Fixed Point Results for Generalized ℱ-Contractions in Modular b-Metric Spaces with Applications

Abstract

The aim of this paper is to generalize the $\mathcal{F}$-contractive condition in the framework of $\alpha -\nu$-complete modular b-metric spaces. Some results in ordered modular b-metric spaces are also presented. Moreover, an illustrative example and some related applications are presented to support the obtained results.

1. Introduction

The concept of a b-metric space has been introduced by Bakhtin [1] and Czerwik [2,3]. So far, many interesting results about the existence of fixed points in b-metric spaces have been presented (see, e.g., [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18].)

Definition 1

([2]). Let Ω be a nonempty set and $s\ge 1$ be a given real number. A function $B:\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty )$ is a b-metric on Ω if, for all $\rho ,\varrho ,\sigma \in \mathrm{\Omega }$, the following assertions hold:

(b${}_1$) 

$B(\rho ,\varrho )=0$ iff $\rho =\varrho$;

(b${}_2$) 

$B(\rho ,\varrho )=B(\varrho ,\rho )$;

(b${}_3$) 

$B(\rho ,\sigma )\le s[B(\rho ,\varrho )+B(\varrho ,\sigma )]$.

The pair $(\mathrm{\Omega },B)$ is called a b-metric space.

Note that a b-metric is not continuous in its two variables. On the other hand, a modular metric space is an applicable extension of classical modulars over linear spaces. The concept of a modular metric space has been introduced in [19,20,21,22]. Here, we deal with modular metric spaces as the nonlinear version of the classical one introduced by Nakano [23] on a vector space and a modular function space presented by Musielak [24] and Orlicz [25].

Let $\mathrm{\Omega }$ be a nonempty set and let $\xi :(0,\infty )\times \mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty ]$ be a function. For simplicity, we will write

$${\xi }_{\lambda }(\rho ,\varrho )=\xi (\lambda ,\rho ,\varrho ),$$

for all $\lambda >0$ and for all $\rho ,\varrho \in \mathrm{\Omega }$.

Definition 2

([19,20]). A function $\xi :(0,\infty )\times \mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty ]$ is called a modular metric on Ω if, for all $\rho ,\varrho ,\sigma \in \mathrm{\Omega }$ and $\lambda >0$, the following assertions hold:

(i) 

$\rho =\varrho$ iff ${\xi }_{\lambda }(\rho ,\varrho )=0$;

(ii) 

${\xi }_{\lambda }(\rho ,\varrho )={\xi }_{\lambda }(\varrho ,\rho )$;

(iii) 

${\xi }_{\lambda +\mu }(\rho ,\varrho )\le {\xi }_{\lambda }(\rho ,\sigma )+{\xi }_{\mu }(z,\varrho )$.

A modular metric $\xi$ on $\mathrm{\Omega }$ is called regular if the following weaker version of $(i)$ holds:

$$\rho =\varrho \mathrm{iff}{\xi }_{\lambda }(\rho ,\varrho )=0\mathrm{for}\mathrm{some}\lambda =\omega >0.$$

Ege and Alaca in [26] introduced the notion of modular b-metric spaces as follows.

Definition 3

([26]). Let Ω be a nonempty set and $s\ge 1$. A mapping $\nu :(0,\infty )\times \mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty ]$ is called a modular b-metric, if for all $\rho ,\varrho ,\sigma \in \mathrm{\Omega }$ and $\lambda >0$, we have the following assertions:

(i) 

${\nu }_{\lambda }(\rho ,\varrho )=0$ iff $\rho =\varrho$;

(ii) 

${\nu }_{\lambda }(\rho ,\varrho )={\nu }_{\lambda }(\varrho ,\rho )$;

(iii) 

${\nu }_{\lambda +\mu }(\rho ,\varrho )\le s[{\nu }_{\lambda }(\rho ,\sigma )+{\nu }_{\mu }(z,\varrho )]$ for all $\lambda ,\mu >0$.

Then, we say that $(\mathrm{\Omega },\nu )$ is a modular b-metric space.

Definition 4.

Let $(\mathrm{\Omega },\nu )$ be a modular b-metric space. Let M be a subset of Ω and ${\left\{{\rho }_n\right\}}_{n\in \mathbb{N}}$ be a sequence in Ω. Then, we have the following statements:

(i) 

${\left\{{\rho }_n\right\}}_{n\in \mathbb{N}}$ is called ν-convergent to ρ if there is $\lambda >0$ (maybe it depends on $\left\{{\rho }_n\right\}$ and ρ), such that ${\nu }_{\lambda }({\rho }_n,\rho )\to 0$, as $n\to \infty$. ρ will be called the ν-limit of $({\rho }_n)$.

(ii) 

${\left\{{\rho }_n\right\}}_{n\in \mathbb{N}}$ is called ν-Cauchy if there is $\lambda >0$ (maybe it depends on the sequence) such that ${\nu }_{\lambda }({\rho }_m,{\rho }_n)\to 0$, as $m,n\to \infty$.

(iii) 

M is called ν-complete if any ν-Cauchy sequence in M is ν-convergent in $M.$

(iv) 

The function $f:(\mathrm{\Omega },\nu )\to (\mathrm{\Omega },\nu )$ is called ν-continuous if ${\nu }_{\lambda }(f{\rho }_n,f\rho )\to 0$, whenever ${\nu }_{\lambda }({\rho }_n,\rho )\to 0$.

Example 1.

Let $(\mathrm{\Omega },\xi )$ be an MbMS and let $p\ge 1$ be a real number. Take ${\nu }_{\lambda }(\rho ,\varrho )={({\xi }_{\lambda }(\rho ,\varrho ))}^p$. Using the convexity of $f(t)=t^p$ for $t\ge 0$, by Jensen inequality, we have

$${(a+b)}^p\le 2^{p-1}(a^p+b^p)$$

(1)

for nonnegative real numbers $a,b$. Thus, $(\mathrm{\Omega },\nu )$ is an MbMS with $s=2^{p-1}$.

Lemma 1.

Let $(\mathrm{\Omega },\nu )$ be an MbMS. Suppose that $\left\{{\rho }_n\right\}$ and $\left\{{\varrho }_n\right\}$ ν-converge to ρ and ϱ, respectively. Then,

$$\begin{array}{cc}\hfill \frac{1}{s^2}{\nu }_{\lambda }(\rho ,\varrho )& \le \underset{n⟶\infty }{lim\; sup}{\nu }_{\frac{\lambda }{4}}({\rho }_n,{\varrho }_n)\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{n⟶\infty }{lim\; sup}{\nu }_{\lambda }({\rho }_n,{\varrho }_n)\le s^2{\nu }_{\frac{\lambda }{4}}(\rho ,\varrho ).\end{array}$$

In particular, if $\rho =\varrho ,$ then ${\displaystyle \underset{n⟶\infty }{lim}}{\nu }_{\lambda }({\rho }_n,{\varrho }_n)=0.$ Moreover, for each $\sigma \in \mathrm{\Omega }$, we have

$$\begin{array}{cc}\hfill \frac{1}{s}{\nu }_{\lambda }(\rho ,\sigma )& \le \underset{n⟶\infty }{lim\; sup}{\nu }_{\frac{\lambda }{2}}({\rho }_n,\sigma )\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{n⟶\infty }{lim\; sup}{\nu }_{\lambda }({\rho }_n,\sigma )\le s{\nu }_{\frac{\lambda }{2}}(\rho ,\sigma ).\end{array}$$

Proof. 

(a) Using the triangular inequality, one writes

$$\begin{array}{cc}\hfill {\nu }_{\lambda }(\rho ,\varrho )& \le s({\nu }_{\frac{\lambda }{2}}(\rho ,{\rho }_n)+{\nu }_{\frac{\lambda }{2}}({\rho }_n,\varrho ))\hfill \\ & \le s({\nu }_{\frac{\lambda }{2}}(\rho ,{\rho }_n)+s({\nu }_{\frac{\lambda }{4}}({\rho }_n,{\varrho }_n)+{\nu }_{\frac{\lambda }{4}}({\varrho }_n,\varrho )))\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\nu }_{\lambda }({\rho }_n,{\varrho }_n)\le s({\nu }_{\frac{\lambda }{2}}({\rho }_n,\rho )+s({\nu }_{\frac{\lambda }{4}}(\rho ,\varrho )+{\nu }_{\frac{\lambda }{4}}(\varrho ,{\varrho }_n))).\end{array}$$

As $n\to \infty$, we have

$$\begin{array}{cc}\hfill {\nu }_{\lambda }(\rho ,\varrho )& \le s^2(\underset{n⟶\infty }{lim\; sup}{\nu }_{\frac{\lambda }{4}}({\rho }_n,{\varrho }_n)).\hfill \end{array}$$

Taking the upper limit as $n\to \infty$ in the second inequality implies that

$$\begin{array}{c}\hfill \underset{n⟶\infty }{lim\; sup}{\nu }_{\lambda }({\rho }_n,{\varrho }_n)\le s^2({\nu }_{\frac{\lambda }{4}}(\rho ,\varrho )).\end{array}$$

(b) Using the triangular inequality, one has

$$\begin{array}{cc}\hfill {\nu }_{\lambda }(\rho ,\sigma )& \le s({\nu }_{\frac{\lambda }{2}}(\rho ,{\rho }_n)+{\nu }_{\frac{\lambda }{2}}({\rho }_n,\sigma ))\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\nu }_{\lambda }({\rho }_n,\sigma )\le s({\nu }_{\frac{\lambda }{2}}({\rho }_n,\rho )+{\nu }_{\frac{\lambda }{2}}(\rho ,\sigma )).\end{array}$$

We find that

$$\begin{array}{cc}\hfill {\nu }_{\lambda }(\rho ,\sigma )& \le s(\underset{n⟶\infty }{lim\; sup}{\nu }_{\frac{\lambda }{2}}({\rho }_n,\sigma ))\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{n⟶\infty }{lim\; sup}{\nu }_{\lambda }({\rho }_n,\sigma )\le s({\nu }_{\frac{\lambda }{2}}(\rho ,\sigma )).\end{array}$$

 □

Definition 5

([27]). Given $\alpha :\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty )$, where Ω is a nonempty set, the self-mapping T on Ω is said to be α-admissible if

$$\rho ,\varrho \in \mathrm{\Omega },\alpha (\rho ,\varrho )\ge 1⟹\alpha (T\rho ,T\varrho )\ge 1.$$

Definition 6

([28]). Given $\alpha :\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty )$, the mapping $T:\mathrm{\Omega }\to \mathrm{\Omega }$ is said to be triangular α-admissible if

(i) 

$\rho ,\varrho \in \mathrm{\Omega },\alpha (\rho ,\varrho )\ge 1⟹\alpha (T\rho ,T\varrho )\ge 1$;

(ii) 

$\rho ,\varrho ,\sigma \in \mathrm{\Omega },\left\{\begin{array}{c}\alpha (\rho ,\sigma )\ge 1\hfill \\ \alpha (\sigma ,\varrho )\ge 1\hfill \end{array}\right.⟹\alpha (\rho ,\varrho )\ge 1$.

Lemma 2

([28]). Let f be a triangular α-admissible mapping. Assume that there is ${\rho }_0\in \mathrm{\Omega }$ such that $\alpha ({\rho }_0,f{\rho }_0)\ge 1.$ Define ${\rho }_n=f^n{\rho }_0$. Then,

$$\alpha ({\rho }_m,{\rho }_n)\ge 1forallm,n\in \mathbb{N}withm<n.$$

Definition 7.

Let $(\mathrm{\Omega },d)$ be a metric space and let $\alpha :\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty )$ be a function. The metric space Ω is said to be α-complete if every Cauchy sequence $\left\{{\rho }_n\right\}$ in Ω with $\alpha ({\rho }_n,{\rho }_{n+1})\ge 1$ for all $n\in \mathbb{N}$, converges in Ω.

Remark 1.

If $(\mathrm{\Omega },d)$ is a complete metric space, then Ω is also an α-complete metric space. The converse is not true.

Definition 8

([29]). Let $(\mathrm{\Omega },d)$ be a metric space. Given $\alpha :\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty )$, the mapping $T:\mathrm{\Omega }\to \mathrm{\Omega }$ is said to be α-continuous on $(\mathrm{\Omega },d)$, if for given $\rho \in \mathrm{\Omega }$ and sequence $\left\{{\rho }_n\right\}$,

$${\rho }_n\to \rho \mathit{as}n\to \infty \mathit{and}\alpha ({\rho }_n,{\rho }_{n+1})\ge 1\mathit{for}\mathit{all}n\in \mathbb{N},$$

we have $T{\rho }_n\to T\rho$ as $n\to \infty$.

We extend Definitions 7 and 8 to modular b-metric spaces.

Definition 9.

Let $(\mathrm{\Omega },\nu )$ be an MbMS and Υ be a self-mapping on Ω. Then, we have the following statements:

(i) 

$(\mathrm{\Omega },\nu )$ is said to be $\alpha -\nu$-complete if every ν-Cauchy sequence $\left\{{\rho }_n\right\}$ in Ω with $\alpha ({\rho }_n,{\rho }_{n+1})\ge 1$ for all $n\in \mathbb{N}$ is ν-convergent to some ρ in Ω.

(ii) 

Υ is said to be $\alpha -\nu$-continuous on $(\mathrm{\Omega },\nu )$, if for a ν-convergent sequence $\left\{{\rho }_n\right\}$ to some $\rho \in \mathrm{\Omega }$ so that $\alpha ({\rho }_n,{\rho }_{n+1})\ge 1$ for all $n\in \mathbb{N}$, we have $\left\{\mathrm{{\rm Y}}{\rho }_n\right\}$ is ν-convergent to $\mathrm{{\rm Y}}\rho$.

Wardowski [30] presented a new type of contractions called $\mathcal{F}$-contractions and proved a new fixed point theorem as a generalization of the Banach contraction principle.

In this paper, we prove some fixed point results for generalized $\mathcal{F}$-contractive mappings in the setup of modular b-metric spaces. An example is presented to verify the effectiveness of our obtained results. Some applications are also presented at the end.

2. Main Results

Motivated by Wardowski [30] (see also [31]), we denote by $\Delta$ the set of all functions $\mathcal{F}:(0,\infty )\to \mathbb{R}$ such that:

(${\mathcal{F}}_1$)

$\mathcal{F}$ is a continuous and strictly increasing mapping;

(${\mathcal{F}}_2$)

for each sequence $\left\{t_n\right\}\subseteq (0,\infty )$, $\underset{n\to \infty }{lim}{t}_n=0$ iff $\underset{n\to \infty }{lim}\mathcal{F}(t_n)=-\infty .$

Example 2.

Let $t>0$. The following functions:

• $\mathcal{F}(t)=ln(t)$;
• $\mathcal{F}(t)=1-\frac{1}{t^p}$ (with $p>0$);
• $\mathcal{F}(t)=1-\frac{1}{e^t-1}$;
• $\mathcal{F}(t)=\frac{1}{e^{-t}-e^t}$

belong to Δ.

The class $\Delta$ is different from the class of functions introduced by Wardowski [30]. It suffices to $\mathcal{U}(t)=-\frac{1}{t}+t$ for $t>0$. Note that ${lim}_{\alpha \to 0^{+}}{\alpha }^k\mathcal{U}(\alpha )=-\infty$ ($0<k<1$), that is, $\mathcal{U}\in \Delta$, but it is not a Wardowski mapping.

Let $\mathrm{\Theta }$ denote the set of all functions $\vartheta :\mathbb{R}\to \mathbb{R}$ satisfying:

• (${\vartheta }_1$) $\underset{n\to \infty }{lim}{\vartheta }^n\left(t\right)=-\infty$ for all $t>0$;
• (${\vartheta }_2$) $\vartheta (t)<t$ for all $t\ge 0$;
• (${\vartheta }_3$) $\vartheta$ is an increasing continuous function.

Example 3.

The following functions: $\vartheta (t)=t-\delta$ (with $\delta >0$) and $\vartheta =\sqrt[3]{t}-1$ are elements in Θ.

Definition 10.

Let $(\mathrm{\Omega },\nu )$ be an MbMS and Υ be a self-mapping on Ω. Given $\alpha :\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty )$. We say that Υ is an $\alpha -\vartheta$-$\mathcal{F}$-contraction if the following inequality:

$$\mathcal{F}\left(s^3·{\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho )\right)\le \vartheta \left(\mathcal{F}({\nu }_{\lambda }(\rho ,\varrho )\right),$$

(2)

holds for all $\rho ,\varrho \in \mathrm{\Omega }$ with $\alpha (\rho ,\varrho )\ge 1$ and ${\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho )>0$, where $\mathcal{F}\in \Delta$ and $\vartheta \in \mathrm{\Theta }$.

From now on, assume that $(\mathrm{\Omega },\nu )$ is regular.

Theorem 1.

Let $\alpha :\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty )$ be a function and let $(\mathrm{\Omega },\nu )$ be an α-${\nu }_{\lambda }$-complete MbMS. Assume that $\mathrm{{\rm Y}}:\mathrm{\Omega }\to \mathrm{\Omega }$ is such that

(i) 

Υ is triangular α-admissible;

(ii) 

Υ is an α-ϑ-$\mathcal{F}$-contraction;

(iii) 

there is ${\rho }_0\in \mathrm{\Omega }$ such that $\alpha ({\rho }_0,\mathrm{{\rm Y}}{\rho }_0)\ge 1$;

(iv) 

Υ is $\alpha -\nu$-continuous.

Then, Υ has a fixed point. In addition, Υ has a unique fixed point, provided that $\alpha (\rho ,\varrho )\ge 1$ for all $\rho ,\varrho \in Fix(\mathrm{{\rm Y}})$.

Proof. 

Let ${\eta }_0\in \mathrm{\Omega }$ satisfy $\alpha ({\eta }_0,\mathrm{{\rm Y}}{\eta }_0)\ge 1$. Define a sequence $\left\{{\eta }_n\right\}$ by ${\eta }_n={\mathrm{{\rm Y}}}^n{\eta }_0=\mathrm{{\rm Y}}{\eta }_{n-1}$. Since $\mathrm{{\rm Y}}$ is $\alpha$-admissible, $\alpha ({\eta }_1,{\eta }_2)=\alpha (\mathrm{{\rm Y}}{\eta }_0,\mathrm{{\rm Y}}{\eta }_1)\ge 1$. Continuing this process, we have

$$\alpha ({\eta }_{n-1},{\eta }_n)\ge 1$$

for all $n\in \mathbb{N}$. According to the triangular approach in assumption (i), one writes that

$$\alpha ({\eta }_m,{\eta }_n)\ge 1\mathrm{for}\mathrm{all}m,n\in \mathbb{N},m\ne n.$$

(3)

Suppose that there is $n_0\in \mathbb{N}$ so that ${\eta }_{n_0}={\eta }_{n_0+1}$. Then, ${\eta }_{n_0}$ is a fixed point of $\mathrm{{\rm Y}}$. Hence, suppose that ${\eta }_n\ne {\eta }_{n+1}$, i.e., ${\nu }_{\lambda }(\mathrm{{\rm Y}}{\eta }_{n-1},\mathrm{{\rm Y}}{\eta }_n)>0$ for all $n\in \mathbb{N}$.

We will show that

$$\underset{n\to \infty }{lim}{\nu }_{\lambda }({\eta }_n,{\eta }_{n+1})=0,$$

(4)

for all $\lambda >0$. Since $\mathrm{{\rm Y}}$ is an $\alpha$-$\vartheta$-$\mathcal{F}$-contraction, we have

$$\begin{array}{cc}\hfill & \mathcal{F}\left({\nu }_{\lambda }({\eta }_n,{\eta }_{n+1})\right)=\mathcal{F}\left({\nu }_{\lambda }(\mathrm{{\rm Y}}{\eta }_{n-1},\mathrm{{\rm Y}}{\eta }_n)\right)\le \vartheta \left(\mathcal{F}({\nu }_{\lambda }({\eta }_{n-1},{\eta }_n)\right),\hfill \end{array}$$

which implies that

$$\mathcal{F}\left({\nu }_{\lambda }({\eta }_n,{\eta }_{n+1})\right)\le {\vartheta }^n\left(\mathcal{F}({\nu }_{\lambda }({\eta }_0,{\eta }_1)\right)<\mathcal{F}({\nu }_{\lambda }({\eta }_0,{\eta }_1).$$

(5)

Taking the limit as $n\to \infty$ in Label (5) and using $({\vartheta }_1)$, we have ${lim}_{n\to \infty }\mathcal{F}\left({\nu }_{\lambda }({\eta }_n,{\eta }_{n+1})\right)=-\infty$. In view of $\mathcal{F}\in \Delta$, we get

$$\underset{n\to \infty }{lim}{\nu }_{\lambda }({\eta }_n,{\eta }_{n+1})=0.$$

Hence, Label (4) is proved for all $\lambda >0$.

Next, we show that $\left\{{\eta }_n\right\}$ is a $\nu$-Cauchy sequence in $\mathrm{\Omega }$, that is, there is some $\lambda >0$ so that ${lim}_{n,m}{\nu }_{\lambda }({\eta }_{m_i},{\eta }_{n_i})=0$.

Suppose there is $\epsilon >0$ for which for all $\lambda >0$, we find $\left\{{\eta }_{m_i}\right\}$ and $\left\{{\eta }_{n_i}\right\}$ of $\left\{{\eta }_n\right\}$ so that $n_i$ is the smallest index corresponding to

$$n_i>m_i>i\mathrm{and}{\nu }_{\lambda }({\eta }_{m_i},{\eta }_{n_i})\ge \epsilon ,$$

(6)

for all $\lambda >0.$ This means that

$${\nu }_{\lambda }({\eta }_{m_i},{\eta }_{n_i-1})<\epsilon .$$

(7)

From Label (6) and using the modular inequality, we get

$$\epsilon \le {\eta }_{4\lambda }({\eta }_{m_i},{\eta }_{n_i})\le s{\eta }_{2\lambda }({\eta }_{m_i},{\eta }_{m_i+1})+s[s{\nu }_{\lambda }({\eta }_{m_i+1},{\eta }_{n_i+1})+s{\nu }_{\lambda }({\eta }_{n_i+1},{\eta }_{n_i})].$$

Letting $i\to \infty$ and using Label (4), we get

$$\frac{\epsilon }{s^2}\le \underset{i\to \infty }{lim\; sup}{\nu }_{\lambda }({\eta }_{m_i+1},{\eta }_{n_i+1}).$$

(8)

We also have

$${\nu }_{\lambda }({\eta }_{m_i},{\eta }_{n_i})\le s{\nu }_{\frac{\lambda }{2}}({\eta }_{m_i},{\eta }_{n_i-1})+s{\nu }_{\frac{\lambda }{2}}({\eta }_{n_i-1},{\eta }_{n_i}).$$

Then, from (4) and (7), we get that

$$\underset{i\to \infty }{lim\; sup}{\nu }_{\lambda }({\eta }_{m_i},{\eta }_{n_i})\le s\epsilon .$$

(9)

Because of (3), we can apply (13) to conclude that

$$\begin{array}{cc}\hfill \mathcal{F}(s^3·{\nu }_{\lambda }({\eta }_{m_i+1},{\eta }_{n_i+1}))& =\mathcal{F}(s^2·{\nu }_{\lambda }(\mathrm{{\rm Y}}{\eta }_{m_i},\mathrm{{\rm Y}}{\eta }_{n_i}))\hfill \\ \hfill & \le \vartheta (\mathcal{F}({\nu }_{\lambda }({\eta }_{m_i},{\eta }_{n_i}))).\hfill \end{array}$$

(10)

Now, taking $i\to \infty$ in (10) and using (${\mathcal{F}}_1$), (8) and (9), we have

$$\begin{array}{cc}\hfill \mathcal{F}\left(s\epsilon \right)=\mathcal{F}\left(s^3·\frac{\epsilon }{s^2}\right)& \le \mathcal{F}(s^3·\underset{i\to \infty }{lim\; sup}{\nu }_{\lambda }({\eta }_{m_i+1},{\eta }_{n_i+1}))\hfill \\ \hfill & \le \vartheta (\mathcal{F}(\underset{i\to \infty }{lim\; sup}{\nu }_{\lambda }({\eta }_{m_i},{\eta }_{n_i})))\hfill \\ \hfill & \le \vartheta (\mathcal{F}\left(s\epsilon \right))<\mathcal{F}\left(s\epsilon \right),\hfill \end{array}$$

which is a contradiction due to the property $({\vartheta }_2)$.

Thus, $\left\{{\eta }_n\right\}$ is $\nu$-Cauchy in the MbMS $(\mathrm{\Omega },{\nu }_{\lambda })$, that is, $\alpha -\nu$-complete, so since

$$\alpha ({\eta }_{n-1},{\eta }_n)\ge 1$$

for all $n\in \mathbb{N}$, the sequence $\left\{{\eta }_n\right\}$ is $\nu$-convergent to some $z\in \mathrm{\Omega }$. Thus, there is some $\lambda >0$ (without loss of generality, we choose $\lambda =\frac{\omega }{2}>0$, where $\omega$ was given to ensure the regularity of $(\mathrm{\Omega },\nu )$), so that ${\displaystyle \underset{n\to \infty }{lim}{\nu }_{\lambda }({\eta }_n,z)=:\underset{n\to \infty }{lim}{\nu }_{\frac{\omega }{2}}({\eta }_n,z)=0}$.

If $z\ne \mathrm{{\rm Y}}z$, then, using Lemma 1 and the $\alpha -\nu$-continuity of $\mathrm{{\rm Y}}$, we have

$${\nu }_{\omega }(z,\mathrm{{\rm Y}}z)\le s[{\nu }_{\frac{\omega }{2}}(z,\mathrm{{\rm Y}}{\eta }_n)+{\nu }_{\frac{\omega }{2}}(\mathrm{{\rm Y}}{\eta }_n,\mathrm{{\rm Y}}z)].$$

Taking the limit as $n\to \infty$ and using again $\alpha -\nu$-continuity of $\mathrm{{\rm Y}}$, we get that the the right-hand side goes to 0, so ${\nu }_{\omega }(z,\mathrm{{\rm Y}}z)=0$. Using regularity of $(\mathrm{\Omega },\nu )$, we obtain that $z=\mathrm{{\rm Y}}z$.

Let $\rho ,\varrho \in \mathcal{F}ix(T)$ where $\rho \ne \varrho$ and $\alpha (\rho ,\varrho )\ge 1$. We have

$$\mathcal{F}\left({\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho )\right)\le \vartheta \left(\mathcal{F}\left({\nu }_{\lambda }(\rho ,\varrho )\right)\right)<\mathcal{F}\left({\nu }_{\lambda }(\rho ,\varrho )\right)).$$

It is a contradiction. We deduce that $\rho =\varrho$. Therefore, $\mathrm{{\rm Y}}$ has a unique fixed point. □

An MbMS $(\mathrm{\Omega },\nu )$ is said to have the $\alpha -\nu$-sequential limit comparison property if, for each sequence $\left\{{\eta }_n\right\}$ in $\mathrm{\Omega }$ so that $\alpha ({\eta }_n,{\eta }_{n+1})\ge 1$ and $\nu$-converges to $\rho \in \mathrm{\Omega }$, one has $\alpha ({\eta }_n,\rho )\ge 1$ for all $n\in \mathbb{N}$.

Theorem 2.

Let $\alpha :\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty )$ be a function and let $(\mathrm{\Omega },\nu )$ be an $\alpha -{\nu }_{\lambda }$-complete MbMS. Let $\mathrm{{\rm Y}}:\mathrm{\Omega }\to \mathrm{\Omega }$ satisfy the following conditions:

(i) 

Υ is triangular α-admissible;

(ii) 

Υ is an $\alpha -\vartheta$-$\mathcal{F}$-contraction;

(iii) 

there is ${\eta }_0\in \mathrm{\Omega }$ such that $\alpha ({\eta }_0,\mathrm{{\rm Y}}{\eta }_0)\ge 1$;

(iv) 

$(\mathrm{\Omega },\nu )$ enjoys the $\alpha -\nu$-sequential limit comparison property.

Then, Υ has a fixed point. Furthermore, this fixed point is unique provided that $\alpha (\rho ,\varrho )\ge 1$ for all $\rho ,\varrho \in Fix(\mathrm{{\rm Y}})$.

Proof. 

Let ${\eta }_0\in \mathrm{\Omega }$ be so that $\alpha ({\eta }_0,\mathrm{{\rm Y}}{\eta }_0)\ge 1$. As in the proof of Theorem 1, one has

$$\alpha ({\eta }_n,{\eta }_{n+1})\ge 1\mathrm{and}{\eta }_n\to {\rho }^{\ast }\mathrm{as}n\to \infty ,$$

where ${\eta }_{n+1}=\mathrm{{\rm Y}}{\eta }_n$. Thus,

$$\alpha ({\eta }_{n+1},{\rho }^{\ast })\ge 1$$

for all $n\in \mathbb{N}$. For $\lambda =\omega >0$, recall that $(\mathrm{\Omega },\nu )$ is regular. By (13), we find that

$$\mathcal{F}\left({\nu }_{\omega }(\mathrm{{\rm Y}}{\eta }_n,\mathrm{{\rm Y}}{\rho }^{\ast })\right)\le \vartheta \left(\mathcal{F}\left({\nu }_{\omega }({\eta }_n,{\rho }^{\ast })\right)\right),$$

which implies

$$\mathcal{F}\left({\nu }_{\omega }(\mathrm{{\rm Y}}{\eta }_n,\mathrm{{\rm Y}}{\rho }^{\ast })\right)\le \mathcal{F}\left({\nu }_{\omega }({\eta }_n,{\rho }^{\ast })\right).$$

Using (${\mathcal{F}}_1$), we have

$${\nu }_{\omega }({\eta }_{n+1},\mathrm{{\rm Y}}{\rho }^{\ast })\le {\nu }_{\omega }({\eta }_n,{\rho }^{\ast }).$$

Suppose that ${\rho }^{\ast }\ne \mathrm{{\rm Y}}{\rho }^{\ast }$. By Lemma 1, we have

$$\frac{1}{s}{\nu }_{\omega }({\rho }^{\ast },\mathrm{{\rm Y}}{\rho }^{\ast })\le \underset{n\to \infty }{lim\; sup}{\nu }_{\omega }({\eta }_{n+1},\mathrm{{\rm Y}}{\rho }^{\ast })\le \underset{n\to \infty }{lim\; sup}{\nu }_{\omega }({\nu }_n,{\rho }^{\ast })=0.$$

Thus, ${\nu }_{\omega }({\rho }^{\ast },\mathrm{{\rm Y}}{\rho }^{\ast })=0$. The regularity of $(\mathrm{\Omega },\nu )$ implies that ${\rho }^{\ast }=\mathrm{{\rm Y}}{\rho }^{\ast }$. Thus, ${\rho }^{\ast }$ is a fixed point of $\mathrm{{\rm Y}}$. Its uniqueness comes as in Theorem 1. □

Taking $\vartheta (t)=t-\tau$ ($\tau >0$), an extension of Wardowski’s result (Theorem 2.1 [30]) to the class of an MbMS is as follows.

Corollary 1.

Let $(\mathrm{\Omega },\nu )$ be an $\alpha -\nu$-complete MbMS and $\mathrm{{\rm Y}}:\mathrm{\Omega }\to \mathrm{\Omega }$ be a self-mapping. Suppose that the following inequality:

$$\tau +\mathcal{F}\left(s^3·{\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho )\right)\le \mathcal{F}\left({\nu }_{\lambda }(\rho ,\varrho )\right)$$

holds for all $\rho ,\varrho \in \mathrm{\Omega }$ with ${\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho )>0$, where $\tau >0$. Then, Υ has a fixed point, if it satisfies the following conditions:

(iii) 

Υ is $\alpha -\nu$-continuous, or

(iii${}^{\prime }$) 

$(\mathrm{\Omega },\nu )$ enjoys the $\alpha -\nu$-sequential limit comparison property.

By considering various functions $\mathcal{F}\in \Delta$ given in [30] and $\vartheta \in \mathrm{\Theta }$, other results could be derived.

We present an illustrative example.

Example 4.

Let $\mathrm{\Omega }=[0,\infty )$. Take the modular b-metric

$${\nu }_{\lambda }(\rho ,\varrho )=\left\{\begin{array}{cc}\frac{{({\rho }^2+{\varrho }^2)}^2}{\lambda },\hfill & \mathrm{if}\rho \ne \varrho ,\hfill \\ 0,\hfill & \mathrm{if}\rho =\varrho ,\hfill \end{array}\right.$$

for all $\rho ,\varrho \in \mathrm{\Omega }$ and $\lambda >0$. Define $\mathrm{{\rm Y}}:\mathrm{\Omega }\to \mathrm{\Omega }$, $\alpha :\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty )$, $\vartheta :\mathbb{R}\to \mathbb{R}$ and $\mathcal{F}:(0,\infty )\to \mathbb{R}$ by

$$\mathrm{{\rm Y}}\rho =\left\{\begin{array}{cc}2{\rho }^2+1,\hfill & \mathrm{if}\rho \in [0,0.2),\hfill \\ \frac{1}{4}{\rho }^2,\hfill & \mathrm{if}\rho \in [0.2,1],\hfill \\ 3\rho -1,\hfill & \mathrm{if}\rho \in (1,2),\hfill \\ 6{\rho }^{10}\hfill & \mathrm{if}\rho \in [2,\infty ),\hfill \end{array}\right.$$

$$\alpha (\rho ,\varrho )=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\rho ,\varrho \in [0.2,1],\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

$$\vartheta (\rho )=\left\{\begin{array}{cc}{\rho }^3-1,\hfill & \mathrm{if}\rho \in (-\infty ,1],\hfill \\ \rho -1,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

and $\mathcal{F}(t)=-\frac{1}{t}+t$, respectively.

Note that $(\mathrm{\Omega },\nu )$ is a modular b-metric space with $s=2.$ Here, $\mathrm{{\rm Y}}$ is triangular $\alpha$-admissible.

Let $\left\{{\eta }_n\right\}$ be in $\mathrm{\Omega }$ so that $\alpha ({\eta }_n,{\eta }_{n+1})\ge 1$ with ${\eta }_n\to \rho$ as $n\to \infty$, then ${\eta }_n\in [0.2,1]$ for all $n\in \mathbb{N}$. Thus, $\rho \in [0.2,1]$. This yields that $\alpha ({\eta }_n,\rho )\ge 1$ for all $n\in \mathbb{N}.$ In addition, $\alpha (1,\mathrm{{\rm Y}}1)\ge 1$. Moreover, $-\frac{1}{\frac{8\rho }{256}}+\frac{8\rho }{256}\le {(-\frac{1}{\rho }+\rho )}^3-1$ for all $\rho \ge 0.2$. Now, for all $\rho ,\varrho$ with $\alpha (\rho ,\varrho )\ge 1$, we have

$$\begin{array}{cc}\hfill \mathcal{F}(s^3{\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho ))& =-\frac{1}{s^3{\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho )}+s^3{\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho )\hfill \\ & =-\frac{1}{8\frac{{(\mathrm{{\rm Y}}{\rho }^2+\mathrm{{\rm Y}}{\varrho }^2)}^2}{\lambda }}+8\frac{{(\mathrm{{\rm Y}}{\rho }^2+\mathrm{{\rm Y}}{\varrho }^2)}^2}{\lambda }\hfill \\ & =-\frac{1}{8\frac{{({(\frac{1}{4}{\rho }^2)}^2+{(\frac{1}{4}{\varrho }^2)}^2)}^2}{\lambda }}+8\frac{{({(\frac{1}{4}{\rho }^2)}^2+{(\frac{1}{4}{\varrho }^2)}^2)}^2}{\lambda }\hfill \\ & \le -\frac{1}{8\frac{{({(\frac{1}{4}\rho )}^2+{(\frac{1}{4}\varrho )}^2)}^2}{\lambda }}+8\frac{{({(\frac{1}{4}\rho )}^2+{(\frac{1}{4}\varrho )}^2)}^2}{\lambda }\hfill \\ & \le -\frac{1}{\frac{8\frac{{({\rho }^2+{\varrho }^2)}^2}{\lambda }}{256}}+\frac{8\frac{{({\rho }^2+{\varrho }^2)}^2}{\lambda }}{256}\hfill \\ & \le -\frac{1}{\frac{8{\nu }_{\lambda }(\rho ,\varrho )}{256}}+\frac{8{\nu }_{\lambda }(\rho ,\varrho )}{256}\hfill \\ & \le {[-\frac{1}{{\nu }_{\lambda }(\rho ,\varrho )}+{\nu }_{\lambda }(\rho ,\varrho )]}^3-1\hfill \\ & =\vartheta (\mathcal{F}({\nu }_{\lambda }(\rho ,\varrho ))).\hfill \end{array}$$

Thus, $\mathrm{{\rm Y}}$ is an $\alpha -\vartheta$-$\mathcal{F}$ contraction. All the hypotheses of Theorem 2 are verified, so $\mathrm{{\rm Y}}$ has a fixed point.

A self-mapping $\mathrm{{\rm Y}}$ has the property P if $Fix({\mathrm{{\rm Y}}}^n)=Fix(\mathrm{{\rm Y}})$ for all $n\in \mathbb{N}$.

Theorem 3.

Let $(\mathrm{\Omega },\nu )$ be an MbMS and $\mathrm{{\rm Y}}:\mathrm{\Omega }\to \mathrm{\Omega }$ be an $\alpha -\nu$-continuous self-mapping. Assume that there are $\vartheta \in \mathrm{\Theta }$ and $\mathcal{F}\in \Delta$ such that

$$\begin{array}{c}\hfill \begin{array}{c}\mathcal{F}\left(s^3{\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,{\mathrm{{\rm Y}}}^2\rho )\right)\le \vartheta (\mathcal{F}\left({\nu }_{\lambda }(\rho ,\mathrm{{\rm Y}}\rho )\right))\hfill \end{array}\end{array}$$

(11)

for all $\rho \in \mathrm{\Omega }$ with ${\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,{\mathrm{{\rm Y}}}^2\rho )>0$. If Υ is α-admissible and there exists ${\eta }_0\in \mathrm{\Omega }$ in order that $\alpha ({\eta }_0,\mathrm{{\rm Y}}{\eta }_0)\ge 1$, then Υ has the property P.

Proof. 

Let $n>1.$ Assume contrarily that $w\in Fix({\mathrm{{\rm Y}}}^n)$ and $w\notin Fix(\mathrm{{\rm Y}})$. Then, ${\nu }_{{\lambda }_0}(w,\mathrm{{\rm Y}}w)>0$ for some ${\lambda }_0>0$. Now, we have

$$\begin{array}{ccc}\hfill \mathcal{F}({\nu }_{{\lambda }_0}(w,\mathrm{{\rm Y}}w))& =& \mathcal{F}({\nu }_{{\lambda }_0}(\mathrm{{\rm Y}}({\mathrm{{\rm Y}}}^{n-1}w)),{\mathrm{{\rm Y}}}^2({\mathrm{{\rm Y}}}^{n-1}w)))\hfill \\ & \le & \vartheta (\mathcal{F}({\nu }_{{\lambda }_0}({\mathrm{{\rm Y}}}^{n-1}w),{\mathrm{{\rm Y}}}^{n}w)))\hfill \\ & \le & {\vartheta }^2(\mathcal{F}({\nu }_{{\lambda }_0}({\mathrm{{\rm Y}}}^{n-2}w),{\mathrm{{\rm Y}}}^{n-1}w)))\le \cdots \hfill \\ & \le & {\vartheta }^n({\nu }_{{\lambda }_0}(w,\mathrm{{\rm Y}}w)).\hfill \end{array}$$

Taking the limit as $n\to \infty$ in the above inequality, we have $\mathcal{F}({\nu }_{{\lambda }_0}(w,\mathrm{{\rm Y}}w))=-\infty$. Hence, by (${\mathcal{F}}_2$), we get ${\nu }_{{\lambda }_0}(w,\mathrm{{\rm Y}}w)=0$, which is a contradiction. Therefore, $Fix({\mathrm{{\rm Y}}}^n)=Fix(\mathrm{{\rm Y}})$ for all $n\in \mathbb{N}$. □

3. Results in Ordered Modular b-Metric Spaces

Let $(\mathrm{\Omega },\nu ,⪯)$ be a partially ordered MbMS and let $\mathrm{{\rm Y}}$ be a self-mapping on $\mathrm{\Omega }$.

Definition 11.

$(i)$$(\mathrm{\Omega },\nu )$ is said to be $⪯-\nu$-complete if every ν-Cauchy sequence $\left\{{\rho }_n\right\}$ in Ω with ${\rho }_n⪯{\rho }_{n+1}$ for all $n\in \mathbb{N}$, ν-converges in Ω.

$(ii)$ Υ is said to be $⪯-\nu$-continuous on $(\mathrm{\Omega },\nu )$, if, for a ν-convergent sequence $\left\{{\rho }_n\right\}$ to some $\rho \in \mathrm{\Omega }$ so that ${\rho }_n⪯{\rho }_{n+1}$ for all $n\in \mathbb{N}$, we have $\left\{\mathrm{{\rm Y}}{\rho }_n\right\}$ is ν-convergent to $\mathrm{{\rm Y}}\rho$.

$(iii)$$(\mathrm{\Omega },\nu )$ is said to have the $⪯-\nu$-sequential limit comparison property if, for each sequence $\left\{{\eta }_n\right\}$ in Ω so that ${\eta }_n⪯{\eta }_{n+1}$ and is ν-convergent to $\rho \in \mathrm{\Omega }$, one has ${\eta }_n⪯\rho$ for all $n\in \mathbb{N}$.

Now, we say that $\mathrm{{\rm Y}}$ is an ⪯-$\vartheta$-$\mathcal{F}$-contraction if for all $\rho ,\varrho \in \mathrm{\Omega }$ with $\rho ⪯\varrho$ and ${\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho )>0$, we have

$$\mathcal{F}\left(s^3·{\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho )\right)\le \vartheta \left(\mathcal{F}({\nu }_{\lambda }(\rho ,\varrho )\right),$$

(12)

where $\mathcal{F}\in \Delta$ and $\vartheta \in \mathrm{\Theta }$.

Using the above statements and applying Theorem 1, we have the following result.

Theorem 4.

Let $(\mathrm{\Omega },\nu ,⪯)$ be an $⪯-\nu$-complete partially ordered MbMS. Assume that

(i) 

Υ is a $⪯-\vartheta$-$\mathcal{F}$-contraction;

(ii) 

Υ is nondecreasing;

(iii) 

there is ${\eta }_0\in \mathrm{\Omega }$ so that ${\eta }_0⪯\mathrm{{\rm Y}}{\eta }_0;$

(iv) 

either Υ is $⪯-\nu$-continuous, or $(\mathrm{\Omega },\nu ,⪯)$ possesses the $⪯-\nu$-sequential limit comparison property.

Then, Υ has a fixed point.

Again, we apply Theorem 2 to state the following result.

Theorem 5.

Let $(\mathrm{\Omega },\nu ,⪯)$ be an $⪯-\nu$-complete partially ordered MbMS. Assume that

(i) 

the inequality (11) holds for all $\rho \in \mathrm{\Omega }$ with ${\nu }_{\lambda }(\mathrm{{\rm Y}}\rho ,{\mathrm{{\rm Y}}}^2\rho )>0$.

(ii) 

Υ is nondecreasing;

(iii) 

there is ${\eta }_0\in \mathrm{\Omega }$ such that ${\eta }_0⪯\mathrm{{\rm Y}}{\eta }_0;$

(iv) 

either Υ is $⪯-\nu$-continuous, or $(\mathrm{\Omega },\nu ,⪯)$ possesses the $⪯-\nu$-sequential limit comparison property.

Then, Υ has the property P.

4. Applications

In [17], Hussain and Salimi presented the relationship between modular metrics and fuzzy metrics and deduced certain fixed point results in triangular partially ordered fuzzy metric spaces.

Definition 12

([32]). A binary operation $\star :\left[0,1\right]\times [0,1]\to [0,1]$ is called a continuous t-norm if it satisfies the following assertions:

(T1) 

$\star$is commutative and associative;

(T2) 

$\star$is continuous;

(T3) 

$a\star 1=a$ for all $a\in \left[0,1\right];$

(T4) 

$a\star b\le c\star d$ when $a\le c$ and $b\le d$, with $a,b,c,d\in \left[0,1\right]$.

Definition 13.

A 3-tuple $\left(X,M,\ast \right)$ is said to be a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm and M is a fuzzy set on $X^2\times \left(0,\infty \right)$ satisfying the following conditions, for all $x,y,z\in X$ and for all $t,s>0$,

(i) 

$M\left(x,y,t\right)>0$;

(ii) 

$M\left(x,y,t\right)=1$ for all $t>0$ if and only if $x=y$;

(iii) 

$M\left(x,y,t\right)=M\left(y,x,t\right)$;

(iv) 

$M\left(x,y,t\right)\ast M\left(y,z,s\right)\le M\left(x,z,t+s\right)$;

(v) 

$M\left(x,y,.\right):\left(0,\infty \right)\to [0,1]$ is continuous.

The function $M\left(x,y,t\right)$ denotes the degree of nearness between x and y with respect to t.

Definition 14

([33]). A fuzzy b-metric space is an ordered triple $\left(X,B,\star \right)$ such that X is a nonempty set, $\star$ is a continuous t-norm and B is a fuzzy set on $X\times X\times \left(0,\infty \right)$ satisfying the following conditions, for all $x,y,z\in X$ and for all $t,s>0$:

(F1) 

$B\left(x,y,t\right)>0$;

(F2) 

$B\left(x,y,t\right)=1$ if and only if $x=y$;

(F3) 

$B\left(x,y,t\right)=B\left(y,x,t\right)$;

(F4) 

$B\left(x,y,t\right)\star B\left(y,z,s\right)\le B\left(x,z,b\left(t+s\right)\right)$ where $b\ge 1$;

(F5) 

$B\left(x,y,·\right):\left(0,\infty \right)\to \left(0,1\right]$ is left-continuous.

Definition 15

([33]). Let $\left(X,B,\star \right)$ be a fuzzy b-metric space (in short, FbMS). Then,

(i) 

a sequence $\left\{x_n\right\}$ converges to $x\in X$, if and only if ${lim}_{n\to \infty }B\left(x_n,x,t\right)=1$ for all $t>0;$

(ii) 

a sequence $\left\{x_n\right\}$ in X is a Cauchy sequence if and only if, for all $ϵ\in \left(0,1\right)$ and for all $t>0,$ there exists $n_0$ such that $B\left(x_n,x_m,t\right)>1-ϵ$ for all $m,n\ge n_0$;

(iii) 

the fuzzy b-metric space is called complete if every Cauchy sequence converges to some $x\in X$.

Definition 16

([33]). The fuzzy b-metric space $\left(X,B,\ast \right)$ is called triangular whenever

$$\frac{1}{B\left(x,y,t\right)}-1\le s\left[\frac{1}{B\left(x,z,t\right)}-1+\frac{1}{B\left(z,y,t\right)}-1\right]$$

for all $x,y,z\in X$ and for all $t>0.$

Motivated by Lemmas 33 and 34 of [34], we present the following.

Remark 2.

Let $\left(X,B,\ast \right)$ be a triangular fuzzy b-metric space. Define $\nu :X\times X\times \left(0,\infty \right)\to [0,\infty )$ by $\nu \left(x,y,t\right)=s[\frac{1}{B\left(x,y,t\right)}-1]$. Then, ν is a modular b-metric.

In view of Remark 2 and applying the results established in Section 2, we can deduce the following results in fuzzy b-metric spaces.

Definition 17.

Let $\left(\mathrm{\Omega },B,\ast \right)$ be an FbMS and Υ be a self-mapping on Ω. We say that Υ is a fuzzy α-ϑ-$\mathcal{F}$-contraction if, for all $\rho ,\varrho \in \mathrm{\Omega }$ with $\alpha (\rho ,\varrho )\ge 1$ and $B(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho ,t)<1$ $(t>0)$, we have

$$\mathcal{F}\left(\frac{s^4}{B\left(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho ,t\right)}-s^4\right)\le \vartheta \left(\mathcal{F}(\frac{s}{B(\rho ,\varrho ,t}-s)\right),$$

(13)

where $\mathcal{F}\in \Delta$ and $\vartheta \in \mathrm{\Theta }$.

In addition, note that Definition 9 could be derived for FbMS.

Theorem 6.

Let $\alpha :\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty )$ be a function and let $\left(\mathrm{\Omega },B,\ast \right)$ be an α-ν-complete FbMS. Assume that $\mathrm{{\rm Y}}:\mathrm{\Omega }\to \mathrm{\Omega }$ is such that

(i) 

Υ is triangular α-admissible;

(ii) 

Υ is a fuzzy α-ϑ-$\mathcal{F}$-contraction;

(iii) 

there is ${\rho }_0\in \mathrm{\Omega }$ such that $\alpha ({\rho }_0,\mathrm{{\rm Y}}{\rho }_0)\ge 1$;

(iv) 

Υ is $\alpha -B$-continuous.

Then, Υ has a fixed point. In addition, Υ has a unique fixed point, provided that $\alpha (\rho ,\varrho )\ge 1$ for all $\rho ,\varrho \in$ Fix($\mathrm{{\rm Y}}$).

Proof. 

It follows from Theorem 1. □

Theorem 7.

Let $\alpha :\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\infty )$ be a function and let $\left(X,B,\ast \right)$ be an α-ν-complete FbMS. Let $\mathrm{{\rm Y}}:\mathrm{\Omega }\to \mathrm{\Omega }$ satisfy the following conditions:

(i) 

Υ is triangular α-admissible;

(ii) 

Υ is a fuzzy α-ϑ-$\mathcal{F}$-contraction;

(iii) 

there is ${\eta }_0\in \mathrm{\Omega }$ such that $\alpha ({\eta }_0,\mathrm{{\rm Y}}{\eta }_0)\ge 1$;

(iv) 

$\left(\mathrm{\Omega },B,\ast \right)$ enjoys the $\alpha -B$-sequential limit comparison property.

Then, Υ has a fixed point. Furthermore, this fixed point is unique provided that $\alpha (\rho ,\varrho )\ge 1$ for all $\rho ,\varrho \in$ Fix$(\mathrm{{\rm Y}})$.

Proof. 

It follows from Theorem 2. □

Theorem 8.

Let $\left(X,B,\ast \right)$ be an FbMS and $\mathrm{{\rm Y}}:\mathrm{\Omega }\to \mathrm{\Omega }$ be an $\alpha -\nu$-continuous self-mapping. Assume that there are $\vartheta \in \mathrm{\Theta }$ and $\mathcal{F}\in \Delta$ such that

$$\begin{array}{c}\hfill \begin{array}{c}\mathcal{F}\left(\frac{s^4}{B\left(\mathrm{{\rm Y}}\rho ,{\mathrm{{\rm Y}}}^2\rho ,t\right)}-s^4)\right)\le \vartheta \left(\mathcal{F}\left(\frac{s}{B\left(\rho ,\mathrm{{\rm Y}}\rho ,t\right)}-s\right)\right)\hfill \end{array}\end{array}$$

(14)

for all $\rho \in \mathrm{\Omega }$ with $B(\mathrm{{\rm Y}}\rho ,{\mathrm{{\rm Y}}}^2\rho ,t)<1$. If Υ is α-admissible and there exists ${\eta }_0\in \mathrm{\Omega }$ in order that $\alpha ({\eta }_0,\mathrm{{\rm Y}}{\eta }_0)\ge 1$, then Υ has the property P.

Proof. 

It follows from Theorem 3. □

Remark 3.

The analogue of Theorem 4 and Theorem 5 could be derived easily in the context of partially ordered fuzzy b-metric spaces.

Now, we consider the following boundary value problem:

$$\left\{\begin{array}{c}{y}^{″}(x)=f(x,y(x)),x\in [0,1],\hfill \\ y(0)=y(1)=0,\hfill \end{array}\right.$$

where $f:[0,1]\times \mathbb{R}⟶\mathbb{R}$ is a continuous function.

The above equation can be transformed to the following Fredholm integral equation:

$$y(x)=-{\int }_0^{1}K(x,t)f(t,y(t))dt,$$

(15)

where the kernel is given by

$$K(x,t)=\left\{\begin{array}{cc}t(1-x),\hfill & \mathrm{if}t\in [0,x],\hfill \\ x(1-t),\hfill & \mathrm{if}t\in [x,1].\hfill \end{array}\right.$$

See [35] for details.

Now, to give an existence theorem for a solution of (15) that belongs to $X=C(I,\mathbb{R})$ (the set of continuous real functions defined on $I=[0,1]$), note that the space X endowed with the b-metric given by

$$d(x,y)=\underset{t\in I}{max}\mid x(t)-y(t){\mid }^2$$

for all $x,y\in X$ is a b-complete b-metric space ($s=2^{2-1}$).

Take on X the partial order $⪯$ given by

$$x⪯y⟺x(t)\le y(t),$$

for all $x,y\in X$ and $t\in I.$

For $\rho \in X$, define

$${\parallel \rho \parallel }_{\infty }=\underset{t\in I}{sup}|\rho (t)|.$$

Here, $(X,\parallel ·{\parallel }_{\infty })$ is a Banach space. The modular metric induced by this norm is

$${\mu }_{\lambda }(\rho ,\varrho )={\displaystyle \frac{{\parallel \rho -\varrho \parallel }_{\infty }}{\lambda }}=\underset{t\in I}{max}{\displaystyle \frac{|\rho (t)-\varrho (t)|}{\lambda }},\lambda >0,$$

for all $\rho ,\varrho \in X$. We consider the modular b-metric $\nu$ given by

$${\nu }_{\lambda }(\rho ,\varrho )={\displaystyle \frac{{\parallel \rho -\varrho \parallel }_{\infty }^2}{{\lambda }^2}}=\underset{t\in I}{max}{\displaystyle \frac{{|\rho (t)-\varrho (t)}^2}{{\lambda }^2}}.$$

Define $\mathrm{{\rm Y}}:X\to X$ by

$$\mathrm{{\rm Y}}\rho (x)=-{\int }_0^{1}K(x,t)f(t,\rho (t))dt,\rho \in X,x\in I.$$

Clearly, a function $u\in X$ is a solution of (15) if and only if it is a fixed point of $\mathrm{{\rm Y}}$.

Consider the following assumptions:

($C1$)

For all $u,v\in \mathbb{R}$ with $u⪯v$ and for all $t\in I$,

$$|f(t,u)-f(t,v){|}^2\le \frac{{\parallel u-v\parallel }_{\infty }^2}{8}.$$

($C2$)

There is ${\eta }_0:I\to \mathbb{R}$ so that

$${\eta }_0(x)\le -{\int }_0^{1}K(x,t)f(t,{\eta }_0(t))dt,x\in I.$$

($C3$)

$f(t,.):\mathbb{R}⟶\mathbb{R}$ is nonincreasing for all $t\in [0,1]$.

Theorem 9.

Assume that above assumptions $(C1)-(C3)$ hold. Then, Equation (15) has a solution in X.

Proof. 

First, by assumption $(C2)$, we have ${\eta }_0⪯\mathrm{{\rm Y}}{\eta }_0$. Clearly, $\mathrm{{\rm Y}}$ is $⪯-\nu$-continuous and nondecreasing.

To show that all the assumptions of Theorem 4 are satisfied, it remains to prove that $\mathrm{{\rm Y}}$ is an $⪯-\vartheta$-$\mathcal{F}$-contraction. Let $\rho ,\varrho \in X$ with $\rho ⪯\varrho$. For each $x\in I$, we have

$$\begin{array}{cc}\hfill & |\mathrm{{\rm Y}}\rho (x)-\mathrm{{\rm Y}}\varrho (x){|}^2=|{\int }_0^{1}K(x,t)f(t,\rho (t))dt-{\int }_0^{1}K(x,t)f(t,\varrho (t))dt{|}^2\hfill \\ \hfill & ={\int }_0^1(K(x,t)|f(t,\rho (t))-{f(t,\varrho (t))|)}^{2}dt\hfill \\ \hfill & \le \left[{\int }_0^1{|K(x,t)|}^{2}dt\right]\left[{\int }_0^1{|f(t,\rho (t))dt-f(t,\varrho (t))|}^{2}dt\right]\hfill \\ \hfill & \le \left[{\int }_0^1{|K(x,t)|}^{2}dt\right]{\int }_0^1\frac{||\rho (t)-{\varrho (t)||}_{\infty }^2}{8}dt\hfill \\ \hfill & \le \left[{\int }_0^1{|K(x,t)|}^{2}dt\right]\frac{||\rho -{\varrho ||}_{\infty }^2}{8}.\hfill \end{array}$$

Via a careful calculation, we get that

$$\begin{array}{c}\hfill {\int }_0^1{|K(x,t)|}^{2}dt=\frac{{(1-x)}^2{x}^3+x^2{(1-x)}^3}{3},x\in [0,1].\end{array}$$

We obtain that

$$|\mathrm{{\rm Y}}\rho (x)-\mathrm{{\rm Y}}\varrho (x){|}^2\le [\frac{{(1-x)}^2{x}^3}{3}+\frac{x^2{(1-x)}^3}{3}]\frac{||\rho -{\varrho ||}_{\infty }^2}{8}.$$

(16)

Taking the supremum on $x\in [0,1]$, we deduce that

$$\begin{array}{c}\hfill |\mathrm{{\rm Y}}\rho -\mathrm{{\rm Y}}\varrho {|}_{\infty }^2\le \frac{21}{1000}\frac{||\rho -{\varrho ||}_{\infty }^2}{8}.\end{array}$$

Now, one writes

$$\begin{array}{cc}\hfill ln(\frac{s^3|\mathrm{{\rm Y}}\rho -\mathrm{{\rm Y}}\varrho {|}_{\infty }^2}{{\lambda }^2})& =ln(\frac{8|\mathrm{{\rm Y}}\rho -\mathrm{{\rm Y}}\varrho {|}_{\infty }^2}{{\lambda }^2})\hfill \\ \hfill & \le ln(\frac{21}{1000})+ln(\frac{{\parallel \rho -\varrho \parallel }_{\infty }^2}{{\lambda }^2})\hfill \\ \hfill & \le ln(\frac{21}{1000})+ln({\nu }_{\lambda }(\rho ,\varrho )).\hfill \end{array}$$

That is,

$$\mathcal{F}\left(s^3·{\eta }_{}(\mathrm{{\rm Y}}\rho ,\mathrm{{\rm Y}}\varrho )\right)\le \vartheta \left(\mathcal{F}({\nu }_{\lambda }(\rho ,\varrho )\right),$$

(17)

where $\mathcal{F}(t)=lnt$ and $\vartheta (t)=t-\delta$ with $\delta =-ln(\frac{21}{1000})>0$ (Example 3). Thus, all the hypotheses of Theorem 4 are fulfilled and we deduce the existence of $u\in X$ such that $u=\mathrm{{\rm Y}}u$. □

5. Conclusions

We presented some fixed point results for generalized $\mathcal{F}$-contractions in the setting of modular b-metric spaces. We also established some related results in fuzzy b-metric spaces. At the end, we resolved a Fredholm type integral equation.