ &= \eta (x,y)\wp (x,y)+\eta (y,z)\wp (y,z). \end{aligned} $$

On the other hand, \((F,\wp )\) is not a b-metric-like space. We argue by contradiction by assuming that \((F ,\wp )\) is a b-metric-like space with a coefficient \(s\geq 1\) (a constant). Then, for any real \(y>0\), we have

$$ \wp (y,y+1)\leq s\bigl[\wp (y,0)+\wp (0,y+1)\bigr]. $$

That is,

$$ 2y+1\leq s\biggl[\frac{y}{1+y}+\frac{y+1}{2+y}\biggr]. $$

Letting \(y\rightarrow \infty \), we get \(+\infty \leq 2s\), which is a contradiction.

Consider the self-mapping ζ on F defined by \(\zeta (x)=\frac{x}{3}\). It is clear that (3.1) holds for all \(x,y\in F\), with \(k=\frac{2}{5}\). Take \(s_{0}=1\). The sequence \(\{s_{n}\}\) given as \(s_{n}=\zeta ^{n}(s_{0})\) is written as

$$ s_{n}=\frac{1}{3^{n}},\quad n=0,1,\ldots . $$

A simple calculation yields

$$ \sup_{m\geq 1}\lim_{i\rightarrow \infty } \frac{\eta (s_{i+1},s_{i+2})}{\eta (s_{i},s_{i+1})}\eta (s_{i+1},s_{m})=\biggl(2+ \frac{1}{3}\biggr)=\frac{7}{3}< \frac{5}{2}= \frac{1}{k}. $$

That is, (3.2) is verified. Moreover, we have, for each \(s\in F\),

$$ \lim_{n\rightarrow \infty }\eta (s_{n},s)=\lim _{n\rightarrow \infty } \eta (s,s_{n})=(2+2s) \quad \text{exists}. $$

All hypotheses of Theorem 3.1 hold. Hence, ζ admits a unique fixed point, which is \(\tau =0\). It verifies \(\wp (\tau ,\tau )=0\).

Definition 3.4

Let \(\zeta :F \longrightarrow F\). For some \(s_{0}\in F\), let \(O (s_{0})=\{s_{0},\zeta s_{0},\zeta ^{2}s_{0},\ldots\}\) be the orbit of \(s_{0}\). Given \(P:F\longrightarrow \mathbb{R}\). Such a P is called ζ-orbitally lower semi-continuous at \(\varrho \in F\) if for \(\{s_{n}\}\subset O (s_{0})\) so that \(s_{n}\longrightarrow \varrho \), we have \(P(\varrho )\leq \lim_{n\rightarrow \infty }\inf P(s_{n})\).

As in [24], using Definition 3.4, a consequence of Theorem 3.1 (it is a generalization of Theorem 1 in [13]) is as follows.

Corollary 3.5

Let \(\zeta \colon F\rightarrow F\) be a self-mapping on a complete controlled metric-like space \((F,\wp )\). Given \(s_{0}\in F\). Assume there is \(k\in (0,1)\) so that

$$ \wp \bigl(\zeta (z),\zeta ^{2}(z)\bigr)\leq k\wp \bigl(z,\zeta (z) \bigr), \quad \textit{for each } z\in O(s_{0}). $$
(3.8)

Take \(s_{n}=\zeta ^{n}(s_{0})\). Suppose that

$$ \sup_{m\geq 1}\lim_{i\rightarrow \infty } \frac{\eta (s_{i+1},s_{i+2})}{\eta (s_{i},s_{i+1})}\eta (s_{i+1},s_{m})< \frac{1}{k}. $$
(3.9)

Then \(s_{n}\rightarrow l\in F \) as \(n\rightarrow \infty \). Further, if the functional \(\delta \mapsto \wp (s,\zeta (\delta ))\) is ζ-orbitally lower semi-continuous at l, we get \(\zeta (l)=l\).

Controlled metric-like spaces endowed with a graph

Throughout this section, we denote controlled metric-like spaces by \((CMLS)\). Now, we present \((CMLS)\) endowed with a graph. Figure 1 is an example of a controlled metric-like space endowed with a graph.

Figure 1
figure1

Controlled metric-like space as in Example 2.5

Consider the \((CMLS)\) \((F,\wp )\), let Δ be the diagonal of \(F^{2}\). A graph G is defined by the pair \((V,E)\) where V is a set of vertices coinciding with F and E is the set of its edges with \(\Delta \subset E\). From now on, assume that G has no parallel edges.

Definition 4.1

([32])

Let t and s be two vertices of a graph G. We define \(q\in \mathbb{N}\cup \{0\}\) to be the length of the path between t and s in G by a sequence \((k_{i})_{i=0}^{q}\) of \(q+1\) distinct vertices where \(k_{0}=t\), \(k_{n}=s\) and \((k_{i},k_{i+1})\in E(G)\) for \(i=1,2,\ldots,q\).

Note that the graph G can be converted to a weighted graph and associate to each edge the value of \((CMLS)\).

Notation

Let \(F^{g}=\{x\in F/ (x,gx)\in E(G) \text{ or } (gx,x)\in E(G)\}\).

Definition 4.2

Let \((F,\wp )\) be a complete \((CMLS)\) endowed with a graph G. The mapping \(\zeta \colon F\rightarrow F\) is said to be a \(G_{\phi }\)-contraction if

  • $$ \text{for all } t,s\in F, \quad (t,s)\in E(G) \quad \Longrightarrow \quad (\zeta t, \zeta s)\in E(G); $$
    (4.1)
  • there is \(\phi :[0,\infty )\longrightarrow [0,\infty )\) so that

    $$ \xi \bigl(\zeta t,\zeta ^{2}t\bigr)\leq \phi \bigl(\xi (t,\zeta t) \bigr)\quad \forall t \in X^{\zeta }, $$
    (4.2)

    where ϕ is nondecreasing and \(\{\phi ^{n}(t)\}_{n\in \mathbb{N}}\rightarrow 0\) \(\forall t>0\).

Definition 4.3

The mapping \(\zeta :F\longrightarrow F\) is called orbitally G-continuous if for all \(\tau ,\rho \in X\) and \(\{s_{n}\}_{n\in \mathbb{N}}\) a positive sequence,

$$ \zeta ^{s_{n}}\tau \longrightarrow \rho , \quad\quad \bigl(\zeta ^{s_{n}}\tau , \zeta ^{s_{n+1}}\tau \bigr)\in E(G) \quad \Longrightarrow \quad \zeta \bigl(\zeta ^{s_{n}} \tau \bigr)\longrightarrow \zeta \rho \quad \text{as } n\rightarrow \infty . $$

Lemma 4.4

Let \((F,\wp )\) be a complete \((CMLS)\) equipped by a graph G. Suppose that \(\zeta \colon F\rightarrow F\) be a \(G_{\phi }\)-contraction. If \(t\in X^{\zeta }\) then there is \(q(t)\geq 0\) so that

$$ \wp \bigl(\zeta ^{n}t,\zeta ^{n+1}t\bigr)\leq \phi ^{n}\bigl(q(t)\bigr) \quad \forall n \in \mathbb{N} , $$
(4.3)

where \(q(t)=\wp (t,\zeta t)\).

Proof

Let \(t\in F^{\zeta }\), then \((t,\zeta t)\in E(G)\) or \((\zeta t,t)\in E(G)\). Assume that w.l.o.g. \((t,\zeta t)\in E(G)\). Hence,

$$ \bigl(\zeta ^{n},\zeta ^{n+1}t\bigr)\in E(G)\quad \forall n\in \mathbb{N}. $$
(4.4)

Hence, we deduce

$$\begin{aligned} \wp \bigl(\zeta ^{n}t,\zeta ^{n+1}t\bigr) \leq & \phi \bigl( \wp \bigl(\zeta ^{n-1}t, \zeta ^{n}t\bigr)\bigr) \\ \leq & \phi ^{2}\bigl(\wp \bigl(\zeta ^{n-2}t,\zeta ^{n-1}t\bigr)\bigr) \\ \leq & \vdots \\ \leq & \phi ^{n}\bigl(\wp (t,\zeta t)\bigr)=\phi ^{n} \bigl(q(t)\bigr). \end{aligned}$$

 □

Theorem 4.5

Let \((F,\wp , G)\) be a complete \((CMLS)\) equipped by a graph G. Suppose that \(\zeta \colon F\rightarrow F\) be a \(G_{\phi }\)-contraction, which is orbitally G-continuous. Consider the property \((P)\) as follows: \(\forall \{t_{n}\}_{n\in \mathbb{N}}\) in F, if \(t_{n}\longrightarrow t\) and \((t_{n},t_{n+1})\in E(G)\), then there is \(\{t_{k_{n}}\}_{n\in \mathbb{N}}\) where \((t_{k_{n}},t)\in E(G)\), holds. Further, suppose that, for each \(s\in F\),

$$ \lim_{n\rightarrow \infty }\eta \bigl(\zeta ^{i}s,\zeta ^{n}s\bigr) \quad ( \forall i\geq 1)\quad \textit{and}\quad \lim _{n\rightarrow \infty }\eta \bigl( \zeta ^{n}s,\zeta ^{n+1}s \bigr) \quad \textit{exist and are finite} $$
(4.5)

and

$$ \textit{if} \quad \lim_{n\rightarrow \infty }\zeta ^{n}s \rightarrow u\in F , \quad \textit{then}\quad \wp (\zeta u,u)> \limsup_{n\rightarrow \infty } \eta \bigl( \zeta u,\zeta ^{n}s\bigr)\wp (\zeta u,\zeta u). $$
(4.6)

Thus, the restriction of \(\zeta _{| [s]_{\tilde{G}}}\) to \([s]_{\tilde{G}}\) possesses a fixed point. Moreover, if for every two fixed points \(v_{1}\), \(v_{2}\) we have \(\eta (v_{1}, v_{2})>1\), then we have uniqueness of the fixed point.

Proof

Consider \(s\in F^{\zeta }\). By Lemma 4.4, there is \(q(s)\geq 0\) so that

$$ \wp \bigl(\zeta ^{n}s,\zeta ^{n+1}s\bigr)\leq \phi ^{n}\bigl(q(s)\bigr). $$

First, we show that \((\zeta ^{n}s)_{n\in \mathbb{N}}\) converges to some \(u\in F\). It should be enough to ensure that \(\{\zeta ^{n}t\}_{n\in \mathbb{N}}\) is Cauchy. Take the integers nm. Using again Lemma 4.4, we get

$$\begin{aligned} \wp \bigl(\zeta ^{n}s,\zeta ^{n+m}s\bigr) \leq & \eta \bigl( \zeta ^{n}s,\zeta ^{n+1}s\bigr) \wp \bigl(\zeta ^{n}s,\zeta ^{n+1}s\bigr)+\eta \bigl(\zeta ^{n+1}s, \zeta ^{n+m}s\bigr) \wp \bigl(\zeta ^{n+1}s,\zeta ^{n+m}s\bigr) \\ \leq & \eta \bigl(\zeta ^{n}s,\zeta ^{n+1}s\bigr)\wp \bigl( \zeta ^{n}s,\zeta ^{n+1}s\bigr)+ \eta \bigl(\zeta ^{n+1}s,\zeta ^{n+m}s\bigr)\eta \bigl(\zeta ^{n+1}s, \zeta ^{n+2}s\bigr) \\ & {} \times \wp \bigl(\zeta ^{n+1}s,\zeta ^{n+2}s\bigr) \\ & {} + \eta \bigl(\zeta ^{n+1}s,\zeta ^{n+m}s\bigr)\eta \bigl(\zeta ^{n+2}s,\zeta ^{n+m}s\bigr) \wp \bigl(\zeta ^{n+2}s,\zeta ^{n+m}s\bigr) \\ \leq & \vdots \\ \leq & \eta \bigl(\zeta ^{n}s,\zeta ^{n+1}s\bigr)\wp \bigl( \zeta ^{n}s,\zeta ^{n+1}s\bigr)+ \sum _{i=n+1}^{n+m-2}\Biggl( \prod_{j=n+1}^{i} \eta \bigl(\zeta ^{j}s,\zeta ^{n+m}s\bigr)\Biggr)\eta \bigl( \zeta ^{i}s,\zeta ^{i+1}s\bigr) \\ & {} \times \wp \bigl(\zeta ^{i}s,\zeta ^{i+1}s\bigr)+ \prod_{k=n+1}^{n+m-1} \eta \bigl(\zeta ^{k}s,\zeta ^{n+m}s\bigr)\wp \bigl(\zeta ^{n+m-1}s, \zeta ^{n+m}s\bigr) \\ \leq & \eta \bigl(\zeta ^{n}s,\zeta ^{n+1}s\bigr)\phi ^{n}\bigl(q(s)\bigr)+ \sum_{i=n+1}^{n+m-2} \Biggl( \prod_{j=n+1}^{i} \eta \bigl(\zeta ^{j}s,\zeta ^{n+m}s\bigr)\Biggr)\eta \bigl(\zeta ^{i}s,\zeta ^{i+1}s\bigr) \phi ^{i}\bigl(q(s)\bigr) \\ & {} + \prod_{k=n+1}^{n+m-1}\eta \bigl(\zeta ^{k}s,\zeta ^{n+m}s\bigr) \phi ^{n+m-1}\bigl(q(s) \bigr)\eta \bigl(\zeta ^{n+m-1}s,\zeta ^{n+m}s\bigr) \\ \leq & \eta \bigl(\zeta ^{n}s,\zeta ^{n+1}s\bigr)\phi ^{n}\bigl(q(s)\bigr)+ \sum_{i=n+1}^{n+m-1} \Biggl( \prod_{j=0}^{i} \eta \bigl(\zeta ^{j}s,\zeta ^{n+m}s\bigr)\Biggr)\eta \bigl(\zeta ^{i}t,\zeta ^{i+1}s\bigr) \phi ^{i}\bigl(q(s)\bigr) \\ \leq & \prod_{j=0}^{n}\eta \bigl(\zeta ^{j}t,\zeta ^{n+m}s\bigr)) \eta \bigl(\zeta ^{n}s, \zeta ^{n+1}s\bigr)\phi ^{n}\bigl(q(s)\bigr) \\ & {} + \sum_{i=n+1}^{n+m-1}\Biggl( \prod _{j=0}^{i} \eta \bigl(\zeta ^{j}s, \zeta ^{n+m}s\bigr)\Biggr)\eta \bigl(\zeta ^{i}s,\zeta ^{i+1}s\bigr) \phi ^{i}\bigl(q(s)\bigr) \\ = & \sum_{i=n}^{n+m-1}\Biggl( \prod _{j=0}^{i} \eta \bigl(\zeta ^{j}s,\zeta ^{n+m}s\bigr)\Biggr)\eta \bigl(\zeta ^{i}s,\zeta ^{i+1}s\bigr) \phi ^{i}\bigl(q(s)\bigr) \\ = & \sum_{i=n}^{n+m-1}\Gamma _{i} \phi ^{i}\bigl(q(s)\bigr),\quad \text{where } \Gamma _{i}=\Biggl( \prod_{j=0}^{i}\eta \bigl(\zeta ^{j}s, \zeta ^{n+m}s\bigr)\Biggr)\eta \bigl(\zeta ^{i}s,\zeta ^{i+1}s\bigr) \\ = & \sum_{i=1}^{m}\Gamma _{i}^{n+i-1}\phi ^{n+i-1}\bigl(q(s)\bigr). \end{aligned}$$

In view of the property of ϕ and using (4.5), we deduce that \(\sum_{i=1}^{m}\Gamma _{i}^{n+i-1}\phi ^{n+i-1}(q(s))\) is convergent to 0 as \(n,m\longrightarrow \infty \). Thus,

$$ \lim_{n,m\rightarrow \infty }\wp \bigl(\zeta ^{n}s,\zeta ^{m}s\bigr)=0, $$
(4.7)

that is, the sequence \(\{\zeta ^{n}s\}\) is Cauchy in \((F,\wp )\). Its completeness entails that \(\{\zeta ^{n}s\}\) converges to some \(u\in F\), so that

$$ \lim_{n\rightarrow \infty }\wp \bigl(\zeta ^{n}s,u \bigr)=\wp (u,u)=\lim_{n,m \rightarrow \infty }\wp \bigl(\zeta ^{n}s,\zeta ^{m}s\bigr)=0. $$
(4.8)

We have \(\wp (u,u)=0\).

Since \(s\in F^{\zeta }\), one writes that \(\zeta ^{n}s\in F^{\zeta }\) for every \(n\in \mathbb{N}\). We suppose that \((s,\zeta s)\in E(G)\), by \((P)\), we deduce that there is \(\{\zeta ^{k_{n}}s\}_{n\in \mathbb{N}}\) of \(\{\zeta ^{n}s\}_{n\in \mathbb{N}}\) so that \((\zeta ^{k_{n}}s,u)\in E(G)\) for any \(n\in \mathbb{N}\). We consider the path in G as follows: \(s,\zeta s,\ldots,\zeta ^{k_{1}}s,u\) and so \(u\in [s]_{\tilde{G}}\). The orbital G-continuity of ζ yields

$$ \lim_{n\rightarrow \infty } \wp \bigl(\zeta \bigl(\zeta ^{k_{n}}s\bigr), \zeta u\bigr)= \wp (\zeta u,\zeta u). $$
(4.9)

Suppose that \(\wp (\zeta u,u)>0\). Applying \((CL3)\), one writes

$$\begin{aligned} \wp (\zeta u,u) \leq \eta (\zeta u,\zeta ^{k_{n}}\wp \bigl(\zeta u,\zeta ^{k_{n}}s\bigr)+ \eta \bigl(\zeta ^{n}s,u\bigr)\wp \bigl(\zeta ^{k_{n}} s,u\bigr). \end{aligned}$$

Letting \(n\rightarrow \infty \) and using (4.5), (4.8) and (4.9), we get

$$ \wp (\zeta u,u)\leq \limsup_{n\rightarrow \infty } \eta \bigl(\zeta u, \zeta ^{k_{n}} s\bigr)\wp (\zeta u,\zeta u). $$

It contradicts (4.6). Hence, \(\wp (\zeta u,u)=0\), so \(\zeta u=u\), that is, u is a fixed point of \(\zeta| _{[s]_{\tilde{G}}}\).

For its uniqueness, suppose there are two fixed points \(v_{1}\) and \(v_{2}\), i.e., \(\zeta v_{1}=v_{1}\) and \(\zeta v_{2}=v_{2}\). We have

$$\begin{aligned} \wp (v_{1},v_{2}) \leq & \eta (v_{1},v_{1}) \wp (v_{1},v_{1})+ \eta (v_{1},v_{2}) \wp (v_{1},v_{2}) \\ \leq & \frac{\eta (v_{1},v_{1})}{1-\eta (v_{1},v_{2})}\wp (v_{1},v_{1}). \end{aligned}$$

Since \(\eta >1\), we have \(\wp (v_{1},v_{2})\leq 0\), and so \(v_{1}=v_{2}\). □

Conclusion

The \((CMLS)\) endowed with a graph introduced in this paper can be successfully used in different branches of scientific knowledge, for example when studying the kinetics of biochemical reaction networks, where we have the serious problem of the choice of “appropriate distance” in the analysis of a “K-angle general kinetic reaction system”; see [34].

Availability of data and materials

The data used to support the findings of this study are available from the corresponding author upon request.

References

  1. 1.

    Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux equations integrales. Fundam. Math. 3, 133–181 (1922)

    Article  Google Scholar 

  2. 2.

    Abdeljawad, T., Jarad, F., Baleanu, D.: On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives. Sci. China Math. 51, 1775–1786 (2008)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Abdeljawad, T., Baleanu, D., Jarad, F.: Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives. J. Math. Phys. 88, 8 (2018)

    Google Scholar 

  4. 4.

    Abdeljawad, T.: Meir–Keeler contractive fixed and common fixed point theorems. Fixed Point Theory Appl. 2013, 19 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Abodayeh, K., Mlaiki, N., Abdeljawad, T., Shatanawi, W.: Relations between partial metric spaces and M-metric spaces, Caristi Kirk’s theorem in M-metric spaces. J. Math. Anal. 7, 1–12 (2016)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Abdeljawad, T., Abodayeh, K., Mlaiki, N.: On fixed point generalizations to partial b-metric spaces. J. Comput. Anal. Appl. 19, 883–891 (2015)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Afshari, H., Aydi, H., Karapinar, E.: On generalized αψ-Geraghty contractions on b-metric spaces. Georgian Math. J. 27, 9–21 (2020)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Ameer, E., Aydi, H., Arshad, M., De la Sen, M.: Hybrid Ćirić type graphic \((\Upsilon,\Lambda)\)-contraction mappings with applications to electric circuit and fractional differential equations. Symmetry 12, 467 (2020)

    Article  Google Scholar 

  9. 9.

    Patle, P., Patel, D., Aydi, H., Radenović, S.: On \(H^{+}\)-type multivalued contractions and applications in symmetric and probabilistic spaces. Mathematics 7, 144 (2019)

    Article  Google Scholar 

  10. 10.

    Aydi, H., Bota, M.F., Karapinar, E., Moradi, S.: A common fixed point for weak ϕ-contractions on b-metric spaces. Fixed Point Theory 13, 337–346 (2012)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Aydi, H., Felhi, A., Karapinar, E., Sahmim, S.: A Nadler-type fixed point theorem in dislocated spaces and applications. Miskolc Math. Notes 19, 111–124 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Bakhtin, I.A.: The contraction mapping principle in almost metric spaces. Funct. Anal. 30, 26–37 (1989)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Qawaqneh, H., Noorani, M.S.M., Shatanawi, W., Aydi, H., Alsamir, H.: Fixed point results for multi-valued contractions in b-metric spaces and an application. Mathematics 7, 132 (2018)

    Article  Google Scholar 

  14. 14.

    Czerwik, S.: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav. 1, 5–11 (1993)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Karapinar, E., Czerwik, S., Aydi, H.: \((\alpha ,\psi )\)-Meir–Keeler contraction mappings in generalized b-metric spaces. J. Funct. Spaces 2018, 3264620 (2018)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Kilbas, A.A., Srivastava, M.H., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. North Holland Mathematics Studies, vol. 204 (2006)

    Google Scholar 

  17. 17.

    Khamsi, M.A., Kirk, W.A.: An Introduction to Metric Spaces and Fixed Point Theory. Wiley, New York (1996)

    Google Scholar 

  18. 18.

    Mlaiki, N., Mukheimer, A., Rohen, Y., Souayah, N., Abdeljawad, T.: Fixed point theorems for αψ-contractive mapping in \(\Upsilon _{b}\)-metric spaces. J. Math. Anal. 8, 40–46 (2017)

    MathSciNet  Google Scholar 

  19. 19.

    Patel, D.K., Abdeljawad, T., Gopal, D.: Common fixed points of generalized Meir–Keeler contractions. Fixed Point Theory Appl. 2013, 260 (2013)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Souayah, N., Mlaiki, N., Mrad, M.: The \(G_{M}\)-contraction principle for mappings on M-metric spaces endowed with a graph and fixed point theorems. IEEE Access 6, 25178–25184 (2018)

    Article  Google Scholar 

  21. 21.

    Souayah, N., Mlaiki, N.: A fixed point theorem in \(\Upsilon _{b}\) metric spaces. J. Math. Comput. Sci. 16, 131–139 (2016)

    Article  Google Scholar 

  22. 22.

    Rashid, M., Bibi, R., Kalsoom, A., Baleanu, D., Ghaffar, A., Nisar, K.S.: Multidimensional fixed points in generalized distance spaces. Adv. Differ. Equ. 2020, Article ID 571 (2020)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Baleanu, D., Etemad, S., Rezapour, Sh.: On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators. Alex. Eng. J. 59(5), 3019–3027 (2020)

    Article  Google Scholar 

  24. 24.

    Kamran, T., Samreen, M., Ain, Q.U.L.: A generalization of b-metric space and some fixed point theorems. Mathematics 5, 19 (2017)

    Article  Google Scholar 

  25. 25.

    Samreen, M., Ullah, W., Karapinar, E.: Multivalued φ-contractions on extended b-metric spaces. J. Funct. Spaces 2020, Article ID 5989652 (2020)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Alghamdi, M., Gulyaz-Ozyurt, S., Karapinar, E.: A note on extended Z-contraction. Mathematics 8, 195 (2020)

    Article  Google Scholar 

  27. 27.

    Chifu, C., Karapinar, E.: On contractions via simulation function on extended b-metric spaces. Miskolc Math. Notes 21(1), 127–141 (2020)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Abdeljawad, T., Karapinar, E., Panda, S.K., Mlaiki, N.: Solutions of boundary value problems on extended-Branciari b-distance. J. Inequal. Appl. 2020, 103 (2020)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Alqahtani, B., Fulga, A., Karapinar, E., Rakocevic, V.: Contractions with rational inequalities in the extended b-metric space. J. Inequal. Appl. 2019, 220 (2019)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Alqahtani, B., Karapinar, E., Ozturk, A.: On \((\alpha ,\psi )\)K-contractions in the extended b-metric space. Filomat 32(15), 5337–5345 (2018)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Mlaiki, N., Aydi, H., Souayah, N., Abdeljawad, T.: Controlled metric type spaces and the related contraction principle. Mathematics 6, 194 (2018)

    Article  Google Scholar 

  32. 32.

    Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136, 1359–1373 (2008)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Abdeljawad, T., Mlaiki, N., Aydi, H., Souayah, N.: Double controlled metric type spaces and some fixed point results. Mathematics 6, 320 (2018)

    Article  Google Scholar 

  34. 34.

    Kyurkchiev, N., Markov, S.: On the numerical solution of the general kinetic “K-angle” reaction system. J. Math. Chem. 54(3), 792–805 (2016)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The first and third authors would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17. The fourth author thank Sefako Makgatho Health Sciences University.

Funding

This research received no external funding.

Author information

Affiliations

Authors

Contributions

All authors contributed equally in writing this article. All authors read and approved the final manuscript.

Corresponding authors

Correspondence to Thabet Abdeljawad or Hassen Aydi.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mlaiki, N., Souayah, N., Abdeljawad, T. et al. A new extension to the controlled metric type spaces endowed with a graph. Adv Differ Equ 2021, 94 (2021). https://doi.org/10.1186/s13662-021-03252-9

Download citation

MSC

  • 47H10
  • 54H25

Keywords

  • b-metric spaces
  • Extended b-metric space
  • Controlled metric type space
  • Controlled metric-like space
  • Fixed point