Double controlled quasi metric-like spaces and some topological properties of this space

1.   Introduction

Among various generalizations of concept of metric, Matthews ^[15] introduced a special kind of a partial metric space where the self-distance $d(x, x)$ is not necessarily zero. On the other hand, Amini-Harandi ^[4] redefined a dislocated metric of Hitzler and Seda ^[11] and introduced metric-like spaces. Combining these two concepts we get quasi-metric-like spaces. The study of partial metric spaces has wide area of application, especially in computer science ^[14,18]. Therefore, we can find many fixed point results in the setting of partial metric spaces ^{[3,4,5,6,8,9,10]}.

The b-metric space ^[6,7] and its partial versions, which extends the metric space by modifying the triangle equality metric axiom by inserting a constant multiple $s > 1$ to the right-hand side, is one of the most applied generalizations for metric spaces (see ^[1,12]).

Very recently, the authors in ^[13] introduced a type of extended b-metric spaces by replacing the constant $s$ by a function $\theta(x, y)$ depending on the parameters of the left-hand side of the triangle inequality.

In this paper we introduce a new type of generalized metric space, which we call as a double controlled quasi metric-Like type space. We also prove the corresponding Banach fixed point theorem on this metric space and we provide an illustrating example.

2.   Preliminary assertions

In 2017 Kamran et al. ^[13] initiated the concept of extended b-metric spaces.

Definition 2.1. ^[13] Let $\Upsilon$ be a non empty set and $\theta:\Upsilon \times \Upsilon \to [1, \infty)$. An extended b-metric is a function $\varpi:\Upsilon \times \Upsilon \to [0, \infty)$ such that for all $v, t, r \in \Upsilon$ the following conditions hold:

$(1)$ $\varpi(v, t) = 0\Leftrightarrow v = t$;

$(2)$ $\varpi(v, t) = \varpi(t, v)$;

$(3)$ $\varpi(v, t)\leq \theta(v, t)\big[\varpi(v, r)+\varpi(r, t)\big]$,

for all $v, t, r \in \Upsilon$. The pair $(\Upsilon, \varpi)$ is called an extended b-metric space.

Mlaiki et al. ^[17] generalized the notion of b-metric spaces.

Definition 2.2. ^[17] Given a nonempty set $\Upsilon$ and $\theta:\Upsilon \times \Upsilon \to [1, \infty)$. The function $\varpi:\Upsilon \times \Upsilon \to [0, \infty)$ is called a controlled metric type if

$(1)$ $\varpi(v, t) = 0\Leftrightarrow v = t$;

$(2)$ $\varpi(v, t) = \varpi(t, v)$;

$(3)$ $\varpi(v, t)\leq \theta(v, r)\varpi(v, r)+\theta(r, t)\varpi(r, t)\big]$

for all $v, t, r \in \Upsilon$. The pair $(\Upsilon, \varpi)$ is called a controlled metric type space.

Next we present the definition of double controlled metric-type spaces.

Definition 2.3. ^[2] Let there be given two non-comparable functions $\beta, \rho:\Upsilon \times \Upsilon \to [1, \infty)$. Let $\varpi:\Upsilon \times \Upsilon \to [0, \infty)$ be a function satisfying

$(1)$ $\varpi(v, t) = 0\Leftrightarrow v = t$;

$(2)$ $\varpi(v, t) = \varpi(t, v)$;

$(3)$ $\varpi(v, t)\leq \beta(v, r)\varpi(v, r)+\rho(r, t)\varpi(r, t)$,

for all $v, t, r \in \Upsilon$. Then $\varpi$ is called a double controlled metric type by $\beta$ and $\rho$ and the pair $(\Upsilon, \varpi)$ is a double controlled metric type space.

The following definition is a generalization of double controlled metric-type spaces to double controlled metric-like-type spaces, where the condition (1) is replaced by a weaker one.

Definition 2.4. ^[16] Consider a set $\Upsilon$ be a non empty set and non-comparable functions $\beta, \rho:\Upsilon \times \Upsilon \to [1, \infty)$. Suppose that a function $\varpi:\Upsilon \times \Upsilon \to [0, \infty)$ satisfies the following conditions for all $v, t, r \in \Upsilon$:

$(1)$ $\varpi(v, t) = 0\Rightarrow v = t$;

$(2)$ $\varpi(v, t) = \varpi(t, v)$;

$(3)$ $\varpi(v, t)\leq \beta(v, r)\varpi(v, r)+\rho(r, t)\varpi(r, t)\big]$.

Then the pair $(\Upsilon, \varpi)$ is called a double controlled metric-like space.

3.   Double controlled quasi metric-Like spaces and some topological properties of this space

In this section we present our generalization of the double controlled quasi metric-like-type spaces. This concept is extension of the double controlled metric-like spaces, so called double controlled quasi metric-like spaces "assuming that the self-distance may not be zero".

Definition 3.1. Let $\Upsilon$ be a non empty set and consider non-comparable functions

Suppose that a function $\varpi:\Upsilon \times \Upsilon \to [0, \infty)$, for all $v, t, r \in \Upsilon$, satisfies the following conditions:

$(\varpi_{1})$ $\varpi(v, t) = \varpi(t, v) = 0\Rightarrow v = t$;

$(\varpi_{2})$ $\varpi(v, t)\leq \varpi(v, r)\beta(v, r)+\varpi(r, t)\rho(r, t)$.

Then the pair $(\Upsilon, \varpi)$ is called a double controlled quasi metric-like space or shortly ($DCQMLS$).

Definition 3.2. Let $(\Upsilon, \varpi)$ be a $DCQMLS$ and $(v_{n})$ be a sequence in $\Upsilon$. Then we say

$\bf(i)$ $(v_{n})$ converges to $v \in \Upsilon$ if and only if

In this case $v$ is called a double controlled quasi like-limit or shortly ($\varpi$-limit) of $(v_{n})$, and we write $\lim\limits_{n\longrightarrow +\infty}v_{n} = v$.

$\bf(ii)$ A sequence $(v_{n})$ is a $\varpi$-Cauchy sequence if both $\lim\limits_{n, m\longrightarrow +\infty}\varpi(v_{n}, v_{m})$ and $\lim\limits_{n, m\longrightarrow +\infty} \varpi (v_{m}, v_{n})$ exist and are finite.

$\bf(iii)$ $(\Upsilon, \varpi)$ is $\varpi$-complete if for any $\varpi$-Cauchy sequence $(v_{n})$, there exists some $v \in \Upsilon$ such that

$\bf(iv)$ The mapping $\Xi : \Upsilon \rightarrow \Upsilon$ is said to be continuous at $v\in \Upsilon$ if for any sequence $(v_{n})$ converging to $v$, we have $\lim\limits_{n\longrightarrow +\infty} \Xi v_{n} = \Xi v$, that is,

Remark 3.1.

$\bf(i)$ Topology of ($DCQMLS$) is not necessarily a Hausdorff topology, so the limit of convergent sequence is not always unique.

$\bf(ii)$ There are convergent sequences in ($DCQMLS$) that are not Cauchy sequences.

Example 3.1. Let $\Upsilon = \{0, 1, 2\}$ and $\varpi: \Upsilon \times \Upsilon \rightarrow [0, +\infty)$ defined with

Consider the following $\beta, \rho:\Upsilon \times \Upsilon \to [1, \infty)$:

and

Thus, $(\Upsilon, \varpi)$ is a ($DCQMLS$).

The constant sequence $(v_{n} = 1)_{n\in\mathbb{N}}$ is convergent with both 1 and 2 as limits since

Consider the sequence $t_{2n} = 1, t_{2n-1} = 0, n \in \mathbb{N}$. Obviously, $(t_{n})$ is not a Cauchy sequence, but

implying that $\lim\limits_{n\rightarrow +\infty}t_{n} = 2$.

Definition 3.3. Let $(\Upsilon, \varpi)$ be a ($DCQMLS$) with $\varepsilon > 0$, $v_{0} \in \Upsilon$. The set $\delta(v_{0}, \varepsilon) = \lbrace v/v \in \Upsilon, \max \big(\varpi(v_{0}, v), \varpi(v, v_{0})\big) < \varepsilon \rbrace$ is called $\varpi$-open ball of radius $\varepsilon$, center $v_{0}$ and $B_{\varepsilon}(v_{0}) = \lbrace v_{0}\rbrace \cup \delta(v_{0}, \varepsilon)$. The set $\overline{\delta}(v_{0}, \varepsilon) = \lbrace v/v \in \Upsilon, \max \big(\varpi(v_{0}, v), \varpi(v, v_{0})\big) \leq\varepsilon \rbrace$ is called $\varpi$-closed ball of radius $\varepsilon$, center $v_{0}$ and $\overline{B}_{\varepsilon}(v_{0}) = \lbrace v_{0}\rbrace \cup \overline{\delta}(v_{0}, \varepsilon)$.

4.   Main results

Theorem 4.1. Let $(\Upsilon, \varpi)$ be a complete ($DCQMLS$) defined by functions $\beta, \rho:\Upsilon \times \Upsilon \to [1, \infty)$. Let $\Xi : \Upsilon \rightarrow \Upsilon$ be a mapping such that

for all $v, t \in \Upsilon$, where $h \in (0, 1)$. For $v_{0} \in \Upsilon$, take $v_{n} = \Xi^{n}v_{0}$. Suppose that

Also assume that, for every $v\in \Upsilon$, we have

Then $\Xi$ has a unique fixed point.

Proof. Let ${v_{n} = \Xi^{n}v_{0}}$ in $\Upsilon$ be a sequence that satisfies the conditions of our theorem. By using (4.1) we get

Let $n, m \in \mathbb{N}$ be such that $n < m$. Then

Note that we are using the fact that $\beta(v, t) \geq 1$ and $\rho(v, t) \geq 1$. Let

Then we have

By condition (4.2), using the ratio test, we see that $\lim\limits_{n\rightarrow +\infty}\Phi_{n}$ exists, and hence the real sequence $(\Phi_{n})$ a Cauchy sequence. Finally, if we take the limit in inequality (4.5) as $n, m\rightarrow +\infty$, we deduce that

Similarly proceeding we have

Hence the sequence $(v_{n})$ is $\varpi$-Cauchy in $(\Upsilon, \varpi)$, which is a complete ($DCQMLS$), so $(v_{n})$ converges to some $v^{\ast}\in\Upsilon$, that is,

Then $\varpi(v^{\ast}, v^{\ast}) = 0$. Next, we show that $\Xi v^{\ast} = v^{\ast}$. By the triangle inequality and (4.1) we have

and

Taking the limit as $n\rightarrow +\infty$, by (4.3) and (4.7) we deduce that $\varpi(v^{\ast}, \Xi v^{\ast}) = \varpi(\Xi v^{\ast}, v^{\ast}) = 0$, that is, $\Xi v^{\ast} = v^{\ast}$. Finally, assume that $\Xi$ has two fixed points, say $\nu$ and $\xi$. Then

which leads us to a contradiction. Therefore $\varpi(\nu, \xi) = \varpi(\xi, \nu) = 0$, so $\nu = \xi$. Hence $\Xi$ has a unique fixed point.

Remark 4.1. Note that condition (4.3) in Theorem 4.1 can be changed by the assumption that $\Xi$ and the ($DCQMLS$) $\varpi$ are continuous. To see this, the continuity gives us that if $v_{n} \rightarrow v^{\ast}$, then $\Xi v_{n} \rightarrow \Xi v^{\ast}$, and hence we have

then

Next we show that $\varpi(\Xi v^{\ast}, \Xi v^{\ast}) = 0$. In fact by (4.1) we have

Taking the limit as $n\rightarrow +\infty$, by (4.3), (4.7) and (4.8) we deduce that $\varpi(\Xi v^{\ast}, \Xi v^{\ast}) = \varpi(v^{\ast}, \Xi v^{\ast}) = \varpi(\Xi v^{\ast}, v^{\ast}) = 0$, so $\Xi v^{\ast} = v^{\ast}$.

Definition 4.1. Let $\Xi: \Upsilon\rightarrow \Upsilon$. For some $v_{0}\in \Upsilon$, consider $O(v_{0}) = \{v_{0}, \Xi v_{0}, \Xi^{2}v_{0}, \cdots\}$ to be the orbit of $v_{0}$. We say that a function $\varphi$ is $\Xi$- orbitally lower semicontinuous at $u\in\Upsilon$ if for $(v_{n}) \subset O(v_{0})$ such that $v_{n}\rightarrow u$, we have $\varphi(u) \leq \lim\limits_{n\rightarrow +\infty} \inf \varphi(v_{n})$.

Corollary 4.1. Let $(\Upsilon, \varpi)$ be a complete ($DCQMLS$) defined by functions $\beta, \rho:\Upsilon \times \Upsilon \to [1, \infty)$. Let $\Xi : \Upsilon \rightarrow \Upsilon$. Let $v_{0} \in \Upsilon$ and $0 < h < 1$ be such that

Take $v_{n} = \Xi^{n}v_{0}$. Suppose that

Then $\lim\limits_{n\rightarrow +\infty} v_{n} = u \in \Upsilon$. Moreover, $\Xi u = u\Leftrightarrow u \mapsto \varpi(u, \Xi u)$ is $\Xi$- orbitally lower semicontinuous at $u$.

Next, we present the nonlinear case.

Theorem 4.2. Let $(\Upsilon, \varpi)$ be a complete ($DCQMLS$) defined by functions $\beta, \rho:\Upsilon \times \Upsilon \to [1, \infty)$ and assume that there exists a nondecreasing and continuous function $\psi: \mathbb{R_{+}}\rightarrow \mathbb{R_{+}}$ such that

and

for all $v, t \in \Upsilon$. Moreover, assume that for each $v_{0}\in\Upsilon$, we have

where $v_{n} = \Xi^{n}v_{0}, n \in \mathbb{N}$. If the ($DCQMLS$) $\varpi$ and $\Xi$ are continuous, then $\Xi$ admits a unique fixed point $v^{\ast} \in\Upsilon$ with $\Xi^{n}v\rightarrow v^{\ast}$ for each $v \in \Upsilon$.

Proof. Assume that there exists $k\in \mathbb{ N}$ such that $v_{k} = v_{k+1} = \Xi v_{k}$, which implies that $v_{k}$ is a fixed point. So we may assume that $v_{n+1}\neq v_{n}$ for each $n$. From condition (4.11) we have

where $\Theta(v_{n-1}, v_{n}) = \max\{\varpi(v_{n-1}, v_{n}), \varpi(v_{n}, v_{n+1})\}$. If for some $n$ we accept that

$\Theta (v_{n-1}, v_{n}) = \varpi(v_{n}, v_{n+1})$, then by (4.13) and the assumption $\psi(t) < t$ for all $t > 0$, we deduce that

which is a contradiction. Thus, for all $n \in \mathbb{N}$, we obtain $\Theta (v_{n-1}, v_{n}) = \varpi(v_{n-1}, v_{n})$. It follows that $0 < \varpi(v_{n}, v_{n+1}) \leq \psi\big(\varpi(v_{n-1}, v_{n})\big)$. By using induction we easily see that for all $n \geq 0$,

By the properties of $\psi$ we can easily deduce that

Using the argument in the proof of Theorem 4.1, for $n, m \in\mathbb{ N}$ such that $n < m$, we can easily deduce that

and

By condition (4.12), using the ratio test, we can easily deduce that the sequence $(v_{n})$ is $\varpi$-Cauchy. Since $(\Upsilon, \varpi)$ is a complete ($DCQMLS$), if $v_{n} \rightarrow r$ as $n \rightarrow +\infty$, then $\lim\limits_{n\rightarrow +\infty} \varpi(v_{n}, r) = \lim\limits_{n\rightarrow +\infty} \varpi(r, v_{n}) = 0$. Hence by Remark 4.1 we conclude that $\Xi r = r$. Finally, assume that $r$ and $t$ are two fixed points of $\Xi$ such that $r\neq t$. From assumption (4.11) we have

and

which leads to a contradiction. Therefore $r = t$, as desired.

Remark 4.2. Note that if $\psi (v) = \alpha v, \; 0 < \alpha < 1$, then condition (4.11) in Theorem 4.2 becomes

Next, we prove the following result for mappings satisfying Kannan-type contraction.

Theorem 4.3. Let $(\Upsilon, \varpi)$ be a complete ($DCQMLS$) defined by functions $\beta, \rho:\Upsilon \times \Upsilon \to [1, \infty)$. Let $\Xi: \Upsilon\rightarrow \Upsilon$ be a Kannan mapping defined as follows:

for $v, t \in \Upsilon$, where $\delta \in (0, \frac{1}{2})$. For $v_{0}\in \Upsilon$, take $v_{n} = \Xi^{n}v_{0}$. Suppose that

Also, assume that for every $v\in\Upsilon$, we have

Then $\Xi$ has a fixed point. Moreover, if for every fixed point $z$, we have $\varpi(z, z) = 0$, then the fixed point is unique.

Proof. Consider the sequence $(v_{n} = \Xi v_{n-1})$ in $\Upsilon$ satisfying hypotheses (4.18) and (4.19). From (4.17) we obtain

Then $\varpi(v_{n}, v_{n+1}) \leq \frac{\delta}{1-\delta}\varpi(v_{n-1}, v_{n})$. By induction we get

Next we show that $(v_{n})$ is a $\varpi$-Cauchy sequence. For two natural numbers $n < m$, we have

Similarly to the proof of Theorem 4.1, we get

Similarly proceeding we have

Since $0\leq\delta < \frac{1}{2}$, we have $0 < \frac{\delta}{1-\delta} < 1$, and similarly to the argument in the proof of Theorem 4.1, we obtain that $(v_{n})$ is a $\varpi$-Cauchy sequence in the complete ($DCQMLS$) $(\Upsilon, \varpi)$. Thus $(v_{n})$ converges to some $z\in\Upsilon$. Suppose that $\Xi z\neq z$. Then

and

Taking the limit in both sides of these inequalities and using (4.19), we deduce that $0 < \varpi(z, \Xi z) < \varpi(z, \Xi z)$ and $0 < \varpi(\Xi z, z) < \varpi(\Xi z, z)$, which is a contradiction. Hence $\Xi z = z$. Now assume that for every fixed point $w$, we have $\varpi(z, z) = 0$ and suppose that $\Xi$ has more than one fixed point, say $z$ and $\eta$. Then

and

Thereby $z = \eta$, as required.

Remark 4.3. It will be interesting to find more applications to our current paper in other fields see ^{[19,20,21,22,23]}.

Acknowledgments

The first author (A. M. Zidan) extends his appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia for funding this work through research groups program under grant number R.G.P-2/142/42.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.