Fuzzy Triple Controlled Metric Spaces and Related Fixed Point Results

Abstract

In this study, we introduce fuzzy triple controlled metric space that generalizes certain fuzzy metric spaces, like fuzzy rectangular metric space, fuzzy rectangular $b$-metric space, fuzzy $b$-metric space, and extended fuzzy $b$-metric space. We use $f,g,h,$ three noncomparable functions as follows: ${\mathfrak{m}}_q\mu ,\eta ,t+s+w\ge {\mathfrak{m}}_q\mu ,\nu ,t/f\mu ,\nu \ast {\mathfrak{m}}_q\nu ,\xi ,s/g\nu ,\xi \ast {\mathfrak{m}}_q\xi ,\eta ,w/h\xi ,\eta$. We prove Banach fixed point theorem in the settings of fuzzy triple controlled metric space that generalizes Banach fixed point theorem for aforementioned spaces. An example is presented to support our main results. We also apply our technique to the uniqueness for the solution of an integral equation.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and Preliminaries

Banach contraction mapping principle (BCMP) [1] has many applications in various scientific fields ([2–6]). BCMP was proved in 1922 and has been investigated by many researchers in different ways ([7–11]). In 1965, Zadeh [12] defined a fuzzy set that generalizes the definition of a crisp set by associating all elements with membership values between the interval $0,1$. Since then, the fuzzy set theory has been used extensively in mathematics ([13–18]) and many other areas ([19–21]). The definition of fuzzy metric space was given by Kramosil and Michálek [22] in 1975. The fuzzy version of BCMP was proved by Grabiec [23]. He also extended the Edelstein theorem to fuzzy metric spaces. George and Veeramani [24] introduced the Hausdorff topology in fuzzy metric space and modified the definition of fuzzy metric space given in [22]. In 1989, Bakhtin [25] introduced the notion of a $b$-metric space that generalizes the definition of classical metric space. Branciari [26] introduced the rectangular metric space and proved some fixed point results. Roshan et al. [27] introduced the notion of $b$-rectangular metric space that generalizes the definition of a rectangular metric space.

In [28], Nădăban utilized fuzzy sets in $b$-metric spaces and introduced the notion of a fuzzy $b$-metric space. He also discussed the topological properties of this new space. Kamran et al. [29] further generalized the definition of [25] by introducing the idea of extended $b$-metric space while in [30], Mehmood et al. applied the fuzzy sets to the definition in [29] by introducing the notion of an extended fuzzy $b$-metric space and proved BCMP on this space. In [31], authors defined fuzzy version of rectangular $b$-metric space while in [32], Asim et al. introduced the concept of extended rectangular $b$-metric space and proved related fixed point theorem. Recently, Saleem et al. [33] introduced the notion of fuzzy double controlled metric space and proved BCMP on such space. Abdeljawad et al. [34] modified the definition of controlled metric type space defined in [35] by giving the idea of a double controlled metric type space.

Definition 1 ([36]).

Let $\ast :0,1\times 0,1⟶0,1$ be a binary operation, then $\ast$ is said to be continuous triangular norm (in short, continuous $t$-norm), if for all $x_1,x_2,x_3,x_4\in 0,1,$ the following conditions are satisfied:

$\ast 1\ast x_1,x_2=\ast x_2,x_1$;

$\ast 2\ast x_1,\ast x_2,x_3=\ast \ast x_1,x_2,x_3$;

$\ast 3$$\ast$ is continuous;

$\ast 4\ast x,1=x$ for every $x\in 0,1$;

$\ast 5\ast x_1,x_2\le \ast x_3,x_4$ whenever $x_1\le x_3,$$x_2\le x_4$.

Definition 2 ([28]).

Let $F$ be a nonempty set, $K\ge 1$ a real number, $\ast$ a continuous $t$-norm, and $\mathfrak{m}$ be a fuzzy set on $F\times F\times 0,\infty$. Then, $\mathfrak{m}$ is said to be a fuzzy $\text{b}$-metric on $F,$ if for all $\mu ,\nu ,\eta \in F$, $\mathfrak{m}$ satisfies the following:

$b\mathfrak{m}1\mathfrak{m}\mu ,\nu ,0=0$ for $t=0$;

$b\mathfrak{m}2\mathfrak{m}\mu ,\nu ,t=1$ for all $t>0,$ iff $\mu =\nu$;

$b\mathfrak{m}3\mathfrak{m}\mu ,\nu ,t=\mathfrak{m}\nu ,\mu ,t$;

$b\mathfrak{m}4\mathfrak{m}\mu ,\eta ,Kt+s\ge \mathfrak{m}\mu ,\nu ,t\ast \mathfrak{m}\nu ,\eta ,s$ for all $s,t>0$;

$b\mathfrak{m}5\mathfrak{m}\mu ,\nu ,·\text{:}0,\infty ⟶0,1$ is left continuous and ${\lim }_{t⟶\infty }\mathfrak{m}\mu ,\nu ,t=1$.

The quadruple $F,\mathfrak{m},\ast ,K$ is called fuzzy $b$-metric space.

Definition 3 ([30]).

Let $F$ be a nonempty set,$f:F\times F⟶1,\infty$, $\ast$ a continuous $t$-norm, and ${\mathfrak{m}}_f$ is a fuzzy set on $F\times F\times 0,\infty$. Then, ${\mathfrak{m}}_f$ is extended fuzzy $b$-metric if for all $\mu ,\nu ,\eta \in F$, the following conditions are satisfied:

${\mathfrak{m}}_{f}1$${\mathfrak{m}}_f\mu ,\nu ,0=0$ for $t=0$;

${\mathfrak{m}}_{f}2$${\mathfrak{m}}_f\mu ,\nu ,t=1$ for all $t>0,$ iff $\mu =\nu$;

${\mathfrak{m}}_{f}3$${\mathfrak{m}}_f\mu ,\nu ,t={\mathfrak{m}}_f\nu ,\mu ,t$;

${\mathfrak{m}}_{f}4$${\mathfrak{m}}_f\mu ,\eta ,f\mu ,\eta t+s\ge {\mathfrak{m}}_f\mu ,\nu ,t\ast {\mathfrak{m}}_f\nu ,\eta ,s$ for all $s,t>0$;

${\mathfrak{m}}_{f}5$${\mathfrak{m}}_f\mu ,\nu ,·\text{:}0,\infty ⟶0,1$ is left continuous.

Then, $F,{\mathfrak{m}}_f,\ast ,f$ is known as an extended fuzzy $b$-metric space.

Definition 4 ([37]).

Let $F$ be a nonempty set, $\ast$ a continuous $t$-norm, and ${\mathfrak{m}}_r$ is a fuzzy set on $F\times F\times 0,\infty$. Then, ${\mathfrak{m}}_r$ is called a fuzzy rectangular metric if for any $\mu ,\nu \in F$ and all distinct points $u,\eta \in F\backslash \mu ,\nu$, the following conditions are satisfied:

${\mathfrak{m}}_{r}1$${\mathfrak{m}}_r\mu ,\nu ,0=0$ for $t=0$;

${\mathfrak{m}}_{r}2$${\mathfrak{m}}_r\mu ,\nu ,t=1$ for all $t>0$ iff $\mu =\nu$;

${\mathfrak{m}}_{r}3$${\mathfrak{m}}_r\mu ,\nu ,t={\mathfrak{m}}_r\nu ,\mu ,t$;

${\mathfrak{m}}_{r}4$${\mathfrak{m}}_r\mu ,\nu ,t+s+w\ge {\mathfrak{m}}_r\mu ,\xi ,t\ast {\mathfrak{m}}_r\xi ,\eta ,s\ast {\mathfrak{m}}_r\eta ,\nu ,w$ for all $t,s,w>0$;

${\mathfrak{m}}_{r}5$${\mathfrak{m}}_r\mu ,\nu ,·\text{:}0,\infty ⟶0,1$ is left continuous, and ${\lim }_{t⟶\infty }{\mathfrak{m}}_r\mu ,\nu ,t=1$.

Then, $F,{\mathfrak{m}}_r,\ast$ is called a fuzzy rectangular metric space.

Definition 5 ([31]).

Let $F$ be a nonempty set, $b\ge 1$ a real number, $\ast$ a continuous $t$-norm, and ${\mathfrak{m}}_{rb}$ be a fuzzy set on $F\times F\times 0,\infty$. Then, ${\mathfrak{m}}_{rb}$ is called fuzzy rectangular $b$-metric if for any $\mu ,\nu \in F$ and all distinct points $\xi ,\eta \in F\backslash \mu ,\nu$, the following conditions are satisfied:

${\mathfrak{m}}_{rb}1$${\mathfrak{m}}_{rb}\mu ,\nu ,0=0$;

${\mathfrak{m}}_{rb}2$${\mathfrak{m}}_{rb}\mu ,\nu ,t=1$ for all $t>0$ iff $\mu =\nu$;

${\mathfrak{m}}_{rb}3$${\mathfrak{m}}_{rb}\mu ,\nu ,t={\mathfrak{m}}_{rb}\nu ,\mu ,t$;

${\mathfrak{m}}_{rb}4$${\mathfrak{m}}_{rb}\mu ,\nu ,bt+s+w\ge {\mathfrak{m}}_{rb}\mu ,\xi ,t\ast {\mathfrak{m}}_{rb}\xi ,\eta ,s\ast {\mathfrak{m}}_{rb}\eta ,\nu ,w$ for all $t,s,w>0$;

${\mathfrak{m}}_{rb}5$${\mathfrak{m}}_{rb}\mu ,\nu ,·\text{:}0,\infty ⟶0,1$ is left continuous, and ${\lim }_{t⟶\infty }{\mathfrak{m}}_{rb}\mu ,\nu ,t=1$.

Then, $F,{\mathfrak{m}}_{rb},\ast$ is known as fuzzy rectangular $b$-metric space.

Definition 6 ([34]).

Let$f,g:F\times F⟶1,\infty$ be two noncomparable functions, if for all $\mu ,\nu ,\eta \in F,$$q:F\times F⟶0,\infty$ satisfies the following:

$q_{fg}1$$q\mu ,\nu =0$ iff $\mu =\nu$;

$q_{fg}2$$q\mu ,\nu =q\nu ,\mu$;

$q_{fg}3$$q\mu ,\nu \le f\mu ,\eta q\mu ,\eta +g\eta ,\nu q\eta ,\nu$.

Then, $F,q$ is known as double controlled metric type space.

2. Main Results

We first give the definition of a fuzzy triple controlled metric space as follows.

Definition 7.

Let $f,g,h:F\times F⟶1,\infty$ be three noncomparable functions, $\ast$ a continuous $t$-norm, and ${\mathfrak{m}}_q$ is a fuzzy set on $F\times F\times 0,\infty$. Then, ${\mathfrak{m}}_q$ is called fuzzy triple controlled metric if for any $\mu ,\nu \in F$ and all distinct $\xi ,\eta \in F\backslash \mu ,\nu$, the following conditions are satisfied:

${\mathfrak{m}}_{q}1$${\mathfrak{m}}_q\mu ,\nu ,t>0$;

${\mathfrak{m}}_{q}2$${\mathfrak{m}}_q\mu ,\nu ,t=1$ for all $t>0$ iff $\mu =\nu$;

${\mathfrak{m}}_{q}3$${\mathfrak{m}}_q\mu ,\nu ,t={\mathfrak{m}}_q\nu ,\mu ,t$;

${\mathfrak{m}}_{q}4$${\mathfrak{m}}_q\mu ,\nu ,t+s+w\ge {\mathfrak{m}}_q(\mu ,\xi ,t/f\mu ,\xi \ast {\mathfrak{m}}_q\xi ,\eta ,s/g\xi ,\eta \ast {\mathfrak{m}}_q\eta ,\nu ,w/h\eta ,\nu$, for all $t,s,w>0$;

${\mathfrak{m}}_{q}5$${\mathfrak{m}}_q\mu ,\nu ,·\text{:}0,\infty ⟶0,1$ is continuous.

Then, $F,{\mathfrak{m}}_q,\ast$ is called a fuzzy triple controlled metric space.

Remark 8.

(i) Taking $f\mu ,\xi =g\xi ,\eta =h\eta ,\nu =1$ in (${\mathfrak{m}}_q$4), then we get the definition of fuzzy rectangular metric space [37].

(ii) Taking $f\mu ,\xi =g\xi ,\eta =h\eta ,\nu =b\ge 1$ in (${\mathfrak{m}}_q$4), then our definition reduces to the definition of fuzzy rectangular $b$-metric space [31].

(iii) Taking $\eta =\nu$ and $s+w=t^{\prime }$ in (${\mathfrak{m}}_q$4), then our definition reduces to the definition of fuzzy double controlled metric space defined in [33].

(iv) Taking $\eta =\nu$, $f\mu ,\xi =g\xi ,\eta$ and $s+w=t^{\prime }$ in (${\mathfrak{m}}_q$4), then our definition reduces to the definition of extended fuzzy $b$-metric space defined in [30].

(v) Taking $\eta =\nu$, $f\mu ,\xi =g\xi ,\eta =b\ge 1$ and $s+w=t^{\prime }$ in (${\mathfrak{m}}_q$4), then our definition reduces to the definition of fuzzy $b$-metric space defined in [28].

The following example justifies Definition 7.

Example 1.

Consider $F=1,2,3,4$ and let $f,g,h:F\times F⟶1,\infty$ be three noncomparable functions defined as $f\mu ,\nu =1+\mu +\nu ,g\mu ,\nu ={\mu }^2+\nu +1$ and $h\mu ,\nu ={\mu }^2+{\nu }^2-1$. Then, ${\mathfrak{m}}_q:F\times F\times 0,\infty ⟶0,1$ defined by $$\begin{array}{}\text{(1)}& {\mathfrak{m}}_q\mu ,\nu ,t=\frac{\min \mu ,\nu +t}{\max \mu ,\nu +t},\end{array}$$with product $t$-norm, that is, $t_1\ast t_2=t_1{t}_2$, $F,{\mathfrak{m}}_q,\ast$ is a fuzzy triple controlled metric space.

Now, $$\begin{array}{c}\text{(2)}& f1,2=f2,1=4,f1,3=f3,1=5,f1,4=f4,1=6,f2,3=f3,2=6,f2,4=f4,2=7,f3,4=f4,3=8,f1,1=3,f2,2=5,f3,3=7,f4,4=9,g1,1=3,g1,2=4,g1,3=5,g1,4=6,g2,1=6,g2,2=7,g2,3=8,g2,4=9,g3,1=11,g3,2=12,g3,3=13,g3,4=14,g4,1=18,g4,2=19,g4,3=20,g4,4=21,\\ h1,2=h2,1=4,h1,3=h3,1=9,h1,4=h4,1=16,h2,3=h3,2=12,h2,4=h4,2=19,h3,4=h4,3=24,h1,1=1,h2,2=7,h3,3=17,h4,4=31.\end{array}$$

Axioms (${\mathfrak{m}}_q$1) to (${\mathfrak{m}}_q$3) and (${\mathfrak{m}}_q$5) can easily be verified, and we check only (${\mathfrak{m}}_q$4).

Let $\mu =1,\eta =4,$ then either $\nu =2$ and $\xi =3$ or $\nu =3$ and $\xi =2$. We prove for $\nu =2$ and $\xi =3$. The proof for $\nu =3$ and $\xi =2$ is similar. $$\begin{array}{}\text{(3)}& {\mathfrak{m}}_{q}1,4,t+s+w=\frac{\min 1,4+t+s+w}{\max 1,4+t+s+w}=\frac{1+t+s+w}{4+t+s+w}.\end{array}$$

Now, $$\begin{array}{c}\text{(4)}& {\mathfrak{m}}_{q}1,2,\frac{t}{f1,2}=\frac{\min 1,2+t/f1,2}{\max 1,2+t/f1,2}=\frac{1+t/4}{2+t/4}=\frac{4+t}{8+t},{\mathfrak{m}}_{q}2,3,\frac{s}{g2,3}=\frac{\min 2,3+s/g2,3}{\max 2,3+s/g2,3}=\frac{2+s/8}{3+s/8}=\frac{16+s}{24+s},\\ {\mathfrak{m}}_{q}3,4,\frac{w}{h3,4}=\frac{\min 3,4+w/h3,4}{\max 3,4+w/h3,4}=\frac{3+w/24}{4+w/24}=\frac{72+w}{92+w}.\end{array}$$

Clearly, $$\begin{array}{}\text{(5)}& {\mathfrak{m}}_{q}1,4,t+s+w\ge {\mathfrak{m}}_{q}1,2,\frac{t}{f1,2}\ast {\mathfrak{m}}_{q}2,3,\frac{s}{g2,3}\ast {\mathfrak{m}}_{q}3,4,\frac{w}{h3,4}.\end{array}$$

Working like same steps, remaining cases can easily be proved. Hence, $F,{\mathfrak{m}}_q,\ast$ is a fuzzy triple controlled metric space.

Remark 9.

(i) In Example 1, $F,{\mathfrak{m}}_q,\ast$ is not a fuzzy triple controlled metric space, if we use minimum $t$-norm, i.e., $t_1\ast t_2=\min t_1,t_2$ instead of product $t$-norm.

Next, we define the convergence of a sequence as well as Cauchy sequence in the context of fuzzy triple controlled metric space.

Definition 10.

Consider a fuzzy triple controlled metric space $F,{\mathfrak{m}}_q,\ast$. Then, a sequence ${\mu }_n$ in $F$ is said to be

(1) convergent, if for all $t>0,$ there exists $\mu$ in $F$ such that

$$\begin{array}{}\text{(6)}& \underset{n⟶\infty }{\lim }{\mathfrak{m}}_q{\mu }_n,\mu ,t=1,\end{array}$$

(2) Cauchy iff for all $t>0$ and for each $\epsilon >0,t>0,$ there exists $n_0\in \mathbb{N}$ such that

$$\begin{array}{}\text{(7)}& {\mathfrak{m}}_q{\mu }_n,{\mu }_m,t\ge 1-\epsilon ,\text{for}\text{all}m,n\ge n_0.\end{array}$$

$F,{\mathfrak{m}}_q,\ast$ is called complete fuzzy triple controlled metric space, if every Cauchy sequence in $F$ converges to some point $\mu$ in $F.$

Definition 11.

Let $F,{\mathfrak{m}}_q,\ast$ be a fuzzy triple controlled metric space. Then, an open ball $Bx,r,t$, with center $x$, radius $r,r\in 0,1$, and $t>0$, is given by $$\begin{array}{}\text{(8)}& Bx,r,t=y\in F:m_{q}x,y,t>1-r,\end{array}$$and the corresponding topology is defined as $$\begin{array}{}\text{(9)}& {\tau }_{{\mathfrak{m}}_q}=S\subset F:Bx,r,t\subset S.\end{array}$$

Next example shows that a fuzzy triple controlled metric space is not Hausdorff.

Example 2.

Consider the fuzzy triple controlled metric space as given in Example 1. Then, the open ball centered at $1$, radius $0.3$, and $t=5$ is given by $$\begin{array}{}\text{(10)}& B\mathrm{1,0.3,5}=y\in F:{\mathfrak{m}}_{q}1,y,5>0.7.\end{array}$$

Now, $$\begin{array}{c}\text{(11)}& {\mathfrak{m}}_{q}1,2,5=\frac{1+5}{2+5}=\frac{6}{7}=0.8571,{\mathfrak{m}}_{q}1,3,5=\frac{1+5}{3+5}=\frac{6}{8}=0.75,\\ {\mathfrak{m}}_{q}1,4,5=\frac{1+5}{4+5}=\frac{6}{9}=0.6666.\end{array}$$

Thus, $B\mathrm{1,0.3,5}=2,3$. Now, consider the open ball $B\mathrm{2,0.2,7}$ with centered at $2$, radius $r=0.2$, and $t=7$. Then, $$\begin{array}{c}\text{(12)}& B\mathrm{2,0.2,7}=y\in F:{\mathfrak{m}}_{q}2,y,7>0.8,{\mathfrak{m}}_{q}2,1,7=\frac{1+7}{2+7}=\frac{8}{9}=0.8888,\\ {\mathfrak{m}}_{q}2,3,7=\frac{2+7}{3+7}=\frac{9}{10}=0.9,\\ {\mathfrak{m}}_{q}2,4,7=\frac{2+7}{4+7}=\frac{9}{11}=0.8181.\end{array}$$

Thus, $B\mathrm{2,0.2,7}=1,3,4$ and so $B\mathrm{1,0.3,5}\cap B\mathrm{2,0.2,7}=2,3\cap 1,3,4=3\ne \varnothing .$ Hence, fuzzy triple controlled metric space $F,{\mathfrak{m}}_q,\ast$ is not Hausdorff.

Theorem 12.

Let $f,g,h:F\times F⟶1,1/k$ be three noncomparable functions $k\in 0,1$ and $F,{\mathfrak{m}}_q,\ast$ be a complete fuzzy triple controlled metric space such that $$\begin{array}{}\text{(13)}& \underset{t⟶\infty }{\lim }{\mathfrak{m}}_q\mu ,\nu ,t=1.\end{array}$$

Let $T:F⟶F$ be a given mapping such that $$\begin{array}{}\text{(14)}& {\mathfrak{m}}_{q}T\mu ,T\nu ,kt\ge {\mathfrak{m}}_q\mu ,\nu ,t,\mathit{\text{for}}all\mu ,\nu \in F.\end{array}$$

Then, $T$ has a unique fixed point.

Proof.

Choose $a_0$ an arbitrary point in $F$. If $T{a}_0=a_0$, then we are done. If $T{a}_0\ne a_0$ then $T{a}_0=a_1\in F$. Continuing in this way, we have $$\begin{array}{c}\text{(15)}& T{a}_1=TT{a}_0=T^2{a}_0=a_2,T{a}_2=TT{a}_1=T^2{a}_1=T^{2}T{a}_0=T^3{a}_0=a_3,\end{array}$$and so on, $$\begin{array}{}\text{(16)}& T{a}_n=TT{a}_{n-1}=T^2{a}_{n-1}=\cdots =T^{n+1}{a}_0=a_{n+1}.\end{array}$$

So, we have iterative sequence $a_n=T{a}_{n-1}=T^n{a}_0$. Applying (14) successively, we get $$\begin{array}{}\text{(17)}& {\mathfrak{m}}_q{a}_n,a_{n+1},t={\mathfrak{m}}_{q}T{a}_{n-1},T{a}_n,t\ge {\mathfrak{m}}_q{a}_{n-1},a_n,\frac{t}{k}={\mathfrak{m}}_{q}T{a}_{n-2},T{a}_{n-1},\frac{t}{k}\ge {\mathfrak{m}}_q{a}_{n-2},a_{n-1},\frac{t}{k^2}\ge {\mathfrak{m}}_q{a}_{n-3},a_{n-2},\frac{t}{k^3}\ge \cdots \ge {\mathfrak{m}}_q{a}_0,a_1,\frac{t}{k^n}.\end{array}$$

Now, writing $t=t/3+t/3+t/3$ and for any sequence $a_n$, using rectangular property, we have the following cases:

Case 1.

When $p=2m+1,$ i.e., $p$ is odd, then $$\begin{array}{c}\text{(18)}& {\mathfrak{m}}_q{a}_n,a_{n+2m+1},t\ge {\mathfrak{m}}_q{a}_n,a_{n+1},\frac{t/3}{f{a}_n,a_{n+1}}\ast {\mathfrak{m}}_q{a}_{n+1},a_{n+2},\frac{t/3}{g{a}_{n+1},a_{n+2}}\ast {\mathfrak{m}}_q{a}_{n+2},a_{n+2m+1},\frac{t/3}{h{a}_{n+2},a_{n+2m+1}}\ge {\mathfrak{m}}_q{a}_n,a_{n+1},\frac{t/3}{f{a}_n,a_{n+1}}\ast {\mathfrak{m}}_q{a}_{n+1},a_{n+2},\frac{t/3}{g{a}_{n+1},a_{n+2}}\\ \ast {\mathfrak{m}}_q{a}_{n+2},a_{n+3},\frac{t/3^2}{f{a}_{n+2},a_{n+3}h{a}_{n+2},a_{n+2m+1}}\\ \ast {\mathfrak{m}}_q{a}_{n+3},a_{n+4},\frac{t/3^2}{g{a}_{n+3},a_{n+4}h{a}_{n+2},a_{n+2m+1}}\\ \ast {\mathfrak{m}}_q{a}_{n+4},a_{n+2m+1},\frac{t/3^2}{h{a}_{n+4},a_{n+2m+1}h{a}_{n+2},a_{n+2m+1}}\ge {\mathfrak{m}}_q{a}_n,a_{n+1},\frac{t/3}{f{a}_n,a_{n+1}}\ast {\mathfrak{m}}_q{a}_{n+1},a_{n+2},\frac{t/3}{g{a}_{n+1},a_{n+2}}\\ \ast {\mathfrak{m}}_q{a}_{n+2},a_{n+3},\frac{t/3^2}{f{a}_{n+2},a_{n+3}h{a}_{n+2},a_{n+2m+1}}\\ \ast {\mathfrak{m}}_q{a}_{n+3},a_{n+4},\frac{t/3^2}{g{a}_{n+3},a_{n+4}h{a}_{n+2},a_{n+2m+1}}\\ \ast {\mathfrak{m}}_q{a}_{n+4},a_{n+5},\frac{t/3^3}{f{a}_{n+4},a_{n+5}h{a}_{n+2},a_{n+2m+1}h{a}_{n+4},a_{n+2m+1}}\\ \ast {\mathfrak{m}}_q{a}_{n+5},a_{n+6},\frac{t/3^3}{g{a}_{n+5},a_{n+6}h{a}_{n+2},a_{n+2m+1}h{a}_{n+4},a_{n+2\text{m}+1}}\\ \ast {\mathfrak{m}}_q{a}_{n+6},a_{n+7},\frac{t/3^4}{f{a}_{n+6},a_{n+7}h{a}_{n+2},a_{n+2m+1}h{a}_{n+4},a_{n+2m+1}h{a}_{n+6},a_{n+2m+1}}\\ \ast {\mathfrak{m}}_q{a}_{n+7},a_{n+8},\frac{t/3^4}{g{a}_{n+7},a_{n+8}h{a}_{n+2},a_{n+2m+1}h{a}_{n+4},a_{n+2m+1}h{a}_{n+6},a_{n+2m+1}}\\ ⋮\\ \ast {\mathfrak{m}}_q{a}_{n+2m-2},a_{n+2m-1},\frac{t/3^m}{f{a}_{n+2m-2},a_{n+2m-1}h{a}_{n+2m-2},a_{n+2m+1}\cdots h{a}_{n+2},a_{n+2m+1}}\\ \ast {\mathfrak{m}}_q{a}_{n+2m-1},a_{n+2m},\frac{t/3^m}{g{a}_{n+2m-1},a_{n+2m}h{a}_{n+2m-2},a_{n+2m+1}\cdots h{a}_{n+2},a_{n+2m+1}}\\ \ast {\mathfrak{m}}_q{a}_{n+2m},a_{n+2m+1},\frac{t/3^m}{h{a}_{n+2m},a_{n+2m+1}h{a}_{n+2m-2},a_{n+2m+1}\cdots h{a}_{n+2},a_{n+2m+1}}.\end{array}$$

Applying (17) on right hand side, we deduce $$\begin{array}{c}\text{(19)}& {\mathfrak{m}}_q{a}_n,a_{n+2m+1},t\ge {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3}{f{a}_n,a_{n+1}{k}^n}\ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3}{g{a}_{n+1},a_{n+2}{k}^{n+1}}\ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^2}{f{a}_{n+2},a_{n+3}h{a}_{n+2},a_{n+2m+1}{k}^{n+2}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^2}{g{a}_{n+3},a_{n+4}h{a}_{n+2},a_{n+2m+1}{k}^{n+3}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^3}{f{a}_{n+4},a_{n+5}h{a}_{n+2},a_{n+2m+1}h{a}_{n+4},a_{n+2m+1}{k}^{n+4}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^3}{g{a}_{n+5},a_{n+6}h{a}_{n+2},a_{n+2m+1}h{a}_{n+4},a_{n+2m+1}{k}^{n+5}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^4}{f{a}_{n+6},a_{n+7}h{a}_{n+2},a_{n+2m+1}h{a}_{n+4},a_{n+2m+1}h{a}_{n+6},a_{n+2m+1}{k}^{n+6}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^4}{g{a}_{n+7},a_{n+8}h{a}_{n+2},a_{n+2m+1}h{a}_{n+4},a_{n+2m+1}h{a}_{n+6},a_{n+2m+1}{k}^{n+7}}\\ ⋮\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^m}{f{a}_{n+2m-2},a_{n+2m-1}h{a}_{n+2m-2},a_{n+2m+1}\cdots h{a}_{n+2},a_{n+2m+1}{k}^{n+2m-2}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^m}{g{a}_{n+2m-1},a_{n+2m}h{a}_{n+2m-2},a_{n+2m+1}\cdots h{a}_{n+2},a_{n+2m+1}{k}^{2m-1}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^m}{h{a}_{n+2m},a_{n+2m+1}h{a}_{n+2m-2},a_{n+2m+1}\cdots h{a}_{n+2},a_{n+2m+1}{k}^{2m}}.\end{array}$$

Case 2.

When $p=2m$, i.e., $p$ is even, then $$\begin{array}{c}\text{(20)}& {\mathfrak{m}}_q{a}_n,a_{n+2m},t\ge {\mathfrak{m}}_q{a}_n,a_{n+1},\frac{t/3}{f{a}_n,a_{n+1}}\ast {\mathfrak{m}}_q{a}_{n+1},a_{n+2},\frac{t/3}{g{a}_{n+1},a_{n+2}}\ast {\mathfrak{m}}_q{a}_{n+2},a_{n+2m},\frac{t/3}{h{a}_{n+2},a_{n+2m}}\ge {\mathfrak{m}}_q{a}_n,a_{n+1},\frac{t/3}{f{a}_n,a_{n+1}}\ast {\mathfrak{m}}_q{a}_{n+1},a_{n+2},\frac{t/3}{g{a}_{n+1},a_{n+2}}\\ \ast {\mathfrak{m}}_q{a}_{n+2},a_{n+3},\frac{t/3^2}{f{a}_{n+2},a_{n+3}h{a}_{n+2},a_{n+2m}}\\ \ast {\mathfrak{m}}_q{a}_{n+3},a_{n+4},\frac{t/3^2}{g{a}_{n+3},a_{n+4}h{a}_{n+2},a_{n+2m}}\\ \ast {\mathfrak{m}}_q{a}_{n+4},a_{n+2m},\frac{t/3^2}{h{a}_{n+4},a_{n+2m}h{a}_{n+2},a_{n+2m}}\ge {\mathfrak{m}}_q{a}_n,a_{n+1},\frac{t/3}{f{a}_n,a_{n+1}}\ast {\mathfrak{m}}_q{a}_{n+1},a_{n+2},\frac{t/3}{g{a}_{n+1},a_{n+2}}\\ \ast {\mathfrak{m}}_q{a}_{n+2},a_{n+3},\frac{t/3^2}{f{a}_{n+2},a_{n+3}h{a}_{n+2},a_{n+2m}}\\ \ast {\mathfrak{m}}_q{a}_{n+3},a_{n+4},\frac{t/3^2}{g{a}_{n+3},a_{n+4}h{a}_{n+2},a_{n+2m}}\\ \ast {\mathfrak{m}}_q{a}_{n+4},a_{n+5},\frac{t/3^3}{f{a}_{n+4},a_{n+5}h{a}_{n+2},a_{n+2m}h{a}_{n+4},a_{n+2m}}\\ \ast {\mathfrak{m}}_q{a}_{n+5},a_{n+6},\frac{t/3^3}{g{a}_{n+5},a_{n+6}h{a}_{n+2},a_{n+2m}h{a}_{n+4},a_{n+2m}}\\ \ast {\mathfrak{m}}_q{a}_{n+6},a_{n+7},\frac{t/3^4}{f{a}_{n+6},a_{n+7}h{a}_{n+2},a_{n+2m}h{a}_{n+4},a_{n+2m}h{a}_{n+6},a_{n+2m}}\\ \ast {\mathfrak{m}}_q{a}_{n+7},a_{n+8},\frac{t/3^4}{g{a}_{n+7},a_{n+8}h{a}_{n+2},a_{n+2m}h{a}_{n+4},a_{n+2m}h{a}_{n+6},a_{n+2m}}\\ ⋮\\ \ast {\mathfrak{m}}_q{a}_{n+2m-4},a_{n+2m-3},\frac{t/3^{m-1}}{f{a}_{n+2m-4},a_{n+2m-3}h{a}_{n+2m-4},a_{n+2m}\cdots h{a}_{n+2},a_{n+2m}}\\ \ast {\mathfrak{m}}_q{a}_{n+2m-3},a_{n+2m-2},\frac{t/3^{m-1}}{g{a}_{n+2m-3},a_{n+2m-2}h{a}_{n+2m-4},a_{n+2m}\cdots h{a}_{n+2},a_{n+2m}}\\ \ast {\mathfrak{m}}_q{a}_{n+2m-2},a_{n+2m},\frac{t/3^{m-1}}{h{a}_{n+2m-2},a_{n+2m}h{a}_{n+2m-4},a_{n+2m+1}\cdots h{a}_{n+2},a_{n+2m}}.\end{array}$$

Now applying (17), we deduce $$\begin{array}{c}\text{(21)}& {\mathfrak{m}}_q{a}_n,a_{n+2m+1},t\ge {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3}{f{a}_n,a_{n+1}{k}^n}\ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3}{g{a}_{n+1},a_{n+2}{k}^{n+1}}\ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3/3^2}{f{a}_{n+2},a_{n+3}h{a}_{n+2},a_{n+2m}{k}^{n+2}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3/3^2}{g{a}_{n+3},a_{n+4}h{a}_{n+2},a_{n+2m}{k}^{n+3}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^3}{f{a}_{n+4},a_{n+5}h{a}_{n+2},a_{n+2m}h{a}_{n+4},a_{n+2m+1}{k}^{n+4}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^3}{g{\text{a}}_{n+5},a_{n+6}h{a}_{n+2},a_{n+2m}h{a}_{n+4},a_{n+2m+1}{k}^{n+5}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^4}{f{a}_{n+6},a_{n+7}h{a}_{n+2},a_{n+2m}h{a}_{n+4},a_{n+2m+1}h{a}_{n+6},a_{n+2m}{k}^{n+6}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^4}{g{a}_{n+7},a_{n+8}h{a}_{n+2},a_{n+2m}h{a}_{n+4},a_{n+2m}h{a}_{n+6},a_{n+2m}{k}^{n+7}}\\ ⋮\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^{m-1}}{f{a}_{n+2m-3},a_{n+2m-2}h{a}_{n+2m-4},a_{n+2m}\cdots h{a}_{n+2},a_{n+2m}{k}^{n+2m-4}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^{m-1}}{g{a}_{n+2m-1},a_{n+2m}h{a}_{n+2m-2},a_{n+2m+1}\cdots h{a}_{n+2},a_{n+2m+1}{k}^{n+2m-3}}\\ \ast {\mathfrak{m}}_q{a}_0,a_1,\frac{t/3^{m-1}}{h{a}_{n+2m-2},a_{n+2m}h{a}_{n+2m-4},a_{n+2m}\cdots h{a}_{n+2},a_{n+2m}{k}^{n+2m-2}}.\end{array}$$

Using (13) for each case, we obtain $$\begin{array}{}\text{(22)}& \underset{n⟶\infty }{\lim }{\mathfrak{m}}_q{a}_n,a_{n+p},t=1\ast 1\ast \cdots \ast 1=1,\end{array}$$which shows that $a_n$ is a Cauchy sequence in $F$ and converges to some $a\in F$ (as $F$ is complete), i.e., $$\begin{array}{}\text{(23)}& \underset{n⟶\infty }{\lim }{\mathfrak{m}}_q{a}_n,a,t=1.\end{array}$$

Now, we show that $a$ is the fixed point of $T$. From (14), $$\begin{array}{}\text{(24)}& {\mathfrak{m}}_{q}a,Ta,t\ge {\mathfrak{m}}_{q}a,a_n,\frac{t/3}{fa,a_n}\ast {\mathfrak{m}}_q{a}_n,a_{n+1},\frac{t/3}{g{a}_n,a_{n+1}}\ast {\mathfrak{m}}_q{a}_{n+1},Ta,\frac{t/3}{h{a}_{n+1},Ta}\ge {\mathfrak{m}}_{q}a,a_n,\frac{t/3}{fa,a_n}\ast {\mathfrak{m}}_{q}T{a}_{n-1},T{a}_n,\frac{t/3}{g{a}_n,a_{n+1}}\ast {\mathfrak{m}}_{q}T{a}_n,Ta,\frac{t/3}{h{a}_{n+1},Ta}\ge {\mathfrak{m}}_{q}a,a_n,\frac{t/3}{fa,a_n}\ast {\mathfrak{m}}_q{a}_{n-1},a_n,\frac{t/3}{g{a}_n,a_{n+1}k}\ast {\mathfrak{m}}_q{a}_n,a,\frac{t/3}{h{a}_{n+1},Tak}⟶1\ast 1\ast 1=1asn⟶\infty ,\end{array}$$which shows that $Ta=a$, i.e., $a$ is a fixed point of $T$. To prove $a$ is unique, we assume $a^{\prime }$ is also a fixed point of $T$, i.e., $T{a}^{\prime }=a^{\prime }$. Then, $$\begin{array}{}\text{(25)}& {\mathfrak{m}}_{q}a,a^{\prime },t={\mathfrak{m}}_{q}Ta,T{a}^{\prime },t\ge {\mathfrak{m}}_{q}a,a^{\prime },\frac{t}{k},\end{array}$$which implies that $a=a^{\prime }$, so the fixed point of $T$ is unique.

Example 3.

Let $F=0,1$ and ${\mathfrak{m}}_q:F\times F\times 0,\infty ⟶0,1$ be defined as ${\mathfrak{m}}_q\mu ,\nu ,t=e^{-\mid \mu -\nu \mid /t}$ for all $t>0$. Let $f,g,h:F\times F⟶1,1/k$ be defined by $f\mu ,\nu =\mu +\nu +1,g\nu ,\xi ={\nu }^2+\xi +1$ and $h\xi ,\eta ={\xi }^2+{\eta }^2+1$. Then, $F,{\mathfrak{m}}_q,\ast$ is a complete fuzzy triple controlled metric space. Let $T:F⟶F$ be given by $T\mu =1-\mu /4$. Then, $$\begin{array}{}\text{(26)}& {\mathfrak{m}}_{q}T\mu ,T\nu ,kt={\mathfrak{m}}_{q}1-\frac{\mu }{4},1-\frac{\nu }{4},kt=e^{-\mid 1-\mu /4-1+\nu /4\mid /kt}=e^{-\mu /4-\nu /4/kt}\ge e^{-\mu -\nu /4kt}\ge e^{-\mu -\nu /t}={\mathfrak{m}}_q\mu ,\nu ,t,\end{array}$$for all $\mu ,\nu \in F,$ where $k\in 13/50,1.$ Since all the conditions of Theorem 12 are satisfied, therefore $T$ has a unique fixed point which is $\mu =4/5$.

Theorem 12 generalizes Theorem 2.1 of [31] as follows.

Corollary 13.

Let $F,{\mathfrak{m}}_q,\ast$ be a complete fuzzy rectangular $b$-metric space with $b\ge 1$ such that $$\begin{array}{}\text{(27)}& \underset{t⟶\infty }{\lim }{\mathfrak{m}}_q\mu ,\nu ,t=1.\end{array}$$

Let $T:F⟶F$ be a mapping such that $$\begin{array}{}\text{(28)}& {\mathfrak{m}}_{q}T\mu ,T\nu ,kt\ge {\mathfrak{m}}_q\mu ,\nu ,t,\mathit{\text{for}}all\mu ,\nu \in F.\end{array}$$

Then, $T$ has a unique fixed point.

Remark 14.

If we choose $f\mu ,\nu =g\mu ,\nu =h\mu ,\nu =1,$ then our main result reduces to the Banach contraction principle for the fuzzy rectangular metric space defined in [37].

3. Application

Consider the integral equation $$\begin{array}{}\text{(29)}& \mu \xi =g\xi +{\int }_0{}^{\xi }G\xi ,\gamma ,\mu \gamma d\gamma ,\end{array}$$where $\xi \in I=0,1$.

Define $f\mu ,\nu =3\mu +\nu +1,g\nu ,\xi =3{\mu }^2+{\nu }^2+1$, and $h\xi ,\eta =3{\mu }^2+{\nu }^2-1.$ Also, for all $t>0$, $\mu ,\nu \in CI,ℝ,$$$\begin{array}{}\text{(30)}& {\mathfrak{m}}_q\mu ,\nu ,t=e^{-\underset{\xi \in I}{\sup }{\mu \xi -\nu \xi }^2/t}.\end{array}$$

Then, $CI,ℝ,{\mathfrak{m}}_q$ is a complete fuzzy triple controlled metric space with product $t$-norm, where $CI,ℝ$ is the space of all real valued continuous functions defined on $I.$

Theorem 15.

Consider an integral operator defined on $CI,ℝ$ as follows: $$\begin{array}{}\text{(31)}& T\mu \xi =g\xi +{\int }_0{}^{\xi }G\xi ,\gamma ,\mu \gamma d\gamma ,g\in CI,ℝ,\end{array}$$where $G$ satisfies the following condition.

There exists $f:I\times I⟶0,\infty$ such that for all $\xi ,\gamma \in I,$$f\xi ,\gamma \in L^{1}I,ℝ,$ and for all $\mu ,\nu \in CI,ℝ$, we have $$\begin{array}{}\text{(32)}& {G\xi ,\gamma ,\mu \gamma -G\xi ,\gamma ,\nu \gamma \mid {}^2\le f^2\xi ,\gamma \mu \gamma -\nu \gamma }^2,\end{array}$$where $$\begin{array}{}\text{(33)}& 0<\underset{\xi \in I}{\sup }{\int }_0{}^{\xi }{f}^2\xi ,\gamma d\gamma \le k<1.\end{array}$$

Then, the integral equation (29) has a unique solution.

Proof.

Let $\mu ,\nu \in CI,ℝ$. Note that $$\begin{array}{}\text{(34)}& {\mathfrak{m}}_{q}T\mu ,T\nu ,kt=e^{-\underset{\xi \in I}{\sup }{T\mu \xi -T\nu \xi }^2/kt}=e^{-\underset{\xi \in I}{\sup }{{\int }_0{}^{\xi }G\xi ,\gamma ,\mu \gamma -G\xi ,\gamma ,\nu \gamma d\gamma }^2/kt}\ge e^{-\underset{\xi \in I}{\sup }{\int }_0{}^{\xi }{G\xi ,\gamma ,\mu \gamma -G\xi ,\gamma ,\nu \gamma }^{2}d\gamma /kt}\ge e^{-\underset{\xi \in I}{\sup }{\int }_0{}^{\xi }{f}^2\xi ,\gamma {\mu \gamma -\nu \gamma }^{2}d\gamma /kt}\ge e^{-{\mu \gamma -\nu \gamma }^2\underset{\xi \in I}{\sup }{\int }_0{}^{\xi }{f}^2\xi ,\gamma d\gamma /kt}\ge e^{-k{\mu \gamma -\nu \gamma }^2/kt}=e^{-{\mu \gamma -\nu \gamma }^2/t}\ge e^{-\underset{\gamma \in I}{\sup }{\mu \gamma -\nu \gamma }^2/t}={\mathfrak{m}}_q\mu ,\nu ,t.\end{array}$$

Thus, ${\mathfrak{m}}_{q}T\mu ,T\nu ,kt\ge {\mathfrak{m}}_q\mu ,\nu ,t$ for all $\mu ,\nu \in CI,ℝ$ and consequently, all the conditions of Theorem 12 are satisfied. Therefore, the integral equation (29) has a unique solution.

4. Conclusion

In this article, the concept of fuzzy triple controlled metric space is given which generalizes fuzzy rectangular $b$-metric space, rectangular fuzzy metric space. We have established Banach fixed point theorem in this space. An example is also presented that illustrates our main result. Our newly defined results can be used for further investigation in many existing results in the literature.

Authors’ Contributions

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

Acknowledgments

Authors are thankful to the editor and anonymous referees for their valuable comments and suggestions.

References

[1] S. Banach, "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales," Fundamenta Mathematicae, vol. 3 no. 1, pp. 133-181, DOI: 10.4064/fm-3-1-133-181, 1922.

[2] M. A. Alghamdi, N. Shahzad, O. Valero, "Fixed point theorems in generalized metric spaces with applications to computer science," Fixed Point Theory and Applications, vol. 2013 no. 1,DOI: 10.1186/1687-1812-2013-118, 2013.

[3] O. Ege, I. Karaca, "Banach fixed point theorem for digital images," Journal of Nonlinear Sciences and Applications, vol. 8 no. 3, pp. 237-245, DOI: 10.22436/jnsa.008.03.08, 2015.

[4] R. Espinola, W. A. Kirk, "Fixed point theorems in R-trees with applications to graph theory," Topology and its Applications, vol. 153 no. 7, pp. 1046-1055, DOI: 10.1016/j.topol.2005.03.001, 2006.

[5] S. E. Han, "Banach fixed point theorem from the viewpoint of digital topology," Journal of Nonlinear Science and Applications, vol. 9 no. 3, pp. 895-905, DOI: 10.22436/jnsa.009.03.19, 2016.

[6] F. Lael, N. Saleem, M. Abbas, "On the fixed points of multivalued mappings in b-metric spaces and their application to linear systems," University Politehnica of Bucharest Scientific Bulletin-Series A-Applied Mathematics and Physics, vol. 82 no. 4, pp. 121-130, 2020.

[7] H. Alolaiyan, N. Saleem, M. Abbas, "A natural selection of a graphic contraction transformation in fuzzy metric spaces," Journal of Nonlinear Sciences and Applications, vol. 11 no. 2, pp. 218-227, DOI: 10.22436/jnsa.011.02.04, 2018.

[8] A. B. Amar, A. Jeribi, M. Mnif, "Some fixed point theorems and application to biological model," Numerical Functional Analysis and Optimization, vol. 29 no. 1-2,DOI: 10.1080/01630560701749482, 2008.

[9] A. T. Bharucha-Reid, "Fixed point theorems in probabilistic analysis," Bulletin of the American Mathematical Society, vol. 82 no. 5, pp. 641-658, DOI: 10.1090/S0002-9904-1976-14091-8, 1976.

[10] E. Karapinar, T. Abdeljawad, F. Jarad, "Applying new fixed point theorems on fractional and ordinary differential equations," Advances in Difference Equations, vol. 2019 no. 1,DOI: 10.1186/s13662-019-2354-3, 2019.

[11] K. Sitthikul, S. Saejung, "Some fixed point theorems in complex valued metric spaces," Fixed Point Theory and Applications, vol. 2012 no. 1,DOI: 10.1186/1687-1812-2012-189, 2012.

[12] L. A. Zadeh, "Fuzzy sets," Information and Control, vol. 8 no. 3, pp. 338-353, DOI: 10.1016/S0019-9958(65)90241-X, 1965.

[13] M. Abbas, F. Lael, N. Saleem, "Fuzzy b-metric spaces: fixed point results for ψ -contraction correspondences and their application," Axioms, vol. 9 no. 2,DOI: 10.3390/axioms9020036, 2020.

[14] J. J. Buckley, T. Feuring, "Introduction to fuzzy partial differential equations," Fuzzy Sets and Systems, vol. 105 no. 2, pp. 241-248, DOI: 10.1016/S0165-0114(98)00323-6, 1999.

[15] O. Kaleva, "Fuzzy differential equations," Fuzzy Sets and Systems, vol. 24 no. 3, pp. 301-317, DOI: 10.1016/0165-0114(87)90029-7, 1987.

[16] M. L. Puri, D. A. Ralescu, "Differentials of fuzzy functions," Journal of Mathematical Analysis and Applications, vol. 91 no. 2, pp. 552-558, DOI: 10.1016/0022-247X(83)90169-5, 1983.

[17] N. Saleem, M. Abbas, M. D. L. Sen, "Optimal approximate solution of coincidence point equations in fuzzy metric spaces," Mathematics, vol. 7 no. 4,DOI: 10.3390/math7040327, 2019.

[18] N. Saleem, M. Abbas, Z. Raza, "Optimal coincidence best approximation solution in non-Archimedean fuzzy metric spaces," Iranian Journal of Fuzzy Systems, vol. 13 no. 3, pp. 113-124, 2016.

[19] R. E. Bellman, L. A. Zadeh, "Decision-making in a fuzzy environment," Management Science, vol. 17 no. 4, pp. B-141-B-273, 1970.

[20] S. G. Gal, "Approximation Theory in Fuzzy Setting," Handbook of Analytic-Computational Methods in Applied Mathematics, pp. 617-666, DOI: 10.1201/9780429123610-13, 2000.

[21] M. L. Puri, D. A. Ralescu, L. Zadeh, "Fuzzy Random Variables," Readings in Fuzzy Sets for Intelligent Systems, pp. 265-271, DOI: 10.1016/B978-1-4832-1450-4.50029-8, 1993.

[22] I. Kramosil, J. Michálek, "Fuzzy metrics and statistical metric spaces," Kybernetika, vol. 11 no. 5, pp. 336-344, 1975.

[23] M. Grabiec, "Fixed points in fuzzy metric spaces," Fuzzy Sets and Systems, vol. 27 no. 3, pp. 385-389, DOI: 10.1016/0165-0114(88)90064-4, 1988.

[24] A. George, P. Veeramani, "On some results in fuzzy metric spaces," Fuzzy Sets and Systems, vol. 64 no. 3, pp. 395-399, DOI: 10.1016/0165-0114(94)90162-7, 1994.

[25] I. A. Bakhtin, "The contraction mapping principle in almost metric spaces," Functional Analysis, vol. 30, pp. 26-37, 1989.

[26] A. Branciari, "A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces," Universitatis Debreceniensis, vol. 57 no. 1–2, pp. 31-37, 2000.

[27] J. R. Roshan, V. Parvaneh, Z. Kadelburg, N. Hussain, "New fixed point results in b-rectangular metric spaces," Nonlinear Analysis: Modelling and Control, vol. 21 no. 5, pp. 614-634, 2016.

[28] S. Nadaban, "Fuzzy b-metric spaces," International Journal of Computers Communications & Control, vol. 11 no. 2, pp. 273-281, DOI: 10.15837/ijccc.2016.2.2443, 2016.

[29] T. Kamran, M. Samreen, Q. U. L. Ain, "A generalization of b -metric space and some fixed point theorems," Mathematics, vol. 5 no. 2,DOI: 10.3390/math5020019, 2017.

[30] F. Mehmood, R. Ali, C. Ionescu, T. Kamran, "Extended fuzzy b-metric spaces," Journal of Mathematical Analysis, vol. 8, pp. 124-131, 2017.

[31] F. Mehmood, R. Ali, N. Hussain, "Contractions in fuzzy rectangular b -metric spaces with application," Journnal of Intelligent & Fuzzy Systems, vol. 37, pp. 1275-1285, DOI: 10.3233/jifs-182719, 2019.

[32] M. Asim, M. Imdad, S. Radenovic, "Fixed point results in extended rectangular b-metric spaces with an application," UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, vol. 81 no. 2, pp. 43-50, 2019.

[33] N. Saleem, H. Işik, S. Furqan, C. Park, "Fuzzy double controlled metric spaces and related results," Journnal of Intelligent & Fuzzy Systems, vol. 40 no. 5, pp. 9977-9985, DOI: 10.3233/jifs-202594, 2021.

[34] T. Abdeljawad, N. Mlaiki, H. Aydi, N. Souayah, "Double controlled metric type spaces and some fixed point results," Mathematics, vol. 6 no. 12,DOI: 10.3390/math6120320, 2018.

[35] N. Mlaiki, H. Aydi, N. Souayah, T. Abdeljawad, "Controlled metric type spaces and the related contraction principle," Mathematics, vol. 6 no. 10,DOI: 10.3390/math6100194, 2018.

[36] B. Schweizer, A. Sklar, "Statistical metric spaces," Pacific Journal of Mathematics, vol. 10, pp. 385-389, 1960.

[37] R. Chugh, S. Kumar, "Weakly compatible maps in generalized fuzzy metric spaces," Journal of Analysis, vol. 10, pp. 65-74, 2002.