Periodic and fixed points for $F$-type contractions in $b$-gauge spaces

1.   Introduction

For a nonempty set $U$, let $T:U\rightarrow U$ be a single valued map. The set of its fixed points is denoted by Fix$(T)$ and is defined by Fix$(T) = \{u\in U:u = T(u)\}$. The set of all periodic points of $T$ denoted by Per$(T)$ is defined by Per$(T) = \{u\in U:u = T^{[k]}(u) \text{ for some $k$ in $\mathbb{N}$ } \}$, where T$^{[k]}$ = $T\circ T\circ T\circ\cdots\circ T$ ($k$-times). Also, for each $z^{0}\in U$, a sequence $\left(z^{m} : m\in \{0\}\cup \mathbb{N}\right)$ starting at $z^{0}$ such that $z^{m} = T^{[m]}(z^{0})$, for all $m\in \{0\}\cup \mathbb{N}$ is called a Picard iteration.

The metric fixed point theory is originated from the concept of Picard successive approximations by Picard (one result in this direction can be found in ^[1]). The famous mathematician Banach placed the underlying idea into an abstract framework, hence presented his most eminent research as the Banach contraction principle. Since then, numerous researchers expanded the Banach contraction principle in different directions by generalizing the metric spaces and the contraction conditions for single valued as well as for multi valued maps. For instance, see ^[2,3,4]. In general, fixed point theory remained successful in solving various problems and has participated significantly to many real world problems, such as optimization theory ^[5], image processing ^[6] and game theory ^[7,8].

A useful generalization of Banach contraction is an $F$-contraction presented by Wardowski ^[30]. Due to the simplicity and effectiveness of $F$-contractions numerous research papers have been published in this direction which can be seen in ^{[9,10,11,12,13,14,15,16]}.

In 1966, Dugundji ^[19] initiated the idea of gauge spaces which generalizes metric spaces (or more generally, pseudo-metric spaces). Gauge spaces have the characteristic that even the distance between two distinct points of the space may be zero. This simple characterization has been the center of interest for many researchers world wide. For further facts on gauge spaces, we recommend the readers to Agarwal et al. ^[22], Frigon ^[20], Chis and Precup ^[21], Chifu and Petrusel ^[23], Lazara and Petrusel ^[26], Cherichi et al. ^[24,25], Jleli et al. ^[27] and Branga ^[28].

In 2013, Wlodarczyk and Plebaniak ^[31] have given the notion of left (right) $\mathcal{J}$-families of generalized quasi-pseudo distances in quasi-gauge spaces that generalizes the structure of a quasi gauge and provides powerful and useful tools to obtain more general results with weaker assumptions.

The aim of this paper is to introduce $\mathcal{J}_{s; \Omega}$-families of generalized pseudo-$b$-distances in $b$-gauge spaces $(U, {Q}_{s; \Omega})$. Moreover, by using these $\mathcal{J}_{s; \Omega}$-families on $U$, we define the $\mathcal{J}_{s; \Omega}$-sequential completeness and construct $F$-type contractions $T:U\rightarrow U$. Furthermore, we develop novel periodic and fixed point results for these mappings in the setting of $b$-gauge spaces using $\mathcal{J}_{s; \Omega}$-families on $U$. The obtained results generalize and improve the existing ones in the literature of fixed point theory. The validity and importance of our theorems are shown through an application by ensuring the existence of a solution of an integral type equation.

2.   Preliminaries

The core reason behind to add this section is to recollect some essential concepts and results which are valuable throughout this paper.

In view of generalizing the concept of Banach contraction, Wardowski ^[30] succeeded to generalize Banach contraction condition by introducing the notions of $F$-contractions. He introduced the family $\mathfrak{F}$ of all functions $F: (0, \infty) \rightarrow \mathbb{R}$ which satisfies the following three conditions:

$(F_{1}) \; F$ is a strictly increasing function, i.e., for any $a, b\in (0, \infty)$ with $a < b$ we have $F(a) < F(b)$.

$(F_{2})$ For any sequence $(b_{n}:n\in \mathbb{N})$ in $\mathbb{R_{+}}$, we have $\lim_{n\rightarrow \infty}b_{n} = 0$ iff $\lim_{n\rightarrow \infty}F(b_{n}) = -\infty$.

$(F_{3})$ There exists $p\in (0, 1)$ such that $\lim_{b\rightarrow 0^{+}}b^{p}F(b) = 0$.

According to Wardowski ^[30], a mapping $T:U \rightarrow U$ on the metric space $(U, d)$ is called an $F$-contraction if there exists $\tau > 0$, such that

The implication (2.1) covers various types of contractions. For instance, the case $F_{y} = \ln y$ corresponds to the Banach contraction.

Wardowski ^[30] stated a fixed point theorem involving $F$-contraction mappings.

Theorem 2.1. Let $(U, d)$ be a complete metric space and let $T:U \rightarrow U$ be an $F$-contraction mapping. Then $T$ has a unique fixed point $z\in U$ and for any $z^{0}\in U$, the sequence $(z^{m} = T^{[m]}(z^{0}):m\in \mathbb{N})$ converges to the fixed point $z$.

Minak et al. ^[15] generalized the above result in the following way:

Theorem 2.2. Let $(U, d)$ be a complete metric space and let $T:U \rightarrow U$ be an $F$-contraction such that

for all $u, v\in U$, with $d(Tu, Tv) > 0$. Then $T$ has a unique fixed point, whenever $T$ or $F$ is continuous.

A generalized $F$-contraction of Hardy-Rogers-type is as follows:

Theorem 2.3. Let $(U, d)$ be a complete metric space and let $T:U \rightarrow U$ be a generalized $F$-contraction of Hardy-Rogers-type, i.e., there exist $\tau > 0$ and $F\in \mathfrak{F}$ such that

for all $u, v\in U$, with $d(Tu, Tv) > 0$ where $a, b, c, d, L\geq 0$, $c\neq 1$, then $T$ has a fixed point. Further, if $a+d+L\leq 1$, such a fixed is unique.

Cosentino et al. ^[14] introduced the family $\mathfrak{F_{s}}$ of all functions $F: (0, \infty) \rightarrow \mathbb{R}$ satisfying the following four conditions:

$(F_{1}) \; F$ is a strictly increasing function, i.e., for any $a, b\in (0, \infty)$ with $a < b$ we have $F(a) < F(b)$.

$(F_{2})$ For any sequence $(b_{n}:n\in \mathbb{N})$ in $\mathbb{R_{+}}$, we have $\lim_{n\rightarrow \infty}b_{n} = 0$ iff $\lim_{n\rightarrow \infty}F(b_{n}) = -\infty$.

$(F_{3})$ For any sequence $(b_{n}:n\in \mathbb{N})$ in $\mathbb{R_{+}}$, we have $\lim_{n\rightarrow \infty}b_{n} = 0$, there exists $p\in (0, 1)$ such that $\lim_{n\rightarrow \infty} b^{p}_{n}F(b_{n}) = 0$.

$(F_{4})$ For any sequence $(b_{n}:n\in \mathbb{N})$ in $\mathbb{R_{+}}$ such that $\tau+F(sb_{n})\leq F(b_{n-1})$ for each $n\in \mathbb{N}$ and some $\tau > 0$, then $\tau+F(s^{n}b_{n})\leq F(s^{n-1}b_{n-1})$.

Some examples of functions belonging to $\mathfrak{F_{s}}$ are given below:

(i) $F_{x} = x+\ln x$ for any $x\in (0, \infty)$.

(ii) $F_{y} = \ln y$ for any $y\in (0, \infty)$.

Recently, Ali et al. ^[18] introduced the notion of $b$-gauge spaces, thus extended the idea of gauge spaces in the setting of $b$-metric spaces. We note down the following definitions of their work.

Definition 2.4. A map $q: U\times U\rightarrow [0, \infty)$ is called as a $b$-pseudo metric, if for all $x, y, z\in U$, there exists $s \geq 1$ satisfying the following conditions:

(a) $q(x, x) = 0$;

(b) $q(x, y) = q(y, x)$;

(c) $q(x, z)\leq s\{q(x, y)+q(y, z)\}$.

The pair $\left(U, q\right)$ is called a $b$-pseudo metric space.

Definition 2.5. Each family ${Q}_{s; \Omega} = \{q_{\beta}: \beta\in \Omega\}$ of $b$-pseudo metrics $q_{\beta}: U\times U\rightarrow [0, \infty)$, is called as a $b$-gauge on $U$.

Definition 2.6. The family ${Q}_{s; \Omega} = \{q_{\beta}: \beta\in \Omega\}$ is called to be separating if for every pair $(x, y)$ where $x\neq y$, there exists $q_{\beta}\in {Q}_{s; \Omega}$ such that $q_{\beta}(x, y) > 0$.

Definition 2.7. Let the family ${Q}_{s; \Omega} = \{q_{\beta}: \beta\in \Omega\}$ be a $b$-gauge on $U$. The topology $\mathcal{T}({Q}_{s; \Omega})$ on $U$ whose subbase is defined by the family $\mathcal{B}({Q}_{s; \Omega}) = \{B(x, \epsilon_{\beta}):x\in U, \epsilon_{\beta} > 0, \beta \in \Omega\}$ of all balls $B(x, \varepsilon_{\beta}) = \{y\in U: q_{\beta}(x, y) < \epsilon_{\beta}\}$, is called the topology induced by ${Q}_{s; \Omega}$. The pair $(U, \mathcal{T}({Q}_{s; \Omega}))$ is called to be a $b$-gauge space and is Hausdorff if ${Q}_{s; \Omega}$ is separating.

The following examples shows that a $b$-pseudo metric space (in fact, a $b$-gauge space) is the generalization of a metric space, a pseudo metric space (in fact, a gauge space) and a $b$-metric space.

Example 2.8. ^[18] Let $U = C([0, \infty), \mathbb{R})$. Describe $q:U\times U \rightarrow [0, \infty)$ by

Then $q$ is a $b$-pseudo metric, but neither a metric, nor a pseudo metric, nor a $b$-metric.

In this regard, consider $x, y, z\in U$ defined by

$y(t) = 3$ for each $t\geq 0$ and $z(t) = -3$ for each $t\geq 0$. We note that $d(y, z) = 36 \nleq 18 = d(y, x)+d(x, z)$.

Therefore, $q$ is neither a metric, nor a pseudo metric on $U$. Also, if $u, v\in U$ are defined by

and

then $u\neq v$, but $q(u, v) = 0$. Therefore, $q$ is not a $b$-metric on $U$.

Example 2.9. ^[18] Let $U = C([0, \infty), \mathbb{R})$. Define the family of $b$-pseudo metrics as $q_{m}(x(t), y(t)) = \sup_{t\in [0, m]}|x(t)-y(t)|^{2}$, $m\in \mathbb{N}$. Obviously, ${Q}_{s; \Omega} = \{q_{m}:m\in \mathbb{N}\}$ defines a b-gauge on $U$. Thus $(U, {Q}_{s; \Omega})$ is a $b$-gauge space.

Note that $(U, {Q}_{s; \Omega})$ is not a gauge space, and hence it is not a metric space (as explained in Example 2.8).

3.   Main results

In this section, we introduce $\mathcal{J}_{s; \Omega}$-families of generalized pseudo-$b$-distances in the $b$-gauge space $(U, {Q}_{s; \Omega})$. The new structure determined by these families of distances is a generalization of $b$-gauges and gives valuable and important tools for inquiring periodic points and fixed points of maps in $b$-gauge spaces. Moreover, by using these $\mathcal{J}_{s; \Omega}$-families on $U$, we define the $\mathcal{J}_{s; \Omega}$-sequential completeness which generalizes the usual ${Q}_{s; \Omega}$-sequential completeness. Furthermore, we develop novel periodic and fixed point results for $F$-type contractions in the setting of $b$-gauge spaces using $\mathcal{J}_{s; \Omega}$-families on $U$, which generalize and improve all the results in ^[17] and some of the results in ^[29].

We now introduce the notion of $\mathcal{J}_{s; \Omega}$-family of generalized pseudo-$b$-distances on $U$ ($\mathcal{J}_{s; \Omega}$-family on $U$ is the generalization of $b$-gauges).

Definition 3.1. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space. The family $\mathcal{J}_{s; \Omega} = \{{{{J}}_{\beta}}: \beta \in \Omega\}$ where ${J}_{\beta} :U\times U\rightarrow [0, \infty), \ \beta \in \Omega$, is said to be the $\mathcal{J}_{s; \Omega}$-family of generalized pseudo-$b$-distances on $U$ (for short, $\mathcal{J}_{s; \Omega}$-family on $U$) if the following statements hold for all $\beta \in \Omega$ and for all $u, v, w\in U$:

$(\mathcal{J}1) \; {J}_{\beta}(u, w)\leq s_{\beta}\{{J}_{\beta}(u, v)+{J}_{\beta}(v, w)\}$; and

$(\mathcal{J}2)$ for each sequences $\left(u_{m} :m\in \mathbb{N}\right)$ and $\left(v_{m} :m\in N\right)$ in $U$ fulfilling

and

the following holds:

Take

Also, we denote

and

Then, of course $U = U^{0}_{\mathcal{J}_{s; \Omega}}\cup U^{+}_{\mathcal{J}_{s; \Omega}}$.

Example 3.2. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space, where $U$ contains at least two distinct elements and suppose ${Q}_{s; \Omega} = \{q_{\beta}:\beta \in \Omega\}$ the family of pseudo-$b$-metrics is a $b$-gauge on $U$.

Let the set $F \subset U$ contain at least two distinct elements, but arbitrary and fixed. Let $d_{\beta}\in (0, \infty)$ satisfy $\delta_{\beta}(F) < d_{\beta}$, where $\delta_{\beta}(F) = \sup \{q_{\beta}(e, f): e, f \in F\}$, for all $\beta\in \Omega$. Let ${J}_{\beta} :U\times U\rightarrow [0, \infty)$ for all $e, f\in U$ and for all $\beta \in \Omega$ be defined by

Then $\mathcal{J}_{s; \Omega} = \{{J}_{\beta}:\beta\in \Omega\}\in \mathbb{J}_{(U, {Q})}$.

We observe that ${J}_{\beta}(e, g)\leq s_{\beta}\{{J}_{\beta}(e, f)+{J}_{\beta}(f, g)\}$, for all $e, f, g\in U$, thus condition $(\mathcal{J}_{1})$ holds. Indeed, condition $(\mathcal{J}_{1})$ will not hold in case if there exist some $e, f, g\in U$ such that ${J}_{\beta}(e, g) = d_{\beta}$, ${J}_{\beta}(e, f) = q_{\beta}(e, f)$, ${J}_{\beta}(f, g) = q_{\beta}(f, g)$ and $s_{\beta}\{q_{\beta}(e, f)+q_{\beta}(f, g)\}\leq d_{\beta}$. However, this implies the existence of $h\in \{e, g\}$ such that $h\notin F$ and on other hand, $e, f, g\in F$, which is impossible.

Now, suppose that (3.1) and (3.2) are satisfied by the sequences $(u_{m}: m\in \mathbb{N})$ and $(v_{m}:m\in \mathbb{N})$ in $U$. Then (3.2) yields that for all $0 < \epsilon < d_{\beta}$, for all $\beta\in \Omega$, there exists $m_{1} = m_{1}(\beta)\in \mathbb{N}$ such that

By (3.5) and (3.4), denoting $m_{2} = \min\{m_{1}(\beta):\beta \in \Omega\}$, we have

and

Thus, (3.3) is satisfied by the sequences $(u_{m}: m\in \mathbb{N})$ and $(v_{m}:m\in \mathbb{N})$. Therefore, $\mathcal{J}_{s; \Omega}$ is a $\mathcal{J}_{s; \Omega}$-family on $U$.

Example 3.3. Let $U = [0, 1]$ and $B = \{\frac{1}{2^{m}}:m\in \mathbb{N}\}$.

Let ${Q}_{s; \Omega} = \{q\}$, where $q:U\times U\rightarrow [0, \infty)$ is a pseudo-$b$-metric on $U$ defined for all $x, y\in U$ by

Then $(U, {Q}_{s; \Omega})$ is a $b$-gauge space.

Let the set $F = [\frac{1}{8}, 1] \subset U$ and let ${J}:U\times U\rightarrow [0, \infty)$ for all $x, y\in U$ be defined by

Then $\mathcal{J}_{s; \Omega} = \{{J}\}$ is a $\mathcal{J}_{s; \Omega}$-family on $U$ (see Example 3.2).

We mention here some trivial properties of $\mathcal{J}_{s; \Omega}$-families in the following remark.

Remark 3.4. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space. Then the following hold:

(a) ${Q}_{s; \Omega}\in \mathbb{J}_{(U, {Q}_{s; \Omega})}$.

(b) Let $\mathcal{J}_{s; \Omega} \in \mathbb{J}_{(U, {Q}_{s; \Omega})}$. If ${J}_{\beta}(v, v) = 0$ and ${J}_{\beta}(u, v) = {J}_{\beta}(v, u)$ for all $\beta\in \Omega$ and for all $u, v\in U$ then for each $\beta \in \Omega$, ${J}_{\beta}$ is a pseudo-$b$ metric.

(c) There exist examples of $\mathcal{J}_{s; \Omega} \in \mathbb{J}_{(U, {Q}_{s; \Omega})}$ which show that the maps ${J}_{\beta}$, $\beta \in \Omega$ are not pseudo-$b$ metrics.

Proposition 3.5. Let $(U, {Q}_{s; \Omega})$ be a Hausdorff $b$-gauge space and $\mathcal{J}_{s; \Omega} = \{{{{J}}_{\beta}}: \beta \in \Omega\}$ be the $\mathcal{J}_{s; \Omega}$-family of generalized pseudo-$b$-distances on $U$. Then for each $e, f\in U$, there exists $\beta \in \Omega$ such that

Proof. Let there be $e, f\in U$ where $e\neq f$ such that ${J}_{\beta}(e, f) = 0 = {J}_{\beta}(f, e)$ for all $\beta\in \Omega$. Then by using property $(\mathcal{J}1)$, we have ${J}_{\beta}(e, e) = 0$, for all $\beta\in \Omega$.

Defining sequences $\left(u_{m} :m\in \mathbb{N}\right)$ and $\left(v_{m} :m\in N\right)$ in $U$ by $u_{m} = f$ and $v_{m} = e$, we see that conditions (3.1) and (3.2) of property $(\mathcal{J}2)$ are satisfied, and therefore condition (3.3) holds, which implies that $q_{\beta}(e, f) = 0$, for all $e, f\in U$ and for all $\beta\in \Omega$. It is a contradiction to the fact that $(U, {Q}_{s; \Omega})$ is a Hausdorff $b$-gauge space. Therefore, our supposition is wrong and there exists $\beta \in \Omega$ such that for all $e, f\in U$

Now, using $\mathcal{J}_{s; \Omega}$-families on $U$, we establish the following concept of $\mathcal{J}_{s; \Omega}$-completeness in the $b$-gauge space $(U, {Q}_{s; \Omega})$ which generalizes the usual ${Q}_{s; \Omega}$-sequential completeness.

Definition 3.6. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space. Let $\mathcal{J}_{s; \Omega} = \{{{{J}}_{\beta}}: \beta \in \Omega\}$ be the $\mathcal{J}_{s; \Omega}$-family on $U$. A sequence $(u_{m}: m\in \mathbb{N})$ is a $\mathcal{J}_{s; \Omega}$-Cauchy sequence in $U$ if

Definition 3.7. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space. Let $\mathcal{J}_{s; \Omega} = \{{{{J}}_{\beta}}: \beta \in \Omega\}$ be the $\mathcal{J}_{s; \Omega}$-family on $U$. The sequence $(u_{m}: m\in \mathbb{N})$ is called to be $\mathcal{J}_{s; \Omega}$-convergent to $u\in U$ if $\lim_{m\rightarrow \infty}^{\mathcal{J}_{s; \Omega}}u_{m} = u$, where

Definition 3.8. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space. Let $\mathcal{J}_{s; \Omega} = \{{{{J}}_{\beta}}: \beta \in \Omega\}$ be the $\mathcal{J}_{s; \Omega}$-family on $U$. If $S_{({u_{m}: m\in \mathbb{N}})}^{\mathcal{J}_{s; \Omega}}\neq \emptyset$, where

Then the sequence $(u_{m}: m\in \mathbb{N})$ in $U$ is $\mathcal{J}_{s; \Omega}$-convergent in $U$.

Definition 3.9. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space. Let $\mathcal{J}_{s; \Omega} = \{{{{J}}_{\beta}}: \beta \in \Omega\}$ be the $\mathcal{J}_{s; \Omega}$-family on $U$. The space $(U, {Q}_{s; \Omega})$ is called $\mathcal{J}_{s; \Omega}$-sequentially complete, if every $\mathcal{J}_{s; \Omega}$-Cauchy in $U$ is a $\mathcal{J}_{s; \Omega}$-convergent in $U$.

Example 3.10. Let $U$, ${Q}_{s; \Omega} = \{q\}$, $F$ and $\mathcal{J}_{s; \Omega} = \{{J}\}$ be as in Example 3.3.

First, we show that $(U, {Q}_{s; \Omega})$ is not ${Q}_{s; \Omega}$-sequential complete.

For this, let $\{v_{m}\} = \{\frac{1}{2^{m}}:m\in \mathbb{N}\}$, then by (3.6) for all $\epsilon > 0$ and for all $n, m\in \mathbb{N}$, there exists $k_{0}\in \mathbb{N}$ such that

Thus, $\{v_{m}:m\in \mathbb{N}\}$ is a ${Q}_{s; \Omega}$-Cauchy sequence. However, this sequence is not ${Q}_{s; \Omega}$-convergent in $U$. Otherwise, suppose that $\lim_{m\rightarrow \infty}v_{m} = v$, for some $v\in U$. We may suppose without loosing generality that for all $0 < \epsilon < 1$, there exists $k_{0}\in \mathbb{N}$ such that

We have the following two cases:

(i) If $v\notin B$, then using (3.6) we can write

It is not possible.

(ii) If $v\in B$, then let $v = \frac{1}{2^{m_{1}}}$, for some $m_{1}\in \mathbb{N}$ and using (3.6), we can write

Taking limit interior as $m\rightarrow \infty$, we get

By (3.8), it is impossible.

Thus, we conclude that $(U, {Q}_{s; \Omega})$ is not ${Q}_{s; \Omega}$-sequential complete. Next, we show that $(U, {Q}_{s; \Omega})$ is $\mathcal{J}_{s; \Omega}$-sequential complete.

Let $\{v_{m}:m\in \mathbb{N}\}$ be a $\mathcal{J}_{s; \Omega}$-Cauchy sequence. Without loosing generality, we may assume that for $0 < \epsilon < 1$, there exists $k_{0}\in \mathbb{N}$ such that

Then by (3.7), (3.6) and (3.9), we obtain

and

From (3.12), we have two cases:

(i) If $v_{m} = v_{m_{0}}$ for all $m_{0}\geq k_{0}$, then $\{v_{m}:m\in \mathbb{N}\}$ represents a constant sequence and by (3.11), (3.7), (3.6) and (3.12) the sequence $\{v_{m}:m\in \mathbb{N}\}$ is $\mathcal{J}_{s; \Omega}$-convergent to $v_{m_{0}}$.

(ii) If $v_{m}\in B$, for all $m_{0}\geq k_{0}$, let $v_{k_{0}+s}\in B$ for all $s\in \mathbb{N}$. This together with (3.10)–(3.12) imply that $v_{k_{0}+s} = \frac{1}{2}$ for all $s\in \mathbb{N}$ or $v_{k_{0}+s} = \frac{1}{4}$ for all $s\in \mathbb{N}$ or $v_{k_{0}+s} = \frac{1}{8}$ for all $s\in \mathbb{N}$. Therefore, the sequence $\{v_{m}:m\in \mathbb{N}\}$ is $\mathcal{J}_{s; \Omega}$-convergent to the point $\frac{1}{2}$ or $\frac{1}{4}$ or $\frac{1}{8}$, respectively.

Thus, we conclude that $(U, {Q}_{s; \Omega})$ is $\mathcal{J}_{s; \Omega}$-sequential complete.

Remark 3.11. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space and let $\mathcal{J}_{s; \Omega} = \{{{{J}}_{\beta}}: \beta \in \Omega\}$ be the $\mathcal{J}_{s; \Omega}$-family on $U$.

(i) Example 3.10 indicates that there exists a $b$-gauge spaces $(U, {Q}_{s; \Omega})$ and $\mathcal{J}_{s; \Omega}$-family on $U$ with $\mathcal{J}_{s; \Omega}\neq {Q}_{s; \Omega}$ such that $(U, {Q}_{s; \Omega})$ is $\mathcal{J}_{s; \Omega}$-sequential complete, but not ${Q}_{s; \Omega}$-sequential complete.

(ii) For each subsequence $(v_{m}:m\in \mathbb{N})$ of $(u_{m}:m\in \mathbb{N})$, where $(u_{m}:m\in \mathbb{N})$ is $\mathcal{J}_{s; \Omega}$-convergent in $U$, we have $S_{({u_{m}: m\in \mathbb{N}})}^{\mathcal{J}_{s; \Omega}}\subset S_{({v_{m}: m\in \mathbb{N}})}^{\mathcal{J}_{s; \Omega}}$.

Definition 3.12. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space. The map $T^{[k]}:U \rightarrow U$ (where $k\in \mathbb{N}$) is called to be a ${Q}_{s; \Omega}$-closed map on $U$ if for each sequence $(x_{m}:m\in \mathbb{N})$ in $T^{[k]}(U)$, which is ${Q}_{s; \Omega}$-converging in $U$, i.e., $S_{({x_{m}: m\in \mathbb{N}})}^{{Q}_{s; \Omega}}\neq \emptyset$ and its subsequences $(v_{m}:m\in \mathbb{N})$ and $(u_{m}:m\in \mathbb{N})$ satisfy $v_{m} = T^{[k]}(u_{m})$, for all $m\in \mathbb{N}$ has the property that there exists $w\in S^{{Q}_{s; \Omega}}_{(x_{m}:m\in \mathbb{N})}$ such that $w\in T^{[k]}(w)$.

Now, we present some fixed and periodic point theorems in the $b$-gauge space $(U, {Q}_{s; \Omega})$, using $\mathcal{J}_{s; \Omega}$-family of generalized pseudo-$b$-distances by incorporating the idea of Cosentino for the family $\mathfrak{F_{s}}$ of all functions $F: (0, \infty) \rightarrow \mathbb{R}$ in the setting of $b$-metric spaces and $F$-contraction of Hardy-Rogers type.

Theorem 3.13. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space. Let $\mathcal{J}_{s; \Omega} = \{{{{J}}_{\beta}}: \beta \in \Omega\}$, where ${J}_{\beta}: U\times U \rightarrow [0, \infty)$, be the $\mathcal{J}_{s; \Omega}$-family of distances generated by ${Q}_{s; \Omega}$ such that $U^{0}_{\mathcal{J}_{s; \Omega}}\neq \emptyset$ and $(U, {Q}_{s; \Omega})$ is $\mathcal{J}_{s; \Omega}$-sequentially complete. Let $T:U\rightarrow U$ be a mapping such that $T(U)\subset U^{0}_{\mathcal{J}_{s; \Omega}}$ and we have $F\in \mathfrak{F_{s}}$ and $\tau > 0$ such that:

for all $\beta \in \Omega$ and for any $u, v \in U$, whenever ${J}_{\beta}(Tu, Tv)\neq0$.

Further, $a_{\beta}, b_{\beta}, c_{\beta}, e_{\beta}, L_{\beta}\geq 0$ are such that $a_{\beta}+b_{\beta}+c_{\beta}+(s_{\beta}+1)e_{\beta} < 1$ for each $\beta \in \Omega$. Moreover, assume that the following conditions hold:

(i) There exists $z^{0}\in U$ such that $\alpha(z^{0}, z^{1})\geq 1$.

(ii) If $\alpha(x, y)\geq 1$, then $\alpha(Tx, Ty)\geq 1$.

(iii) If a sequence $(z^{m}:m\in \mathbb{N})$ in $U$ is such that $\alpha(z^{m}, z^{m+1})\geq1$ and $\lim_{m\rightarrow \infty}^{\mathcal{J}_{s; \Omega}}z^{m} = z$, then $\alpha(z^{m}, z)\geq 1$ and $\alpha(z, z^{m})\geq 1$.

Then the following statements hold:

(I) For each $z^{0}\in U$, $(z^{m}: m\in \{0\}\cup\mathbb{N})$ is ${Q}_{s; \Omega}$-convergent sequence in $U$; thus, $S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{{Q}_{s; \Omega}}\neq \emptyset$.

(II) Furthermore, assume that $T^{[k]}$ for some $k\in \mathbb{N}$, is ${Q}_{s; \Omega}$-closed map on $U$ and $s_{\beta}\{c_{\beta}+e_{\beta}s_{\beta}\} < 1$, for each $\beta\in \Omega$. Then

(a₁) Fix$(T^{[k]})\neq \emptyset$;

(a₂) for all $z^{0}\in U$, there exists $z\in \text{Fix}(T^{[k]})$ such that $z\in S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{{Q}_{s; \Omega}}$; and

(a₃) for all $z\in \text{Fix}(T^{[k]})$, ${J}_{\beta}(z, T(z)) = {J}_{\beta}(T(z), z) = 0$, for all $\beta \in \Omega$.

(III) Furthermore, let Fix$(T^{[k]})\neq \emptyset$ for some $k\in \mathbb{N}$ and $(U, {Q}_{s; \Omega})$ is a Hausdorff space. Then

(b₁) Fix$(T^{[k]})$ = Fix$(T)$;

(b₂) for all $z^{0}\in U$, there exists $z\in \text{Fix}(T)$ such that $z\in S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{L-{Q}_{s; \Omega}}$; and

(b₃) for all $z\in \text{Fix}(T^{[k]}) = \text{Fix}(T)$, ${J}_{\beta}(z, z) = 0$, for all $\beta \in \Omega$.

Proof. (I) We first show that $(z^{m}: m\in \{0\}\cup\mathbb{N})$ is a $\mathcal{J}_{s; \Omega}$-cauchy sequence in $U$.

Using assumption $(i)$, there exists $z^{0}\in U$ such that $\alpha(z^{0}, z^{1})\geq 1$. Now, for each $\beta\in \Omega$, using (3.13) we can write

As $F$ is strictly increasing, we can write from above that

It is written as

That is,

Since $a_{\beta}+b_{\beta}+c_{\beta}+(s_{\beta}+1)e_{\beta} < 1$, we get

hence

Now, using (3.14), we can write

Using assumption $(ii)$, we have $\alpha(Tz^{0}, Tz^{1}) = \alpha(z^{1}, z^{2})\geq 1$. For each $\beta\in \Omega$, using (3.13) we can write

As $F$ is strictly increasing, we can write from above that

We can also write it as

Since $a_{\beta}+b_{\beta}+c_{\beta}+(s_{\beta}+1)e_{\beta} < 1$, we get

Now, using (3.15), we can write

Proceeding in the above manner, we get a sequence $(z^{m}: m\in \{0\}\cup\mathbb{N})\subset U$ such that $z^{m} = Tz^{m-1}, z^{m-1}\neq z^{m}$ and $\alpha(z^{m-1}, z^{m})\geq 1$, for each $m\in \mathbb{N}$. Furthermore,

Using property $(F_{4})$, for all $m\in \mathbb{N}$, we can write

Thus,

Letting $m\rightarrow \infty$, from (3.16) we get $\lim_{m\rightarrow \infty}F(s^{m}_{\beta}{J}_{\beta}(z^{m}, z^{m+1})) = -\infty$ for all $\beta\in \Omega$. Hence, using property $(F_{2})$ we get $\lim_{m\rightarrow \infty}s^{m}_{\beta}{J}_{\beta}(z^{m}, z^{m+1}) = 0$. Let $({J}_{\beta})_{m} = {J}_{\beta}(z^{m}, z^{m+1})$ for all $\beta\in \Omega$ and $m\in \mathbb{N}$. From $(F_{3})$, there exists $p\in (0, 1)$ such that

From (3.16), for all $\beta\in \Omega$ and $m\in \mathbb{N}$, we can write

Applying $m\rightarrow \infty$, we have

This implies there exists $m_{1} = m_{1}(\beta)\in \mathbb{N}$ such that $m(s^{m}_{\beta}({J}_{\beta})_{m})^{p}\leq 1$ for each $m\geq m_{1}$ and for all $\beta\in \Omega$. Hence, we can write

Now, by repeated use of $(\mathcal{J}1)$ and (3.19) for all $m, n \in \mathbb{N}$ such that $n > m > m_{1}$ and for all $\beta \in \Omega$, we get

Since $\sum^{\infty}_{i = 1}\frac{1}{i^{\frac{1}{p}}}$ is a convergent series, we have

Since $(U, {Q}_{s; \Omega})$ is a $\mathcal{J}_{s; \Omega}$-sequentially complete $b$-gauge space, we have $(z^{m}: m\in \{0\}\cup\mathbb{N})$ is $\mathcal{J}_{s; \Omega}$-convergent in $U$, thus for all $z\in S^{\mathcal{J}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}$, we can write

Thus, from (3.20) and (3.21), fixing $z\in S^{\mathcal{J}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}$, defining $(u_{m} = z^{m}: m\in \{0\}\cup\mathbb{N})$ and $(v_{m} = z: m\in \{0\}\cup\mathbb{N})$ and applying $(\mathcal{J}2)$ to these sequences, we get

This implies $S^{{Q}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}\neq \emptyset$.

(II) To prove $(a_{1})$, let $z^{0}\in U$ be arbitrary and fixed. Since $S^{{Q}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}\neq \emptyset$, we have

Thus, defining $(z_{m} = z^{m-1+k}: m\in \mathbb{N})$, we can write

Also,

and

satisfy

and are ${Q}_{s; \Omega}$-convergent to each point $z\in S^{{Q}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}$. Now, using the fact below $S_{({z_{m}: m\in \mathbb{N}})}^{{Q}_{s; \Omega}}\subset S_{({y_{m}: m\in \mathbb{N}})}^{{Q}_{s; \Omega}}$ {and} $S_{({z_{m}: m\in \mathbb{N}})}^{{Q}_{s; \Omega}}\subset S_{({x_{m}: m\in \mathbb{N}})}^{{Q}_{s; \Omega}}$ and the supposition that $T^{[k]}$ for some $k\in \mathbb{N}$, is a ${Q}_{s; \Omega}$-closed map on $U$, there exists $z\in S^{{Q}_{s; \Omega}}_{(z_{m}: m\in \{0\}\cup \mathbb{N})} = S^{{Q}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}$ such that $z\in T^{[k]}(z)$. Thus, $(a_{1})$ holds.

The assertion $(a_{2})$ follows from $(a_{1})$ and the fact that $S^{{Q}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}\neq \emptyset$.

To prove $(a_{3})$, on contrary suppose that ${J}_{\beta}(z, Tz) > 0$ for some $\beta \in \Omega$, there exists $m_{0}\in \mathbb{N}$ such that ${J}_{\beta}(z^{m}, Tz) > 0$ for each $m\geq m_{0}$. Hence, for each $m\geq m_{0}$, use triangular inequality and inequality (3.13) to obtain

Letting $m\rightarrow \infty$, we have

We have assumed that $s_{\beta}\{c_{\beta}+e_{\beta}s_{\beta}\} < 1$, so

It is absurd, thus ${J}_{\beta}(z, Tz) = 0$ for all $\beta \in \Omega$.

Next, we prove that ${J}_{\beta}(Tz, z) = 0$ for all $\beta \in \Omega$. On contrary suppose that ${J}_{\beta}(Tz, z) > 0$ for some $\beta \in \Omega$, there exists $m_{0}\in \mathbb{N}$ such that ${J}_{\beta}(Tz, z^{m}) > 0$ for each $m\geq m_{0}$. Hence, for each $m\geq m_{0}$, use triangular inequality and inequality (3.13) to obtain

Letting $m\rightarrow \infty$, we have

We have proved that ${J}_{\beta}(z, Tz) = 0$ for all $\beta \in \Omega$, so ${J}_{\beta}(Tz, z) = 0$ for all $\beta \in \Omega$. Hence, the assertion $(a_{3})$ holds.

(III) Since $(U, {Q}_{s; \Omega})$ is a Hausdorff space, using Proposition (3.5), assertion $(a_{3})$ suggests that for $z\in \text{Fix}(T^{[k]})$, we have $z = T(z)$. This gives $z\in \text{Fix}(T)$. Hence, $(b_{1})$ is true.

Assertions $(a_{2})$ and $(b_{1})$ imply $(b_{2})$. To prove assertion $(b_{3})$, consider $(\mathcal{J}1)$ and use $(a_{3})$ and $(b_{1})$ to have for all $z\in \text{Fix}(T^{[k]}) = \text{Fix}(T)$,

Theorem 3.14. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space. Let $\mathcal{J}_{s; \Omega} = \{{{{J}}_{\beta}}: \beta \in \Omega\}$, where ${J}_{\beta}: U\times U \rightarrow [0, \infty)$, be the $\mathcal{J}_{s; \Omega}$-family of distances generated by ${Q}_{s; \Omega}$ such that $U^{0}_{\mathcal{J}_{s; \Omega}}\neq \emptyset$ and $(U, {Q}_{s; \Omega})$ is $\mathcal{J}_{s; \Omega}$-sequentially complete. Let $T:U\rightarrow U$ be a mapping such that $T(U)\subset U^{0}_{\mathcal{J}_{s; \Omega}}$ and we have $F\in \mathfrak{F_{s}}$ and $\tau > 0$, so that

for all $\beta \in \Omega$ and for any $u, v \in U$, whenever ${J}_{\beta}(Tu, Tv)\neq0$. Also, $L_{\beta}\geq 0$.

Assume, moreover that, the following conditions hold:

(i) There exists $z^{0}\in U$ such that $\alpha(z^{0}, z^{1})\geq 1$.

(ii) If $\alpha(u, v)\geq 1$, then $\alpha(Tu, Tv)\geq 1$.

Then the following statements hold:

(I) For any $z^{0}\in U$, $(z^{m}: m\in \{0\}\cup\mathbb{N})$ is ${Q}_{s; \Omega}$-convergent sequence in $U$, thus $S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{{Q}_{s; \Omega}}\neq \emptyset$.

(II) Furthermore, assume that $T^{[k]}$ for some $k\in \mathbb{N}$, is a ${Q}_{s; \Omega}$-closed map on $U$. Then

(a₁) Fix$(T^{[k]})\neq \emptyset$;

(a₂) for all $z^{0}\in U$, there exists $z\in \text{Fix}(T^{[k]})$ such that $z\in S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{{Q}_{s; \Omega}}$.

(III) Furthermore, let Fix$(T^{[k]})\neq \emptyset$ for some $k\in \mathbb{N}$ and $T$ be continuous. Then

(b₁) Fix$(T^{[k]})$ = Fix$(T)$;

(b₂) for all $z^{0}\in U$, there exists $z\in \text{Fix}(T)$ such that $z\in S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{L-{Q}_{s; \Omega}}$; and

(b₃) for all $z\in \text{Fix}(T^{[k]}) = \text{Fix}(T)$, ${J}_{\beta}(z, z) = 0$, for all $\beta \in \Omega$.

Proof. (I) We first show that $(z^{m}: m\in \{0\}\cup\mathbb{N})$ is $\mathcal{J}_{s; \Omega}$-cauchy sequence in $U$.

Using assumption $(i)$ there exists $z^{0}\in U$ such that $\alpha(z^{0}, z^{1})\geq 1$. For each $\beta\in \Omega$, using (3.22) we can write

We observe a contradiction if we choose $\max\{{J}_{\beta}(z^{0}, z^{1}), {J}_{\beta}(z^{1}, z^{2})\} = {J}_{\beta}(z^{1}, z^{2})$. Hence, choosing $\max\{{J}_{\beta}(z^{0}, z^{1}), {J}_{\beta}(z^{1}, z^{2})\} = {J}_{\beta}(z^{0}, z^{1})$ for all $\beta\in \Omega$, we get

Using assumption $(ii)$, we have $\alpha(Tz^{0}, Tz^{1}) = \alpha(z^{1}, z^{2})\geq 1$. For each $\beta\in \Omega$, using (3.22) we can write

We observe a contradiction if we choose $\max\{{J}_{\beta}(z^{1}, z^{2}), {J}_{\beta}(z^{2}, z^{3})\} = {J}_{\beta}(z^{2}, z^{3})$. Hence, choosing $\max\{{J}_{\beta}(z^{1}, z^{2}), {J}_{\beta}(z^{2}, z^{3})\} = {J}_{\beta}(z^{1}, z^{2})$ for all $\beta\in \Omega$, we get

Proceeding in the above manner, we get a sequence $(z^{m}: m\in \{0\}\cup\mathbb{N})\subset U$ such that $z^{m} = Tz^{m-1}, z^{m-1}\neq z^{m}$ and $\alpha(z^{m-1}, z^{m})\geq 1$, for each $m\in \mathbb{N}$. Furthermore,

Using property $(F_{4})$, for all $m\in \mathbb{N}$, we get

Furthermore,

Now, letting $m\rightarrow \infty$, from (3.23) we get $\lim_{m\rightarrow \infty}F(s^{m}_{\beta}{J}_{\beta}(z^{m}, z^{m+1})) = -\infty$ for all $\beta\in \Omega$. Hence, using property $(F_{2})$ we get $\lim_{m\rightarrow \infty}s^{m}_{\beta}{J}_{\beta}(z^{m}, z^{m+1}) = 0$. Let $({J}_{\beta})_{m} = {J}_{\beta}(z^{m}, z^{m+1})$ for all $\beta\in \Omega$ and $m\in \mathbb{N}$. From $(F_{3})$, there exists $p\in (0, 1)$ such that

From (3.23), we can write

Applying $m\rightarrow \infty$, we have

This implies there exists $m_{1} = m_{1}(\beta)\in \mathbb{N}$ such that $m(s^{m}_{\beta}({J}_{\beta})_{m})^{p}\leq 1$ for each $m\geq m_{1}$ and for all $\beta\in \Omega$. Hence, we can write

Now, by repeated use of $(\mathcal{J}1)$ and (3.19) for all $m, n \in \mathbb{N}$ such that $n > m > m_{1}$ and for all $\beta \in \Omega$, we get

Since $\sum^{\infty}_{i = 1}\frac{1}{i^{\frac{1}{p}}}$ is a convergent series, we have

Now, since $(U, {Q}_{s; \Omega})$ is a $\mathcal{J}_{s; \Omega}$-sequentially complete $b$-gauge space, we have $(z^{m}: m\in \{0\}\cup\mathbb{N})$ is $\mathcal{J}_{s; \Omega}$-convergent in $U$. Thus for all $z\in S^{\mathcal{J}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}$, we can write

Thus, from (3.27) and (3.28), fixing $z\in S^{\mathcal{J}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}$, defining $(u_{m} = z^{m}: m\in \{0\}\cup\mathbb{N})$ and $(v_{m} = z: m\in \{0\}\cup\mathbb{N})$ and applying $(\mathcal{J}2)$ to these sequences, we get

This implies that $S^{{Q}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}\neq \emptyset$.

(II) To prove $(a_{1})$, let $z^{0}\in U$ be arbitrary and fixed. Since $S^{{Q}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}\neq \emptyset$, and

defining $(z_{m} = z^{m-1+k}: m\in \mathbb{N})$, we can write

Also,

and

satisfy

and are ${Q}_{s; \Omega}$-convergent to each point $z\in S^{{Q}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}$. Now, using the fact below $S_{({z_{m}: m\in \mathbb{N}})}^{{Q}_{s; \Omega}}\subset S_{({y_{m}: m\in \mathbb{N}})}^{{Q}_{s; \Omega}}$, $S_{({z_{m}: m\in \mathbb{N}})}^{{Q}_{s; \Omega}}\subset S_{({x_{m}: m\in \mathbb{N}})}^{{Q}_{s; \Omega}}$ and the supposition that $T^{[k]}$ for some $k\in \mathbb{N}$, is a ${Q}_{s; \Omega}$-closed map on $U$, there exists $z\in S^{{Q}_{s; \Omega}}_{(z_{m}: m\in \{0\}\cup \mathbb{N})} = S^{{Q}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}$ such that $z\in T^{[k]}(z)$. Thus, $(a_{1})$ holds.

The assertion $(a_{2})$ follows from $(a_{1})$ and the fact that $S^{{Q}_{s; \Omega}}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}\neq \emptyset$.

(III) By $(a_{2})$, for all $z^{0}\in U$, there exists $z\in \text{Fix}(T^{[k]})$ such that $z\in S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{{Q}_{s; \Omega}}$, and so we have $\lim_{m\rightarrow \infty}z^{m} = z$.

Now, if $T$ is continuous, then $z = \lim_{m\rightarrow \infty}z^{m+1} = \lim_{m\rightarrow \infty}Tz^{m} = T(\lim_{m\rightarrow \infty}z^{m}) = T(z)$. This gives $z\in \text{Fix}(T)$. Hence, $(b_{1})$ is true. Assertions $(a_{2})$ and $(b_{1})$ imply $(b_{2})$. To prove assertion $(b_{3})$, since $T(U)\subset U^{0}_{\mathcal{J}_{s; \Omega}}$, this implies that $z = T(z)\in U^{0}_{\mathcal{J}_{s; \Omega}}$.

Therefore, ${J}_{\beta}(z, z) = 0$, for all $\beta\in \Omega$.

Example 3.15. Let $U = [0, 1]$ and $B = \{\frac{1}{2^{m}}:m\in \mathbb{N}\}$.

Let ${Q}_{s; \Omega} = \{q\}$, where $q:U\times U\rightarrow [0, \infty)$ is a pseudo-$b$-metric on $U$ defined for all $x, y\in U$ by

Let the set $F = [\frac{1}{8}, 1] \subset U$ and let ${J}:U\times U\rightarrow [0, \infty)$ for all $x, y\in U$ be defined by

Define $\alpha:U\times U\rightarrow [0, \infty)$ by

The single-valued map $T$ is defined by

Note that $T(U) = [\frac{1}{5}, \frac{2}{5}]\subset U^{0}_{\mathcal{J}_{s; \Omega}} = [\frac{1}{8}, 1].$ Also, take $F(x) = \ln(x)$, then $F\in \mathfrak{F_{s}}$.

(I.1) $(U, {Q}_{s; \Omega})$ is a $b$-gauge space, which is also Hausdorff.

(I.2) The family $\mathcal{J}_{s; \Omega} = \{{J}\}$ is $\mathcal{J}_{s; \Omega}$-family on $U$ (see Example 3.2).

(I.3) $(U, {Q}_{s; \Omega})$ is $\mathcal{J}_{s; \Omega}$-sequential complete (follows from Example 3.10).

(I.4) Next, applying $F(x) = \ln(x)$ to condition (3.13), we show that $T$ satisfies the following condition.

for any $x, y \in U$ whenever ${J}(Tx, Ty)\neq0$. It is obvious that above condition holds for $a = b = c = \frac{1}{5}$ and $e = L = 0$.

(I.5) Assumptions $(i)$–$(iii)$ of Theorem 3.13 hold. For $z_{0} = 0$ and $z_{1} = Tz_{0} = \frac{1}{5}$, we have $\alpha(z_{0}, Tz_{0}) > 1$. Also, $\alpha(Tx, Ty) > 1$ if $\alpha(x, y) > 1$. Finally, if a sequence $(z_{m}:m\in \mathbb{N})$ in $U$ is such that $\alpha(z_{m}, z_{m+1})\geq1$ and $\lim_{m\rightarrow \infty}^{\mathcal{J}_{s; \Omega}}z_{m} = z$, then $\alpha(z_{m}, z)\geq 1$ and $\alpha(z, z_{m})\geq 1$.

(I.6) Finally, we show that $T$ is a ${Q}_{s; \Omega}$-closed map on $U$. For this, let $(z_{m}:m\in \mathbb{N})$ be a sequence in $T(U) = [\frac{1}{5}, \frac{2}{5}]$ which is ${Q}_{s; \Omega}$-convergent to each point of $S^{{Q}_{s; \Omega}}_{(z_{m}: m\in \{0\}\cup \mathbb{N})}\neq \emptyset$. Let the subsequences $(v_{m}:m\in \mathbb{N})$ and $(u_{m}:m\in \mathbb{N})$ satisfy $v_{m} = T(u_{m})$, for all $m\in \mathbb{N}$.

Let $z\in S^{{Q}_{s; \Omega}}_{(z_{m}: m\in \{0\}\cup \mathbb{N})}$, then without loosing generality we may assume that for all $0 < \epsilon_{1} < 1$ there exists $k\in \mathbb{N}$ such that

As a result, for $\epsilon = \sqrt{\epsilon_{1}}$, we can also write for all $0 < \epsilon < 1$ there exists $k\in \mathbb{N}$ such that

In particular, this implies that

and we obtain

Since $\mid z-v_{m} \mid \rightarrow 0$, when $m\rightarrow \infty$, we get $\mid \frac{1}{4}-z \mid < \epsilon_{2}$ where $\epsilon_{2} = \frac{\epsilon}{4} < \frac{1}{4}$. This gives $S^{{Q}_{s; \Omega}}_{(z_{m}: m\in \mathbb{N})} = \{\frac{1}{4}\}$ and so there exists $z = \frac{1}{4}\in S^{{Q}_{s; \Omega}}_{(z_{m}: m\in \mathbb{N})}$ such that $\frac{1}{4} = T(\frac{1}{4})$. Hence, $T$ is $a {Q}_{s; \Omega}$-closed map on $U$.

(I.7) As all the assumptions of Theorem 3.13 hold, we have

and

Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space and $G = (V, E)$ be a directed graph such that set of vertices $V$ is equal to $U$ and set of edges $E$ includes $\{(u, u): u\in U\}$, but $G$ includes no parallel edges. We obtain the following corollaries from our theorems by defining $\alpha:U\times U \rightarrow [0, \infty)$ for some $\kappa\geq 1$ in the following way.

Corollary 3.16. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space. Let $\mathcal{J}_{s; \Omega} = \{{{{J}}_{\beta}}: \beta \in \Omega\}$, where ${J}_{\beta}: U\times U \rightarrow [0, \infty)$, be the $\mathcal{J}_{s; \Omega}$-family of distances generated by ${Q}_{s; \Omega}$ such that $U^{0}_{\mathcal{J}_{s; \Omega}}\neq \emptyset$ and $(U, {Q}_{s; \Omega})$ is $\mathcal{J}_{s; \Omega}$-sequentially complete. Let $T:U\rightarrow U$ be a mapping such that $T(U)\subset U^{0}_{\mathcal{J}_{s; \Omega}}$ and for which we have $F\in \mathfrak{F_{s}}$ and $\tau > 0$ such that

for all $\beta \in \Omega$ and for any $u, v \in U$ whenever ${J}_{\beta}(Tu, Tv)\neq0$.

Further, $a_{\beta}, b_{\beta}, c_{\beta}, e_{\beta}, L_{\beta}\geq 0$ are such that $a_{\beta}+b_{\beta}+c_{\beta}+(s_{\beta}+1)e_{\beta} < 1$ for each $\beta \in \Omega$. Assume, moreover that, the following conditions hold:

(i) There exists $z^{0}\in U$ such that $(z^{0}, z^{1})\in E$.

(ii) If $(u, v)\in E$, then $(Tu, Tv)\in E$.

(iii) If a sequence $(z^{m}:m\in \mathbb{N})$ in $U$ is such that $(z^{m}, z^{m+1})\in E$ and $\lim_{m\rightarrow \infty}^{\mathcal{J}_{s; \Omega}}z^{m} = z$, then $(z^{m}, z)\in E$ and $(z, z^{m})\in E$.

Then the following statements hold:

(I) For each $z^{0}\in U$, $(z^{m}: m\in \{0\}\cup\mathbb{N})$ is ${Q}_{s; \Omega}$-convergent sequence in $U$; thus, $S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{{Q}_{s; \Omega}}\neq \emptyset$.

(II) Furthermore, assume that $T^{[k]}$ for some $k\in \mathbb{N}$, is a ${Q}_{s; \Omega}$-closed map on $U$ and $s_{\beta}\{c_{\beta}+e_{\beta}s_{\beta}\} < 1$, for each $\beta\in \Omega$. Then

(a₁) Fix$(T^{[k]})\neq \emptyset$;

(a₂) for all $z^{0}\in U$, there exists $z\in \text{Fix}(T^{[k]})$ such that $z\in S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{{Q}_{s; \Omega}}$; and

(a₃) for all $z\in \text{Fix}(T^{[k]})$, ${J}_{\beta}(z, T(z)) = {J}_{\beta}(T(z), z) = 0$, for all $\beta \in \Omega$.

(III) Furthermore, let Fix$(T^{[k]})\neq \emptyset$ for some $k\in \mathbb{N}$ and $(U, {Q}_{s; \Omega})$ is a Hausdorff space. Then

(b₁) Fix$(T^{[k]})$ = Fix$(T)$;

(b₂) for all $z^{0}\in U$, there exists $z\in \text{Fix}(T)$ such that $z\in S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{L-{Q}_{s; \Omega}}$; and

(b₃) for all $z\in \text{Fix}(T) = \text{Fix}(T^{[k]})$, ${J}_{\beta}(z, z) = 0$, for all $\beta \in \Omega$.

Corollary 3.17. Let $(U, {Q}_{s; \Omega})$ be a $b$-gauge space. Let $\mathcal{J}_{s; \Omega} = \{{{{J}}_{\beta}}: \beta \in \Omega\}$, where ${J}_{\beta}: U\times U \rightarrow [0, \infty)$, is the $\mathcal{J}_{s; \Omega}$-family of distances generated by ${Q}_{s; \Omega}$ such that $U^{0}_{\mathcal{J}_{s; \Omega}}\neq \emptyset$ and $(U, {Q}_{s; \Omega})$ is $\mathcal{J}_{s; \Omega}$-sequentially complete. Let $T:U\rightarrow U$ be a mapping such that $T(U)\subset U^{0}_{\mathcal{J}_{s; \Omega}}$ and for which we have $F\in \mathfrak{F_{s}}$ and $\tau > 0$ such that

for all $\beta \in \Omega$ and for any $u, v \in U$, whenever ${J}_{\beta}(Tu, Tv)\neq0$. Also, $L_{\beta}\geq 0$.

Assume, moreover that, the following conditions hold:

(i) There exists $z^{0}\in U$ such that $(z^{0}, z^{1})\in E$.

(ii) If $(u, v)\in E$, then $(Tu, Tv)\in E$.

Then the following statements hold:

(I) For any $z^{0}\in U$, $(z^{m}: m\in \{0\}\cup\mathbb{N})$ is ${Q}_{s; \Omega}$-convergent sequence in $U$; thus, $S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{{Q}_{s; \Omega}}\neq \emptyset$.

(II) Furthermore, assume that $T^{[k]}$ for some $k\in \mathbb{N}$, is a ${Q}_{s; \Omega}$-closed map on $U$. Then

(a₁) Fix$(T^{[k]})\neq \emptyset$;

(a₂) for all $z^{0}\in U$, there exists $z\in \text{Fix}(T^{[k]})$ such that $z\in S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{{Q}_{s; \Omega}}$.

(III) Furthermore, let Fix$(T^{[k]})\neq \emptyset$ for some $k\in \mathbb{N}$ and $T$ be continuous. Then

(b₁) Fix$(T^{[k]})$ = Fix$(T)$;

(b₂) for all $z^{0}\in U$, there exists $z\in \text{Fix}(T)$ such that $z\in S_{({z^{m}: m\in \{0\}\cup \mathbb{N}})}^{L-{Q}_{s; \Omega}}$; and

(b₃) for all $z\in \text{Fix}(T) = \text{Fix}(T^{[k]})$, ${J}_{\beta}(z, z) = 0$, for all $\beta \in \Omega$.

4.   Application

A volterra integral equation

is the integral equation located in the space $C[0, \infty)$ of all continuous functions defined on the interval $[0, \infty)$, where $K(t, s):[0, \infty)\times [0, \infty)\rightarrow \mathbb{R}$ and $f, g: [0, \infty)\rightarrow \mathbb{R}$ are continuous functions so that $g(t)\geq 0$ for all $t\in [0, \infty)$. Let $U = (C[0, \infty), \mathbb{R})$. Define the family of $b$-pseudo metrics by

Obviously, ${Q}_{s; \Omega} = \{q_{m}:m\in \mathbb{N}\}$ defines a complete Hausdorff $b$-gauge structure on $U$. Here, in particular we consider the case when ${Q}_{s; \Omega} = \mathcal{J}_{s; \Omega} = \{q_{m}:m\in \mathbb{N}\}$. Define the map $\alpha:U \times U \rightarrow [0, \infty)$ for some $\kappa\geq 1$ in the following way:

Theorem 4.1. Define the operator $T:C[0, \infty)\rightarrow C[0, \infty)$ as follows:

where $K(t, s):[0, \infty)\times [0, \infty)\rightarrow \mathbb{R}$ and $f, g: [0, \infty)\rightarrow \mathbb{R}$ are continuous functions so that $g(t)\geq 0$ for all $t\in [0, \infty)$.

Assume, moreover there exist $\gamma: U\rightarrow (0, \infty)$ and $\alpha:U \times U \rightarrow (0, \infty)$ such that the following statements hold:

(i) There is $\tau > 0$ such that

for each $t, s\in[0, \infty)$ and $u, v\in U$. Also,

(ii) There exists $z^{0}\in U$ such that $\alpha(z^{0}, Tz^{0})\geq 1$.

(iii) For $x, y\in U$ with $\alpha(x, y)\geq 1$ we have $\alpha(Tx, Ty)\geq 1$.

(iv) If a sequence $(z^{m}:m\in \mathbb{N})$ in $U$ is such that $\alpha(z^{m}, z^{m+1})\geq1$ and $\lim_{m\rightarrow \infty}^{\mathcal{J}_{s; \Omega}}z^{m} = z$, then $\alpha(z^{m}, z)\geq 1$ and $\alpha(z, z^{m})\geq 1$.

(v) $T$ is ${Q}_{s; \Omega}$-closed map.

Then there exists at least one solution of the integral equation (4.1).

Proof. We first prove that $T$ satisfies condition (3.13). For any $u, v\in U$ with $\alpha(u, v)\geq 1$, we have

From here we can write

This can be written as

Obviously, natural logarithm belong to the family $\mathfrak{F_{s}}$, therefore, taking logarithm on both sides, we have

A simplification leads to the following

This implies that (3.13) holds for $a_{m} = 1$ and $b_{m} = c_{m} = e_{m} = L_{m} = 0$, for all $m\in \mathbb{N}$ and $F(u) = \ln u$. Hence, Theorem 3.13 ensures the existence of a fixed point of the operator $T$, thus, there is at least one solution of the integral equation (4.1).

5.   Concluding remarks

Remark 5.1. The fixed point results concerning $F$-type-contractions in a gauge space in ^[17] require the completeness of the space $(U, d)$. Therefore, our theorems and corollaries for $F$-type-contractions in the $b$-gauge space are new generalizations of the results in ^[17] in which assumptions are weaker and assertions are stronger.

Remark 5.2. Our results for $F$-type-contractions in $b$-gauge spaces deal with about periodic points as well. Hence, they improve the results in ^[17].

Remark 5.3. Theorems 3.13 and 3.14 generalize Theorems 4.2 and 5.2, respectively in ^[29].

Acknowledgments

The authors Aiman Mukheimer and Kamal Abodayeh would like to thank Prince Sultan University for paying APC and for the support through the TAS research LAB.

Conflict of interest

The authors declare that they have no conflicts of interest.