On fixed point results in $\mathcal{F}$-metric spaces with applications

1.   Introduction

The theory of fixed points is investigated to be the utmost fascinating and dynamic field of research in the progress of mathematical analysis. In this way, the conception of metric space ^[1] is one of the basic parts of mathematical sciences. Because of its outstanding and extraordinary improvement in various fields, it has been extended and generalized in different directions.

Recently, many compulsive generalizations (or extensions) of the concept of metric space came into sight. The famous extensions of the concept of metric spaces have been done by Bakhtin ^[2] which was formally defined by Czerwik ^[3] in 1993. Czerwik ^[3] gave the idea of $b$-metric space which broadens the notion of metric space by improving the triangle equality metric axiom by putting a constant $s\geq 1$ multiplied to the right-hand side of the inequality, and is one of the enormously applied generalizations for metric spaces. Berinde et al. ^[4] gave a brief survey on the development of fixed point theory, specially on $b$-metric spaces and discussed some important related aspects of it. Brzdek ^[5] proved and discussed some fixed point results for nonlinear operators, acting on some classes of functions with values in a $b$-metric space. Paluszyński et al. ^[6] discussed quasi-metric space as an extension of classical metric space and its involvement in a part of harmonic analysis related to the theory of spaces of homogeneous type. Khamsi et al. ^[7] reintroduced the notion of $b$-metric space with the name metric-type and proved some theorems in this recently introduced space. In ^[8], Branciari gave the concept of rectangular metric space and generalized the classical metric space by replacing the triangle inequality with more general inequality that is called rectangular inequality. This rectangular inequality consists of a distance between four points. In 2018, Jleli et al. ^[9] introduced a fascinating generalization of classical metric space, $b$-metric space and rectangular metric space that is famous as an $\mathcal{F}$-metric space. Subsequently, Al-Mazrooei et al. ^[10] used the notion of $\mathcal{F}$-metric space and proved some results for rational inequality that includes some non-negative constants.

On the other hand, Samet et al. ^[11] defined the notion of $\alpha$ -admissibility and ($\alpha$-$\psi)$-contraction in the background of metric spaces and proved some results for these contractions. Subsequently, Asl et al. ^[12] generalized the above concept of $\alpha$ -admissibility and gave the notion of $\alpha ^{\ast }$-admissible mappings and established fixed point results for multivalued mappings in 2013. Recently, Hussain et al. ^[13] defined the notion of Ćirić type ($\alpha$-$\psi)$-contraction in the framework of $\mathcal{F}$-metric space and proved fixed point theorems.

In this article, we establish common fixed point results for locally rational contractions concerning control functions of one variable in the background of $\mathcal{F}$-metric spaces. We also establish fixed points of ($\alpha ^{\ast }$-$\psi)$-contractive and generalized ($\alpha ^{\ast }$, $\psi, \delta _{\mathcal{F}})$-contractive multifunctions. An important example is also included to display the originality of our principal result.

2.   Preliminaries

Czerwik ^[3] gave the concept of $b$-metric space in this manner.

Definition 1. (^[3]) Let $\Theta \not = \emptyset$ and $s\geq 1$ be a constant. A function $\kappa :\Theta \times \Theta \rightarrow \; \mathbb{[}0, \infty \mathbb{)}$ is called a $b$-metric if the following assertions hold:

$(b1)\ \kappa (\rho, \hbar)\geq 0$ and $\kappa (\rho, \hbar) = 0$ if and only if $\rho = \hbar,$

$(b2)\ \kappa (\rho, \hbar) = \kappa (\hbar, \rho),$

$(b3) \; \kappa (\rho, \mathfrak{\varphi })\leq s[\kappa (\rho, \hbar)+\kappa (\hbar, \mathfrak{\varphi })],$

for all $\rho, \hbar, \mathfrak{\varphi }\in \Theta.$

Then the pair $(\Theta, \kappa)$ is known as a $b$-metric space.

Jleli et al. ^[9] gave the following notion of $\mathcal{F}$-metric space in this way.

Let $\mathcal{F}$ be the family of continuous functions $f:(0, +\infty)\rightarrow \mathbb{R}$ satisfying

($\mathcal{F}_{1}$) $f$ is non-decreasing,

($\mathcal{F}_{2}$) for each $\{ \rho _{\jmath }\} \subseteq \mathbb{R} ^{+}$, $\lim_{\jmath \rightarrow \infty }\rho _{\jmath } = 0$ if and only if $\lim_{\jmath \rightarrow \infty }f(\rho _{\jmath }) = -\infty.$

Definition 2. (^[9]) Let $\Theta \not = \emptyset$ and $\kappa :\Theta \times \Theta \rightarrow \lbrack 0, +\infty)$ be a function satisfying the following conditions

(D$_{1}$) $(\rho, \hbar)\in \Theta \times \Theta$, $\kappa (\rho, \hbar) = 0$ if and only if $\rho = \hbar$,

(D$_{2}$) $\kappa (\rho, \hbar) = \kappa (\hbar, \rho)$, for all $(\rho, \hbar)\in \Theta \times \Theta,$

(D$_{3}$) for every $(\rho, \hbar)\in \Theta \times \Theta$ and $(\rho _{i})_{i = 1}^{N}\subset \Theta$ with

there exists $(f, \mathfrak{h})\in \mathcal{F}\times \lbrack 0, +\infty)$ such that

for all $N\in \mathbb{N}$ and $N\geq 2$.

Then $(\Theta, \kappa)$ is called an $\mathcal{F}$-metric space.

Example 1. Let $\Theta = \mathbb{R}$ and $\kappa :\Theta \times \Theta \rightarrow \lbrack 0, +\infty)$ be defined by

with $f(\iota) = \ln (\iota)$ and $\mathfrak{h} = \ln (2)$, then ($\Theta, \kappa \mathfrak{)}$ is $\mathcal{F}$-metric space.

Definition 3. (^[9]) Let $(\Theta, \kappa)$ be $\mathcal{F}$-metric space,

(i) a sequence $\{ \rho _{\jmath }\}$ in $\Theta$ is said to be $\mathcal{F}$-convergent to $\rho \in \Theta$ if $\{ \rho _{\jmath }\}$ is convergent to $\rho$ with regard to the $\mathcal{F}$-metric $\kappa$;

(ii) a sequence $\{ \rho _{\jmath }\}$ is $\mathcal{F}$-Cauchy, if

(iii) if every $\mathcal{F}$-Cauchy sequence in $\mathcal{F}$-metric space $(\Theta, \kappa)$ is $\mathcal{F}$-convergent to an element of $\Theta$, then $(\Theta, \kappa)$ is said to be $\mathcal{F}$-complete.

Theorem 1. (^[9]) Let $(\Theta, \kappa)$ be $\mathcal{F}$-complete $\mathcal{F}$ -metric space and $\mathfrak{L}:\Theta \rightarrow \Theta$. Assume that there exists $\alpha \in \lbrack 0, 1)$ such that

for all $\rho, \hbar \in \Theta,$ then $\mathfrak{L}$ has a unique fixed point $\rho ^{\ast }\in \Theta$. Moreover, for any $\rho _{0}\in \Theta$, the sequence $\{ \rho _{\jmath }\} \subset \Theta$ defined by

is $\mathcal{F}$-convergent to $\rho ^{\ast }$.

Subsequently, Hussain et al. ^[13] defined $\alpha$-$\psi$-contraction in the background of $\mathcal{F}$-metric spaces and generalized the main result of Jleli et al. ^[9]. Later on, Ahmad et al. ^[10] defined a rational contraction in $\mathcal{F}$-metric space and proved the following result as generalization of main theorem of Jleli et al. ^[9].

Theorem 2. (^[10]) Let $(\Theta, \kappa)$ be an $\mathcal{F}$-complete $\mathcal{F }$-metric space and $\mathfrak{L}:\Theta \rightarrow \Theta$. Suppose that there exists $\alpha, \beta \in \lbrack 0, 1)$ such that

for all $\rho, \hbar \in \Theta,$ then $\mathfrak{L}$ has a unique fixed point.

For more details in the direction of metric space, $b$-metric space and $\mathcal{F}$-metric space, we refer the researchers ^{[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]}.

3.   Main results

We state our main result in this way.

Theorem 3. Let $\left(\Theta, \kappa \right)$ be an $\mathcal{F}$-complete $\mathcal{F}$-metric space and $\mathfrak{L}_{1}, \mathfrak{L}_{2}:\Theta \; \rightarrow \Theta$. If there exist $r > 0$ and mappings $\alpha, \beta :\Theta \mathfrak{\rightarrow }[0, 1)$ such that

(a) $\alpha \left(\mathfrak{L}_{1}\rho \right) \leq \alpha \left(\rho \right)$ and $\alpha \left(\mathfrak{L}_{2}\rho \right) \leq \alpha \left(\rho \right)$ $\ \ \ \ \ \beta \left(\mathfrak{L}_{1}\rho \right) \leq \beta \left(\rho \right)$ and $\beta \left(\mathfrak{L}_{2}\rho \right) \leq \beta \left(\rho \right),$

(b) $\alpha \left(\rho \right) +\beta \left(\rho \right) < 1,$

(c)

for all $\rho _{0}, \rho, \hbar \in \overline{B(\rho _{0}, r)}$ and

where $\lambda = \frac{\alpha \left(\rho _{0}\right) }{1-\beta \left(\rho _{0}\right) } < 1$, then there exists a unique point $\rho ^{\ast }\in \overline{B(\rho _{0}, r)}$ such that $\mathfrak{L}_{1}\rho ^{\ast } = \mathfrak{L}_{2}\rho ^{\ast } = \rho ^{\ast }$.

Proof. For $\rho _{0}\in \overline{B(\rho _{0}, r)}$, define the sequence {$\rho _{\jmath }$} by

for all $\jmath = 0, 1, 2, ...$ By inequality (3.2), we have

that is, $\rho _{1}\in \overline{B(\rho _{0}, r)}.$ Assume that $\rho _{2}, \rho _{3}, ...\rho _{\jmath }\in \overline{B(\rho _{0}, r)}$ for some $j\in \mathbb{N}.$ Now if, $2k+1\leq \jmath,$ then by inequality (3.1), we have

By the sequence

and hence

Similarly, if $2k\leq \jmath,$ we deduce

Thus by inequalities (3.3) and (3.4), we have

Thus by inequalities (3.5) and (3.6), we have

for some $\jmath \in \mathbb{N}.$ Now

By ($\mathcal{F}_{1}$), we get $\rho _{\jmath +1}\in \overline{B(\rho _{0}, r) }.$ Thus $\rho _{\jmath }\in \overline{B(\rho _{0}, r), }$ for all $\jmath \in \mathbb{N}.$ Then it follows that

for all $\jmath \in \mathbb{N}.$ Let $(f, \mathfrak{h})\in \mathcal{F}\times \lbrack 0, +\infty)$ be such that (D$_{3}$) is satisfied. Let $\epsilon > 0$ be fixed. By ($\mathcal{F}_{2}$), there exists $\delta > 0$ such that

Hence, by (3.7), ($\mathcal{F}_{1})$ and ($\mathcal{F}_{2}$), we have

for $m > \jmath \geq \jmath (\epsilon).$ Using (D$_{3}$) and (3.8), we obtain $\kappa (\rho _{\jmath }, \rho _{m}) > 0, \; m > \jmath \geq \jmath (\epsilon)$ implies

which yields by ($\mathcal{F}_{1}$) that $\kappa (\rho _{\jmath }, \rho _{m}) < \epsilon, \; m > \jmath \geq \jmath (\epsilon).$ It shows that $\{ \rho _{\jmath }\}$ is $\mathcal{F}$-Cauchy sequence in $\overline{B(\rho _{0}, r)}$. Now, $\overline{B(\rho _{0}, r)}$ is $\mathcal{F}$-complete since $\overline{B(\rho _{0}, r)}$ is $\mathcal{F}$-closed in $\Theta,$ so there exists $\rho ^{\ast }\in \overline{B(\rho _{0}, r)}$ such that the sequence $\{ \rho _{\jmath }\}$ is $\mathcal{F}$-convergent to $\rho ^{\ast }$, i.e.,

Now, we show that $\rho ^{\ast }$ is fixed point of $\mathfrak{L}_{1}.$ We contrary suppose that $\kappa \left(\rho ^{\ast }, \mathfrak{L}_{1}\rho ^{\ast }\right) > 0.$ Then from (3.1), $(\mathcal{F}_{1})$ and ($D_{3}$), we have

Taking the limit as $\jmath \rightarrow \infty$ and using ($\mathcal{F}_{2}$) and (8), we have

which implies that $\kappa \left(\rho ^{\ast }, \mathfrak{L}_{1}\rho ^{\ast }\right) = 0,$ a contradiction. Thus $\rho ^{\ast } = \mathfrak{L}_{1}\rho ^{\ast }.$ Now we prove that $\rho ^{\ast }$ is fixed point of $\mathfrak{L} _{2}.$ Then from (3.1), $(\mathcal{F}_{1})$ and ($D_{3}$), we have

Taking the limit as $\jmath \rightarrow \infty$ and using ($\mathcal{F}_{2}$) and (8), we have

which implies that $\kappa \left(\rho ^{\ast }, \mathfrak{L}_{1}\rho ^{\ast }\right) = 0,$ a contradiction. Thus $\rho ^{\ast } = \mathfrak{L} _{2}\rho ^{\ast }.$Thus $\rho ^{\ast }$ is a common fixed point of $\mathfrak{L}_{1}$ and $\mathfrak{L}_{2}.$ Now we prove that $\rho ^{\ast }$ is unique. We suppose that

but $\rho ^{\ast }\not = \rho ^{/}.$ Now from (3.1), we have

This implies that, we have

As $\alpha \left(\rho ^{\ast }\right) < 1,$ we have

Thus $\rho ^{\ast } = \rho ^{/}.$ □

Example 2. Let $\Theta \mathfrak{ = }\left \{ S_{\jmath } = 2\jmath +1:\jmath \in \mathbb{N} \right \}$ be endowed with the $\mathcal{F}$-metric

for all $\rho, \hbar \in \Theta$ and $f(\iota) = \ln \iota \mathfrak{.}$ Then $(\Theta, \kappa)$ is an $\mathcal{F}$-complete $\mathcal{F}$-metric space. Define the mapping $\mathfrak{L}_{1}, \mathfrak{L}_{2}:\Theta \rightarrow \Theta$ by

and

Suppose that $m\not = \jmath,$ then

Thus all the assertions of Theorem 3 hold with $\alpha :\Theta \mathfrak{\times }\Theta \mathfrak{\rightarrow }[0, 1)$ defined by $\alpha \left(S_{\jmath }\right) = 2^{-1}$ and any $\beta :\Theta \mathfrak{ \rightarrow }[0, 1).$ Hence $S_{1}$ is a unique common fixed point of $\mathfrak{L}_{1}$ and $\mathfrak{L}_{2}$.

Corollary 1. Let $\left(\Theta, \kappa \right)$ be an $\mathcal{F}$-complete $\mathcal{F}$-metric space and $\mathfrak{L}:\Theta \; \rightarrow \Theta$. If there exist $r > 0$ and mappings $\alpha, \beta :\Theta \mathfrak{ \rightarrow }[0, 1)$ such that

(a) $\alpha \left(\mathfrak{L}\rho \right) \leq \alpha \left(\rho \right)$ and $\beta \left(\mathfrak{L}\rho \right) \leq \beta \left(\rho \right),$

(b) $\alpha \left(\rho \right) +\beta \left(\rho \right) < 1,$

(c)

for all $\rho _{0}, \rho, \hbar \in \overline{B(\rho _{0}, r)}$ and

where $\lambda = \frac{\alpha \left(\rho _{0}\right) }{1-\beta \left(\rho _{0}\right) } < 1$, then there exists a unique point $\rho ^{\ast }\in \overline{B(\rho _{0}, r)}$ such that $\mathfrak{L}\rho ^{\ast } = \rho ^{\ast }$.

Corollary 2. Let $\left(\Theta, \kappa \right)$ be an $\mathcal{F}$-complete $\mathcal{F}$-metric space and $\mathfrak{L}_{1}, \mathfrak{L}_{2}:\Theta \; \rightarrow \Theta$. If there exist $r > 0$ and mappings $\alpha :\Theta \mathfrak{\rightarrow }[0, 1)$ such that

(a) $\alpha \left(\mathfrak{L}_{1}\rho \right) \leq \alpha \left(\rho \right)$ and $\alpha \left(\mathfrak{L}_{2}\rho \right) \leq \alpha \left(\rho \right)$

(b) $\alpha \left(\rho \right) < 1,$

(c)

for all $\rho _{0}, \rho, \hbar \in \overline{B(\rho _{0}, r)}$ and

where $\lambda = \alpha \left(\rho _{0}\right) < 1$, then there exists a unique point $\rho ^{\ast }\in \overline{B(\rho _{0}, r)}$ such that $\mathfrak{L}_{1}\rho ^{\ast } = \mathfrak{L}_{2}\rho ^{\ast } = \rho ^{\ast }$.

Corollary 3. Let $\left(\Theta, \kappa \right)$ be an $\mathcal{F}$-complete $\mathcal{F }$-metric space and $\mathfrak{L}_{1}, \mathfrak{L}_{2}:\Theta \; \rightarrow \Theta$. If there exist $r > 0$ and mapping $\beta :\Theta \mathfrak{ \rightarrow }[0, 1)$ such that

(a) $\beta \left(\mathfrak{L}_{1}\rho \right) \leq \beta \left(\rho \right)$ and $\beta \left(\mathfrak{L}_{2}\rho \right) \leq \beta \left(\rho \right),$

(b) $\alpha \left(\rho \right) +\beta \left(\rho \right) < 1,$

(c)

for all $\rho _{0}, \rho, \hbar \in \overline{B(\rho _{0}, r)}$ and

where $\lambda = \frac{1}{1-\beta \left(\rho _{0}\right) } < 1$, then there exists a unique point $\rho ^{\ast }\in \overline{B(\rho _{0}, r)}$ such that $\mathfrak{L}_{1}\rho ^{\ast } = \mathfrak{L}_{2}\rho ^{\ast } = \rho ^{\ast }$.

Corollary 4. Let $\left(\Theta, \kappa \right)$ be an $\mathcal{F}$-complete $\mathcal{F }$-metric space and $\mathfrak{L}_{1}, \mathfrak{L}_{2}:\Theta \; \rightarrow \Theta$. If there exist $r > 0$ and $\alpha, \beta \in \lbrack 0, 1)$ such that

(a) $\alpha +\beta < 1,$

(b)

for all $\rho _{0}, \rho, \hbar \in \overline{B(\rho _{0}, r)}$ and

where $\lambda = \frac{\alpha }{1-\beta } < 1$, then there exists a unique point $\rho ^{\ast }\in \overline{B(\rho _{0}, r)}$ such that $\mathfrak{L} _{1}\rho ^{\ast } = \mathfrak{L}_{2}\rho ^{\ast } = \rho ^{\ast }$.

Corollary 5. Let $\left(\Theta, \kappa \right)$ be an $\mathcal{F}$-complete $\mathcal{F}$-metric space and $\mathfrak{L}_{1}, \mathfrak{L}_{2}:\Theta \; \rightarrow \Theta$. If there exist $r > 0$ and $\alpha \in \lbrack 0, 1)$ such that

for all $\rho _{0}, \rho, \hbar \in \overline{B(\rho _{0}, r)}$ and

where $\lambda = \alpha < 1$, then there exists a unique point $\rho ^{\ast }\in \overline{B(\rho _{0}, r)}$ such that $\mathfrak{L}_{1}\rho ^{\ast } = \mathfrak{L}_{2}\rho ^{\ast } = \rho ^{\ast }$.

Corollary 6. Let $\left(\Theta, \kappa \right)$ be an $\mathcal{F}$-complete $\mathcal{F }$-metric space and $\mathfrak{L}_{1}, \mathfrak{L}_{2}:\Theta \; \rightarrow \Theta$. If there exist $r > 0$ and $\beta \in \lbrack 0, 1)$ such that

for all $\rho _{0}, \rho, \hbar \in \overline{B(\rho _{0}, r)}$ and

where $\lambda = \frac{1}{1-\beta } < 1$, then there exists a unique point $\rho ^{\ast }\in \overline{B(\rho _{0}, r)}$ such that $\mathfrak{L}_{1}\rho ^{\ast } = \mathfrak{L}_{2}\rho ^{\ast } = \rho ^{\ast }$.

Remark 1. If we set $\mathfrak{L}_{1} = \mathfrak{L}_{2} = \mathfrak{L}$ in the Corollary 5, the we get the main result of Samet et al. ^[9].

4.   $\alpha ^{\ast }$-$\psi$-contractive multivalued results

Let $\Psi$ represents the set of all nondecreasing functions $\psi :[0, +\infty)\rightarrow \lbrack 0, +\infty)$ such that $\sum_{\jmath = 1}^{\infty }\psi ^{\jmath }(\iota) < +\infty, \; \forall \; \iota > 0$, where $\psi ^{\jmath }$ is the $\jmath$-th iterate of these nondecreasing functions$\ \psi$.

Now we state a lemma which is useful in the sequel.

Lemma 1. For $\psi \in \Psi$, these conditions hold:

(i) ($\psi ^{\jmath }(\iota))_{\jmath \in \mathbb{N} }$ converges to $0$ as $\jmath \rightarrow \infty, \; \forall \iota \in (0, +\infty)$,

(ii) $\psi (\iota) < \iota$, $\forall \; \iota > 0$,

(iii) $\psi (\iota) = 0$ iff $\iota = 0.$

Samet et al. ^[11] gave the theory of $\alpha$-admissibility and proved the following result.

Definition 4. (^[11]) A mapping $\mathfrak{L}:\Theta \rightarrow \Theta$ is called an $\alpha$-admissible if there exists a mapping $\alpha :\Theta \times \Theta \rightarrow \lbrack 0, +\infty)$ satisfying

Theorem 4. (^[11]) Let $(\Theta, \kappa)$ be a complete metric space and $\mathfrak{L}$ be $\alpha$-admissible mapping. Assume that

for all $\rho, \hbar \in \Theta$, where $\psi \in \Psi$, also

(i) there exists $\rho _{0}\in \Theta$ such that $\alpha (\rho _{0}, \mathfrak{L}\rho _{0})\geq 1;$

(ii) either $\mathfrak{L}$ is continuous or for any sequence $\{ \rho _{\jmath }\}$ in $\Theta$ with $\alpha (\rho _{\jmath }, \rho _{\jmath +1})\geq 1$ for all $\jmath \in \mathbb{N}$ and $\rho _{\jmath }\rightarrow \rho$ as $\jmath \rightarrow +\infty$, we have $\alpha (\rho _{\jmath }, \rho)\geq 1 \; \forall \jmath \in \mathbb{N}$.

Then $\mathfrak{L}$ has a fixed point.

In 2013, Asl et al. ^[12] gave the notion of $\alpha ^{\ast }$ -admissible mappings in this way.

Definition 5. (^[12]) Let $\alpha :\Theta \times \Theta \rightarrow \lbrack 0, +\infty)$ be a function and $\mathfrak{L}:\Theta \rightarrow CL(\Theta)$ be multivalued mapping. Then $\mathfrak{L}$ is said to be $\alpha ^{\ast }$ -admissible mapping if

where $\alpha ^{\ast }(\mathfrak{L}(\rho), \mathfrak{L}(\hbar)) = \inf \{ \alpha (a, b):a\in \mathfrak{L}(\rho), b\in \mathfrak{L}(\hbar)\}.$

Lemma 2. Let $(\Theta, \kappa)$ be an $\mathcal{F}$-metric space and let $\Re \in CL(\Theta).$ Then, for each $\rho \in \Theta$ with $\kappa (\rho, \Re) > 0$ and $q > 1,$ there exists an member $\hbar \in \; \Re$ such that

Let $(\Theta, \kappa)$ be an $\mathcal{F}$-metric space. We represent by $N(\Theta)$ by set of non empty subsets of $\Theta,$ by $CL(\Theta)$ the set of all nonempty closed subsets of $\Theta$ and $B(\Theta)$ the set of all nonempty bounded subsets of $\Theta$. Now for $\Re \in N(\Theta)$ and $\rho \in \Theta, \; \kappa (\rho, \Re) = \inf \left \{ \kappa (\rho, \hbar):\hbar \in \Re \right \}.$ Also for $\Re _{1}, \Re _{2}\in B(\Theta), \; \delta _{\mathcal{F}}(\Re _{1}, \Re _{2}) = \sup \left \{ \kappa (\rho, \hbar):\rho \in \Re _{1}, \hbar \in \Re _{2}\right \}.$ Whenever $\Re _{1} = \{ \rho \},$ we represent $\delta _{\mathcal{F}}(\Re _{1}, \Re _{2})$ by $\delta _{\mathcal{F}}(\rho, \Re _{2}).$ Let $(\Theta, \preceq, \kappa)$ be an ordered $\mathcal{F}$-metric space and $\Re _{1}, \Re _{2}\subseteq \Theta.$ We say that $\Re _{1}\prec _{r}\Re _{2},$ if for every $\rho \in \Re _{1}, \hbar \in \Re _{2},$ we have $\rho \preceq \hbar.$

Definition 6. Let $(\Theta, \kappa)$ be an $\mathcal{F}$-metric space. A closed-valued multifunction $\mathfrak{L}:\Theta \rightarrow CL(\Theta)$ is said to be ($\alpha ^{\ast }$-$\psi)$- contractive multifunction if there exists two functions $\alpha :\Theta \times \Theta \rightarrow \lbrack 0, +\infty)$ and $\psi \in \Psi$ such that

for each $\rho \in \Theta$ and $\hbar \in \mathfrak{L}(\rho).$

Theorem 5. Let $(\Theta, \kappa)$ be an $\mathcal{F}$-metric space and $\mathfrak{L}:\Theta \rightarrow CL(\Theta)$ be an $\alpha ^{\ast }$ -admissible and ($\alpha ^{\ast }$-$\psi)$- contractive multifunction. Also suppose that the following assertions holds:

(i) $(\Theta, \kappa)$ is $\mathcal{F}$-complete;

(ii) there exists $\rho _{0} \; \in \; \Theta$ and $\rho _{1} \; \in \; \mathfrak{L}(\rho _{0})$ such that $\alpha (\rho _{0}, \rho _{1})\geq 1.$

Then $\rho$ is a fixed point of $\mathfrak{L}$ iff $g(\xi) = \kappa (\xi, \mathfrak{L}\xi)$ is lower semi-continuous at $\rho$.

Proof. Let $\rho _{0}\in \Theta$ be an arbitrary element. Since $\mathfrak{L}(\rho _{0})\not = \emptyset,$ so there exists $\rho _{1}\in \mathfrak{L}(\rho _{0}).$ If $\rho _{0} = \rho _{1},$ then $\rho _{0}$ is a fixed point of $\mathfrak{L }$ and we have nothing to prove. As $\mathfrak{L}(\rho _{1})\not = \emptyset.$ So if $\rho _{1}\in \mathfrak{L}(\rho _{1}),$ then $\rho _{1}$ is a fixed point of $\mathfrak{L}$. Let $\rho _{1}\not \in \mathfrak{L}(\rho _{1}).$ Since $\mathfrak{L}$ is $\alpha ^{\ast }$-admissible, so $\alpha ^{\ast }(\mathfrak{L}(\rho _{0}), \mathfrak{L}(\rho _{1}))\geq 1$. Thus by (4.1), we have

For given $q \; > 1$ and by Lemma 2, $\exists \rho _{2}\in \mathfrak{ L}(\rho _{1})$ such that

Thus by (4.2) and (4.3), we have

It is clear that $\rho _{2}\not = \rho _{1}.$ As $\kappa (\rho _{1}, \rho _{2}) < q\psi (\kappa (\rho _{0}, \rho _{1})).$ Since $\psi$ is strictly increasing, so $\psi (\kappa (\rho _{1}, \rho _{2})) < \psi (q\psi (\kappa (\rho _{0}, \rho _{1}))).$ Put $q_{1} = \frac{\psi (q\psi (\kappa (\rho _{0}, \rho _{1})))}{\psi (\kappa (\rho _{1}, \rho _{2}))}.$ Then $q_{1} > 1$. If $\rho _{2}\in \mathfrak{L}(\rho _{2}),$ then $\rho _{2}$ is fixed point of $\mathfrak{L}.$ Assume that $\rho _{2}\not \in \mathfrak{L}(\rho _{2}).$ As $\alpha ^{\ast }(\rho _{1}, \rho _{2})\geq 1$ and $\mathfrak{L}$ is $\alpha ^{\ast }$-admissible, so $\alpha ^{\ast }(\mathfrak{L}(\rho _{1}), \mathfrak{L }(\rho _{2}))\geq 1.$ Then from (4.1), we get

For given $q_{1} \; > 1$ and by Lemma 2, $\exists \rho _{3}\in \mathfrak{L}(\rho _{2})$ such that

Thus by (4.5) and (4.6), we have

It is clear that $\rho _{3}\not = \rho _{2}.$ As $\kappa (\rho _{2}, \rho _{3}) < \psi (q\psi (\kappa (\rho _{0}, \rho _{1}))).$ Since $\psi$ is strictly increasing, so $\psi (\kappa (\rho _{2}, \rho _{3})) < \psi ^{2}(q\psi (\kappa (\rho _{0}, \rho _{1}))).$ Put $q_{2} = \frac{\psi ^{2}(q\psi (\kappa (\rho _{0}, \rho _{1})))}{\psi (\kappa (\rho _{2}, \rho _{3}))}.$ Then $q_{2} > 1$. If $\rho _{3}\in \mathfrak{L}(\rho _{3}),$ then $\rho _{3}$ is fixed point of $\mathfrak{L}.$ Assume that $\rho _{3}\not \in \mathfrak{L}\rho _{3}.$ As $\alpha ^{\ast }(\rho _{2}, \rho _{3})\geq 1$ and $\mathfrak{L}$ is $\alpha ^{\ast }$-admissible, so $\alpha ^{\ast }(\mathfrak{L}(\rho _{2}), \mathfrak{L}(\rho _{3}))\geq 1.$ Then from (4.1), we get

For given $q_{2} \; > 1$ and by Lemma 2, $\exists \rho _{4}\in \mathfrak{L}(\rho _{3})$ such that

Thus by (4.7) and (4.8), we have

Pursuing the same, we obtain $\{ \rho _{\jmath }\}$ in $\Theta$ such that $\rho _{\jmath }\in \mathfrak{L}\rho _{\jmath -1}$ and $\rho _{\jmath }\not = \rho _{\jmath -1},$ and

for all $\jmath$, which yields that

Now for $m > \jmath.$ Fix $\epsilon > 0$ and let $\jmath (\epsilon)\in \; \mathbb{N}$ such that $\sum_{\jmath \geq \jmath (\delta)}\psi ^{i-1}(q\psi (\kappa (\rho _{0}, \rho _{1}))) < \epsilon$. Now suppose that $(f, \mathfrak{h})\in \mathcal{F}\times \lbrack 0, +\infty)$ be such that (D$_{3}$) is satisfied. Let $\epsilon > 0$ be fixed. By ($\mathcal{F}_{2}$), $\exists \; \delta > 0$ such that

Hence, by (4.10), (4.11) and ($\mathcal{F}_{1}$), we have

for $m > \jmath \geq \jmath (\epsilon).$ Using ($D_{3}$) and (4.12), we obtain $\kappa (\rho _{\jmath }, \rho _{m}) > 0, \; m > \jmath \geq \jmath (\epsilon)$ implies

which implies by ($\mathcal{F}_{1}$) that $\kappa (\rho _{\jmath }, \rho _{m}) < \epsilon, \; m > \jmath \geq \jmath (\epsilon).$ This proves that {$\rho _{\jmath }$} is $\mathcal{F}$-Cauchy. Since $(\Theta, \kappa)$ is $\mathcal{F}$-complete, there exists $\rho ^{\ast }\in \Theta$ such that $\{ \rho _{\jmath }\}$ is $\mathcal{F}$-convergent to $\rho ^{\ast }$, i.e.,

Suppose $g(\xi) = \kappa (\xi, \mathfrak{L}\xi)$ is lower semi-continuous at $\rho$, then

which implies that $\kappa (\rho ^{\ast }, \mathfrak{L}(\rho ^{\ast })) = 0.$ Therefore $\rho ^{\ast }\in \mathfrak{L}(\rho ^{\ast }).$ By the closedness of $\mathfrak{L}$ it yields that $\rho ^{\ast }\in \mathfrak{L}(\rho ^{\ast })$. Conversely, assume that $\rho ^{\ast }$ is a fixed point of $\mathfrak{L }$, then $\kappa (\rho ^{\ast }, \mathfrak{L}(\rho ^{\ast })) = 0$, which implies that

□

Corollary 7. Let $(\Theta, \preceq, \kappa)$ be an ordered $\mathcal{F}$-metric space, $\psi \in \Psi$ be a strictly increasing mapping and $\mathfrak{L}:\Theta \rightarrow CL(\Theta)$ be a mapping such that for each $\rho \in \Theta$ and $\hbar \in \mathfrak{L}(\rho)$ with $\rho \preceq \hbar,$ we have

Also, assume that

(i) $(\Theta, \kappa)$ is $\mathcal{F}$-complete;

(ii) there exists $\rho _{0} \; \in \; \Theta$ and $\rho _{1} \; \in \; \mathfrak{L}(\rho _{0})$ such that $\rho _{0}\preceq \rho _{1};$ (iii) if $\rho \preceq \hbar,$ then $\mathfrak{L}\rho \prec _{r}\mathfrak{L} \hbar.$

Then $\rho$ is a fixed point of $\mathfrak{L}$ iff $g(\xi) = \kappa (\xi, \mathfrak{L}\xi)$ is lower semi-continuous at $\rho$.

Proof. Define $\alpha :\Theta \times \Theta \rightarrow \lbrack 0, \infty)$ by

By using condition (i) and the definition of $\alpha$, we have $\alpha (\rho _{0}, \rho _{1}) = 1.$ Also, from condition (iii), we have $\rho \preceq \hbar,$ then $\mathfrak{L}\rho \prec _{r}\mathfrak{L}\hbar,$ by using the definitions of $\alpha$ and $\prec _{r},$wehave $\alpha (\rho, \hbar) = 1$ implies $\alpha ^{\ast }(\mathfrak{L}\rho, \mathfrak{L}\hbar) = 1$. Furthermore, it is simple to check that $\mathfrak{L}$ is a strictly generalized $(\alpha ^{\ast }, \psi)$-contractive mapping. Therefore, by Theorem 5, $\rho$ is a fixed point of $\mathfrak{L}$ if and only if $g(\xi) = \kappa (\xi, \mathfrak{L}\xi)$ is lower semi-continuous at $\rho$. □

Definition 7. Let $(\Theta, \kappa)$ be an $\mathcal{F}$-metric space and $\mathfrak{L} :\Theta \rightarrow B(\Theta)$ be a mapping. We say that $\mathfrak{L} :\Theta \rightarrow B(\Theta)$ is said to be generalized ($\alpha ^{\ast }$, $\psi, \delta _{\mathcal{F}})$-contractive mapping if there exists two functions $\alpha :\Theta \times \Theta \rightarrow \lbrack 0, +\infty)$ and $\psi \in \Psi$ such that

for each $\rho \in \Theta$ and $\hbar \in \mathfrak{L}(\rho).$

Theorem 6. Let $(\Theta, \kappa)$ be an $\mathcal{F}$-metric space and $\mathfrak{L}:\Theta \rightarrow B(\Theta)$ be an $\alpha ^{\ast }$ -admissible and generalized ($\alpha ^{\ast }$, $\psi, \delta _{\mathcal{F}})$ -contractive mapping. Also suppose that the following assertions holds:

(i) $(\Theta, \kappa)$ is $\mathcal{F}$-complete;

(ii) there exists $\rho _{0} \; \in \; \Theta$ and $\rho _{1} \; \in \; \mathfrak{L}(\rho _{0})$ such that $\alpha (\rho _{0}, \rho _{1})\geq 1.$

Then there exists $\rho \in \Theta$ such that $\{ \rho \} = \mathfrak{L}(\rho)$ iff $g(\xi) = \kappa (\xi, \mathfrak{L}(\xi))$ is lower semi-continuous at $\rho$.

Proof. By the hypothesis of the theorem, there exist $\rho _{0}\in \Theta$ and $\rho _{0}\in \mathfrak{L}(\rho _{0})$ such that $\alpha (\rho _{0}, \rho _{1})\geq 1.$ Assume that $\rho _{0}\not = \rho _{1}$, for otherwise, $\rho _{0}$ is a fixed point. Let $\rho _{1}\not \in \mathfrak{L}(\rho _{1})$. As $\mathfrak{L}$ is $\alpha ^{\ast }$-admissible, we have $\alpha ^{\ast }(\mathfrak{L}(\rho _{0}), \mathfrak{L}(\rho _{1})) \; \geq 1$. Then

Since $\mathfrak{L}(\rho _{1}) \; \not = \; \emptyset$, there is $\rho _{2}\in \mathfrak{L}(\rho _{1})$. Then

From (4.15) and (4.16), we have

Since $\psi$ is nondecreasing, we have

As $\rho _{2}\in \mathfrak{L}\rho _{1}$, we have $\alpha (\rho _{1}, \rho _{2})\geq 1$. Since $\mathfrak{L}(\rho _{2}) \; \not = \; \emptyset$, there is $\rho _{3} \; \in \; \mathfrak{L}(\rho _{2})$. Assume that $\rho _{2}\not = \rho _{3}$, for otherwise, $\rho _{2}$ is a fixed point of $\mathfrak{L}$. Then

Since $\mathfrak{L}(\rho _{2}) \; \not = \; \emptyset$, there is $\rho _{3}\in \mathfrak{L}(\rho _{2})$. Then

From (4.17) and (4.18), we have

By (4.16), we have

Since $\psi$ is nondecreasing, we have

By continuing in this way, we get a sequence $\{ \rho _{\jmath }\}$ in $\Theta$ such that $\rho _{\jmath +1}\in \mathfrak{L}(\rho _{\jmath })$ and $\rho _{\jmath }\not = \rho _{\jmath +1}$ for $\jmath = 0, 1, 2, ...$. Further we have

This yields that

Now for $m > \jmath.$ Fix $\epsilon > 0$ and let $\jmath (\epsilon)\in \; \mathbb{N}$ such that $\sum_{\jmath \geq \jmath (\delta)}\psi ^{i}(\kappa (\rho _{0}, \rho _{1})) < \epsilon$. Now assume that $(f, \mathfrak{h})\in \mathcal{ F}\times \lbrack 0, +\infty)$ be such that (D$_{3}$) is satisfied. Let $\epsilon > 0$ be fixed. By ($\mathcal{F}_{2}$), $\exists \; \delta > 0$ such that

Hence, by (4.20), (4.21) and ($\mathcal{F}_{1}$), we have

for $m > \jmath \geq \jmath (\epsilon).$ Using (D$_{3}$) and (4.22), we obtain $\kappa (\rho _{\jmath }, \rho _{m}) > 0, \; m > \jmath \geq \jmath (\epsilon)$ implies

which implies by ($\mathcal{F}_{1}$) that $\kappa (\rho _{\jmath }, \rho _{m}) < \epsilon, \; m > \jmath \geq \jmath (\epsilon).$ This proves that {$\rho _{\jmath }$} is $\mathcal{F}$-Cauchy. Since $(\Theta, \kappa)$ is $\mathcal{F}$-complete, there exists $\rho ^{\ast }\in \Theta$ such that $\{ \rho _{\jmath }\}$ is $\mathcal{F}$-convergent to $\rho ^{\ast }$. Letting $\jmath \rightarrow \infty$ in (4.19), we have

Suppose $g(\xi) = \delta _{\mathcal{F}}(\xi, \mathfrak{L}\xi)$ is lower semi-continuous at $\rho$, then by ($\mathcal{F}_{1}$), we have

Therefore $\{ \rho ^{\ast }\} \in \mathfrak{L}(\rho ^{\ast }),$ because $\delta _{\mathcal{F}}(\Re _{1}, \Re _{2}) = 0$ implies $\Re _{1} = \Re _{2} = \{a\}$. Conversely, suppose that $\{ \rho ^{\ast }\} \in \mathfrak{L}(\rho ^{\ast })$, then

□

Corollary 8. Let $(\Theta, \preceq, \kappa)$ be an ordered $\mathcal{F}$ -metric space, $\psi \in \Psi$ be a strictly increasing mapping and $\mathfrak{L}:\Theta \rightarrow B(\Theta)$ be a mapping such that for each $\rho \in \Theta$ and $\hbar \in \mathfrak{L}(\rho)$ with $\rho \preceq \hbar,$ we have

Also, assume that

(i) $(\Theta, \kappa)$ is $\mathcal{F}$-complete,

(ii) there exists $\rho _{0} \; \in \; \Theta$ and $\{ \rho _{0}\} \; \in \; \mathfrak{L}(\rho _{0})$ i.e., there exists $\rho _{1} \; \in \; \mathfrak{L} (\rho _{0})$ such that $\rho _{0}\preceq \rho _{1},$ (iii) if $\rho \preceq \hbar,$ then $\mathfrak{L}\rho \prec _{r}\mathfrak{L} \hbar.$

Then there exists $\rho \in \Theta$ such that $\{ \rho \} = \mathfrak{L}(\rho)$ iff $g(\xi) = \kappa (\xi, \mathfrak{L}(\xi))$ is lower semi-continuous at $\rho$.

Proof. Define $\alpha :\Theta \times \Theta \rightarrow \lbrack 0, \infty)$ by

By using condition (i) and the definition of $\alpha$, we have $\alpha (\rho _{0}, \rho _{1}) = 1.$ Also, from condition (iii), we have $\rho \preceq \hbar,$ then $\mathfrak{L}\rho \prec _{r}\mathfrak{L}\hbar,$ by using the definitions of $\alpha$ and $\prec _{r},$wehave $\alpha (\rho, \hbar) = 1$ implies $\alpha ^{\ast }(\mathfrak{L}\rho, \mathfrak{L}\hbar) = 1$. Furthermore, it is simple to check that $\mathfrak{L}$ is a strictly generalized ($\alpha ^{\ast }$, $\psi, \delta _{\mathcal{F}})$-contractive mapping. Therefore, by Theorem 5, there exists $\rho \in \Theta$ such that $\{ \rho \} = \mathfrak{L}(\rho)$ if and only if $g(\xi) = \delta _{ \mathcal{F}}(\xi, \mathfrak{L}\xi)$ is lower semi-continuous at $\rho$. □

5.   Applications

A representative stability result based on fixed point theory arguments follows a number of basic arguments adapted to the special structure of the equation under consideration. It leads to large number of results in the literature for different classes of equations, see ^[29,30]. In the present section, we investigate the existence of solution of differential equation

We state a lemma of Djoudi et al.^[31] which will be used in proving of our theorem.

Lemma 3. (^[31]) Assume that $r^{/}(\iota)\not = 1 \; \forall \iota \in \mathbb{R}$. Then $\rho (\iota)$ is a solution of (5.1) if and only if

where

Now suppose that $\vartheta :(-\infty, 0]\rightarrow \mathbb{R}$ is a bounded and continuous function, then $\rho (\iota) = \rho (\iota, 0, \vartheta)$ is a solution of (5.1) if $\rho (\iota) = \vartheta (\iota)$ for $\iota \leq 0$ and satisfies (5.1) for $\iota \geq 0.$ Assume that $\mathfrak{C}$ is the collection of $\rho : \mathbb{R} \rightarrow \mathbb{R}$ which are continuous. Define $\aleph _{\vartheta }$ by

Then $\aleph _{\vartheta }$ is a Banach space endowed with $\left \Vert \cdot \right \Vert$.

Lemma 4. (^[13]) The space $(\aleph _{\vartheta }, \parallel \cdot \parallel)$ with the $\mathcal{F}$-metric $d$ defined by

for all $\iota, \iota ^{\ast }\in \aleph _{\vartheta },$ is $\mathcal{F}$ -metric space.

Theorem 7. Let $\mathfrak{L}:\aleph _{\vartheta }\rightarrow \aleph _{\vartheta }$ be a mapping defined by

for all $\rho \in \aleph _{\vartheta }$. Assume that there exists $\alpha :\aleph _{\vartheta }\times \aleph _{\vartheta }\mathfrak{\rightarrow }[0, 1)$ such that

Then $\mathfrak{L}$ has a fixed point.

Proof. It follows from (5.3) that $\mathfrak{L}(\rho), \mathfrak{L}(\hbar)\in \; \aleph _{\vartheta }$. Now from (5.4), we have

Hence,

for any $\beta :\Theta \mathfrak{\rightarrow }[0, 1).$ Thus all the assumptions of Corollary 1 are satisfied and $\mathfrak{L}$ has a unique fixed point in $\aleph _{\vartheta }$ which solves (5.1). □

6.   Conclusions

This article is precised on the notion of $\mathcal{F}$-metric space to prove common fixed points of six mappings for generalized rational contractions involving control functions of one variable. A non-trivial example is also provided to show the validity of obtained results. We also established fixed points of ($\alpha ^{\ast }$-$\psi)$-contractive and generalized ($\alpha ^{\ast }$, $\psi, \delta _{\mathcal{F}})$-contractive multifunctions. As application, we discussed the solution of nonlinear neutral differential equation.

Common fixed points of fuzzy mappings in the background of $\mathcal{F}$ -metric space can be interesting outline for the future work in this direction. Differential and integral inclusions can be investigated as applications of these results.

Acknowledgments

Second author acknowledges with thanks Natural Science Foundation of Hebei Province (Grant No. A2019404009) and The Major Project of Education Department of Hebei Province (No. ZD2021039) to Innovation and improvement project of academic team of Hebei University of Architecture (Mathematics and Applied Mathematics) NO. TD202006 for financial support.

Conflicts of interest

The authors declare that they have no conflicts of interest.