$N_b$-fuzzy metric spaces with topological properties and applications

1.   Introduction and preliminaries

In the literature of mathematical analysis there exist many generalizations of metric space (MS). The famous generalization of MS is $2$-MS introduced by Gähler ^[1] but notice that a $2$-metric is not a continuous function of its variables whereas an ordinary metric is. This led Dhage ^[2] to introduce the notion of $D$-MS. After that Mustafa and Sims ^[3] showed that most of the topological properties of $D$-MS were not correct. Then they introduced the $G$-MS unfortunately Jleli and Samet ^[4] showed that most of the obtained fixed point theorems on $G$-MS can be deduced immediately from fixed point theorems on MS or quasi MS. After that Sedghi et al. ^[5] defined $S$-MS. On the other hand after replacing set of positive real number by ordered Banach space Huang and Zhang ^[6] introduced cone MS and many types of cone MSs are defined in ^[7,8,9,10]. The more generalized idea of the triangle inequality was introduced so that the concept of MS was extended to $b$-MS in 1989 by Bakhtin ^[11]. After that a $S_{b}$-MS ^[12,13,14] is also defined.

One of the most famous generalization of MS is FMS introduced by Kramosil and Michalek ^[15] in 1975. In 1994, George and Veeramani ^[16] modified the concept of FMS. It has wide applications in computer science and engineering like (see ^[17,18]) as in colour image filtering, neural network, data mining, etc. Some of its generalizations are fuzzy metric type space (or fuzzy $b$-MS ^[19,20,21]), $Q$-FMS ^[22], dislocated $G$-FMS ^[23] and $N$-FMS ^[24] and also defined same space in ^[25] in 2020 etc.

Inspired by the work of Nadaban ^[20] we are introducing $N_{b}$-FMS which is the generalization of $N$-FMS.

In this work, after introduction and preliminaries in Section 2, we define $N_{b}$-FMS which is a generalization of $N$-FMS and $S_{b}$-MS. In Section 3, we define various types of $N$-FMS and $N_{b}$-FMS and for the base we have also define different $S$-MS and $S_{b}$-MS such as quasi $S$-MS, with examples and counterexamples. We also prove a decomposition theorem on pseudo $N$-FMS. In Section 4, we define convergent sequence, Cauchy sequence and topology of symmetric $N_{b}$-FMS and prove various theorems related to convergence and topology. In Section 5, we apply q-contraction mapping in symmetric $N_{b}$-FMS and prove celebrated BCP with example and integral analogue. We also provide application of BCP in solution of integral equations and linear equations.

We recall the following definitions which will be needed in the sequel.

Definition 1.1. ^[26] A map $o:[0, 1]^3\rightarrow [0, 1]$ is called continuous $t$-norm if it satisfies the following conditions:

(1) $o(a, 1, 1) = a, o(0, 0, 0) = 0;$

(2) $o(a, b, c) = o(a, c, b) = o(b, c, a);$

(3) $o(a_{1}, b_{1}, c_{1}) \geq o(a_{2}, b_{2}, c_{2})$ for $a_{1} \geq a_{2}, b_{1}\geq b_{2}, c_{1} \geq c_{2}$.

Examples of $t$-norm are

(1) $a\circ b\circ c = abc$ product $t$-norm;

(2) $a\circ b\circ c = \min\{a, b, c\}$ minimum $t$-norm.

Definition 1.2. ^[18] A triplet $(X, N, o)$ is an $N$-FMS, if $X$ is an arbitrary (nonempty) set, $o$ is continuous $t$-norm and $N$ is a fuzzy set on $X^{3}\times (0, \infty)$ satisfying the following conditions for all $\zeta, \eta, \upsilon \in X$ and $r, s, t > 0$.

$(N_{1})$ $N(\zeta, \eta, \upsilon, t) > 0$,

$(N_{2})$ $N(\zeta, \eta, \upsilon, t) = 1$ iff $\zeta = \eta = \upsilon$,

$(N_{3})$ $N(\zeta, \eta, \upsilon, r+s+t)\geq N(\zeta, \zeta, a, r)\circ N(\eta, \eta, a, s)\circ N(\upsilon, \upsilon, a, t)$,

$(N_{4})$ $N(\zeta, \eta, \upsilon, \cdot):(0, \infty)\rightarrow (0, 1]$ is a continuous function.

Definition 1.3. ^[13] Let $X$ be a nonempty set and $k\geq 1$ be a given real number. A function $S_{b}:X^{3}\rightarrow [0, \infty)$ is said to be $S_{b}$-metric iff for all $\zeta, \eta, \upsilon, a\in X$ the following conditions are satisfied:

$(S_{b1})$ $S_{b}(\zeta, \eta, \upsilon) = 0$ iff $\zeta = \eta = \upsilon$,

$(S_{b2})$ $S_{b}(\zeta, \eta, \upsilon)\leq k\Big[S_{b}(\zeta, \zeta, a)+ S_{b}(\eta, \eta, a)+ S_{b}(\upsilon, \upsilon, a)\Big]$.

The pair $(X, S_{b})$ is called an $S_{b}$-MS.

2.   A new type of FMSs

In this section, we will define $N_{b}$-FMS with various examples and prove that $N_{b}$-FMS generalizes the $N$-FMS with the help of counterexamples.

Definition 2.1. A quadruple $(X, N_{b}, o, k)$ is an $N_{b}$-FMS if $X$ is an arbitrary (nonempty) set, $o$ is continuous $t$-norm, $k\geq 1$ is a real number and $N_{b}$ is a fuzzy set on $X^{3} \times (0, \infty)$ satisfying the following conditions for all $\zeta, \eta, \upsilon, a\in X$ and $r, s, t > 0$:

$(N_{b1})$ $N_{b}(\zeta, \eta, \upsilon, t) > 0$,

$(N_{b2})$ $N_{b}(\zeta, \eta, \upsilon, t) = 1$ iff $\zeta = \eta = \upsilon$,

$(N_{b3})$ $N_{b}(\zeta, \eta, \upsilon, k(r +s + t))\geq N_{b}(\zeta, \zeta, a, r)\circ N_{b}(\eta, \eta, a, s)\circ N_{b}(\upsilon, \upsilon, a, t)$,

$(N_{b4})$ $N_{b}(\zeta, \eta, \upsilon, \cdot)(0, \infty)\rightarrow (0, 1]$ is a continuous function.

$N_{b}(\zeta, \eta, \upsilon, t)$ is considered as the degree of nearness of $\zeta, \eta$ and $\upsilon$ with respect to $t$.

Remark 2.2. We note that $N_{b}$-FMS is the generalization of $N$-FMS and $b$-MS or we can say that it is the fuzzy version of $S_{b}$-MS. Indeed each $N$-FMS is a $N_{b}$-FMS for $k = 1$. However each $N_{b}$-FMS is need not be $N$-FMS.

Example 2.3. Let $X = R$ be real line and $S_{b}$ be an $S_{b}$-metric on $X$ defined by

Let $N_{b}$ be the function on $X^{3} \times (0, \infty)$ defined by

for all $\zeta, \eta, \upsilon \in X$ and $t > 0$. Then $(X, N_{b}, k, o)$ is an $N_{b}$-FMS with constant $k$ and product $t$-norm.

Proof. $N_{b1}, N_{b2}$ and $N_{b4}$ are obvious we check only $N_{b3}.$ Now,

Hence $N_{b3}$ satisfied.

Example 2.4. Let $X = R$ be a real line and $S_{b}$ be an $S_{b}$-MS as defined in Example 2.3. Let $N_{b}$ is a function on $X^{3}\times (0, \infty)$ defined by

for all $\zeta, \eta, \upsilon\in X$ and $t$ is product $t$-norm.Then $(X, N_{b}, \cdot, k)$ be an $N_{b}$-FMS with constant $k$.

Proof. $N_{b1}, N_{b2}$ and $N_{b4}$ conditions are standard. So we verify only $N_{b3}$, for this consider

Hence $N_{b3}$ satisfied.

Definition 2.5. Let $(X, N_{b}, o, k)$ be an $N_{b}$-FMS and $k\geq 1$. An $N_{b}$-fuzzy metric $N_{b}$ is called symmetric if

for all $\zeta, \eta\in X$ and $t > 0$.

Remark 2.6. From Definition 2.1 for $k = 1$ $N_{b}$-FMS induced to an $N$-FMS and $N$-fuzzy metric satisfies symmetry property (2.1) automatically (see Proposition 3.5 of ^[24]). However $N_{b}$-fuzzy metric is not symmetric everywhere. So $N_{b}$-FMS is more general structure to $N$-FMS.

Example 2.7. Let $X = R$ and $N_{b}$ be a function on $X^{3}\times (0, \infty)$ defined by

for all $\zeta, \eta, \upsilon\in X$, $t > 0$ and $p > 1$. Noting that $(X, N_{b}, o, k)$ be symmetric $N_{b}$-FMS with $k = 2^{2(p-1)}$ and product $t$ norm $o$.

Example 2.8. Let $X = R^{+}\cup \{0\}$, Define $S_{b}:X^3 \rightarrow [0, \infty)$ by

Then $(X, S_{b}, k)$ is a $S_{b}$-MS where $k\geq 1$. Now we define $N_{b}$-fuzzy metric on $X^{3}\times (0, \infty)$ by

Then $(X, N_{b}, o, k)$ be a $N_{b}$-FMS. But it is not $N$-fuzzy metric for some $\zeta, \eta, \upsilon\in X$. To prove it take $\zeta = 4, \eta = 6, \upsilon = 5$, and $a = 8$ and $r = 0.1, s = 0.7, t = 0.2$ and $o$ by minimum $t$-norm. Then,

It is clear that

So that the usual triangle inequality of $N$-fuzzy metric is not satisfied. However if $k > 2$ then triangle inequality of $N_{b}$-fuzzy metric is satisfied. So above example satisfied all properties of $N_{b}$-FMS but not $N$-FMS. Moreover $N$-FMS satisfied symmetric property Proposition 3.5 of ^[24], i.e.,

But in above example it is clear that

Hence $N_{b}$-FMS does not satisfy symmetric property in general. So we conclude that $N_{b}$-FMS generalize the $N$-FMS.

3.   Types of generalized FMSs

In this section, we will define quasi $N$-FMS, pseudo $N_{b}$-FMS for this we build up a base and define quasi-$S$-MS and pseudo-$S_{b}$-MS with counterexamples. We prove decomposition theorem on pseudo $N_{b}$-FMS.

Definition 3.1. Let $X$ be a nonempty set. A function $S:X^{3}\rightarrow [0, \infty)$ is said to be quasi-$S$-metric iff for all $\zeta, \eta, \upsilon, a\in X$ the following conditions are satisfied:

$(S_{q1})$ $S(\zeta, \eta, \upsilon)\geq 0$,

$(S_{q2})$ $S(\zeta, \eta, \upsilon) = S (P\{\zeta, \eta, \upsilon\}) = 0$ iff $\zeta = \eta = \upsilon$, where $P$ is permutation,

$(S_{q3})$ $S(\zeta, \eta, \upsilon)\leq \Big[S(\zeta, \zeta, a)+S(\eta, \eta, a)+S(\upsilon, \upsilon, a)\Big]$.

The pair $(X, S)$ is called a quasi $S$-MS.

Example 3.2. Let $X = R^{+}\bigcup \{0\}$. Define $S:X^{3}\rightarrow [0, \infty)$ by

Then $(X, S)$ is a quasi $S$-MS.

Proof. Note that $S_{q1}$ is obvious and for $S_{q2}$:

that is if $\zeta \neq \eta \neq \upsilon$ then by definition of $S$ in above example $S(\zeta, \eta, \upsilon)\neq S(P\{\zeta, \eta, \upsilon\})$. Moreover $S(\zeta, \zeta, \eta)\neq S({\eta, \eta, \upsilon})$. Hence quasi $S$-metric is not symmetric in general. However, $S$-MS is satisfied symmetric property automatically (see Lemma 2.5 of ^[5]). Hence quasi $S$-MS generalizes $S$-MS.

Now, for $S_{q3}$

Hence $S_{q3}$ also satisfied.

Remark 3.3. Pseudo $S$-MS is already defined in ^[27].

Definition 3.4. A triplet $(X, S_{bp}, k)$ is said to be pseudo $S_{b}$-MS if $X$ is arbitrary (nonempty) set, $k\geq 1$ is given real number satisfying the following conditions:

$(S_{bp1})$ $\forall\; \zeta, \eta, \upsilon\in X, \; \; S_{bp}(\zeta, \eta, \upsilon)\geq 0$,

$(S_{bp2})$ $\forall\; \zeta, \eta, \upsilon\in X$ and $S_{bp}(\zeta, \eta, \upsilon) = 0$ if $\zeta = \eta = \upsilon$,

$(S_{bp3})$ $\forall\; \zeta, \eta, \upsilon, a\in X$ and $k\geq 1$,

Example 3.5. Let $X = R$, define a function $S_{bp}$ on $X^3$ to $[0, \infty)$ by

for all $\zeta, \eta, \upsilon\in X$, then $(X, S_{bp}, k)$ is pseudo $S_{b}$-MS.

Proof. It is clear that $S_{p1}, S_{p2}$ and $S_{p3}$ satisfied by above example, so it is pseudo $S_{b}$-MS but it is not $S_{b}$-MS since if

Hence it is not satisfied $S_{b_{1}}$ property of Definition 1.3.

Remark 3.6. Every $S_{b}$-MS is pseudo $S_{b}$-MS but converse is not true in general, see above example.

Definition 3.7. A quasi $N$-FMS is a triplet $(X, N_{q}, o)$ where $X$ is a nonempty set, $o$ is continuous $t$-norm and $N_{q}$ is a fuzzy set on $X^3\times (0, \infty)$ satisfying the following conditions:

$(N_{q1})$ $N_{q}(\zeta, \eta, \upsilon, t) > 0$,

$(N_{q2})$ $N_{q}(\zeta, \eta, \upsilon, t) = N_{q}\Big(P(\zeta, \eta, \upsilon), t\Big) = 1$ iff $\zeta = \eta = \upsilon$ where $P$ is permutation,

$(N_{q3})$ $N_{q}\Big(\zeta, \eta, \upsilon, (r+s+t)\Big)\geq N_{q}(\zeta, \zeta, a, r)\circ N_{q}(\eta, \eta, a, s)\circ N_{q}(\upsilon, \upsilon, a, t)$,

$(N_{q4})$ $N_{q}(\zeta, \eta, \upsilon, \cdot):(0, \infty)\rightarrow (0, 1]$ is a continuous function.

Example 3.8. Let $(X, S)$ be a quasi $S$-MS (as defined in Example 3.2). Define a function $N_{q}$ on $X^3\times(0, \infty)$ such that

Then $(X, N_{q}, o)$ is a quasi-$N$-FMS.

Proof. Let $t$ be a product $t$-norm then $N_{q1}, N_{q2}, N_{q3}$ and $N_{q4}$ are obvious. But in above Example $N_{q}(1, 1, 2, t)\neq N_{q}(2, 2, 1, t)$.

Remark 3.9. $N$-FMS satisfies symmetric property automatically (see Proposition 3.5 of ^[24]), but quasi $N$-FMS is not symmetric, i.e., in quasi $N$-FMS for some $\zeta, \eta\in X$

Hence quasi $N$-FMS are different FMS.

Definition 3.10. A quadruple $(X, N_{bp}, o, k)$ is said to be pseudo $N_{b}$-FMS if $X$ is an arbitrary (nonempty) set, $o$ is a continuous $t$-norm, $k\geq 1$ is a given real number and $N_{bp}$ is a fuzzy set on $X^{3}\times (0, \infty)$ satisfying the following conditions:

$(N_{bp1})$ $\forall\; \zeta, \eta, \upsilon\in X$ and $\forall\; t > 0, N_{bp}(\zeta, \eta, \upsilon, t) > 0$,

$(N_{bp2})$ $\forall\; \zeta, \eta, \upsilon\in X$ and $\forall\; t > 0, N_{bp}(\zeta, \eta, \upsilon, t) = 1$ if $\zeta = \eta = \upsilon$,

$(N_{bp3})$ $\forall\; \zeta, \eta, \upsilon, a \in X$ and $\forall\; r, s, t > 0$

$(N_{bp4})$ $\forall\; \zeta, \eta, \upsilon\in X, \; N_{bp}(\zeta, \eta, \upsilon, \cdot):(0, \infty)\rightarrow (0, 1]$ is continuous function.

Example 3.11. Consider $R$ with the usual metric. Let $X = \{\{\zeta_{n}\}: \{\zeta_{n}\} \mbox{ is convergent in } R\}$. Define $a\circ b\circ c = abc$ for all $a, b, c\in [0, 1]$ and

Noting that $(X, N_{bp}, o, k)$ is a pseudo $N_{b}$-FMS but not $N_{b}$-FMS.

To see this, let $\{\zeta_{n}\} = \big\{\frac{2}{n}\big\}, \{\eta_{n}\} = \big\{\frac{3}{n}\big\}, \{\upsilon_{n}\} = \big\{\frac{5}{n}\big\}$.

Then $\{\zeta_{n}\}\neq \{\eta_{n}\}\neq \{\upsilon_{n}\}$ for $\{\zeta_{n}\}, \{\eta_{n}\}$ and $\{\upsilon_{n}\}\in X$ but $N_{bp}(\zeta_{n}, \eta_{n}, \upsilon_{n}, t) = 1$.

Remark 3.12. Every $N_{b}$-FMS is a pseudo $N_{b}$-FMS denoted by $(X, N_{bp}, o, k)$ but converse is not true, see above example.

Theorem 3.13. Let $(X, N_{bp}, o, k)$ be a pseudo $N_{b}$-FMS and

Then $S = \{S_{bp}(x)\}_{x\in (0, 1)}$ is an ascending family of pseudo $S_{b}$-metric on $X$.

Proof. $S_{bp(x)}(\zeta, \zeta, \zeta) = \inf \big\{t > 0; N_{bp}(\zeta, \zeta, \zeta, t) > x\big\} = 0.$

Now,

It remain to prove that

Let $x_{1}\leq x_{2},$ then

Thus,

namely

Example 3.14. Let $X = [-1, 1]$, define $N_{bp}(\zeta, \eta, \upsilon, t)$ by

for all $\zeta, \eta, \upsilon\in X$, $o$ is continuous $t$-norm, then $(X, N_{bp}, o, k)$ is pseudo $N_{b}$-FMS. Define

Take $x_{1} = \frac{1}{4}$ and $x_{2} = \frac{1}{2}$ in (0, 1), then it is quiet natural

Now, by property of infimum

Hence,

Similarly, we can find for other distinct members in (0, 1). Then $S = \{S_{bp}(x)\}_{x\in (0, 1)}$ is an ascending family of pseudo $S_{b}$-metric on $X$.

We now define convergent sequence, Cauchy sequence, $F$-bounded set, $q$-contraction, continuous function and proved proposition and lemmas related to convergence and $q$-contraction.

Definition 3.15. Let $(X, N_{b}, o, k)$ be a symmetric $N_{b}$-FMS. A sequence $\{\zeta_{n}\}$ in $(X, N_{b}, o, k)$ converges to $\zeta\in X$, if $N_{b}(\zeta_{n}, \zeta_{n}, \zeta, t)\rightarrow 1$ or $N_{b}(\zeta, \zeta, \zeta_{n}, t)\rightarrow 1$ as $n\rightarrow \infty$ for each $t > 0$. That is, for $r > 0$ and $t > 0$ there exists $n_{0}\in N$ such that for all $n\geq n_{0}$, $N_{b}(\zeta_{n}, \zeta_{n}, \zeta, t) > 1-r$ or $N_{b}(\zeta, \zeta, \zeta_{n}, t) > 1-r$.

Lemma 3.16. Let $(X, N_{b}, o, k)$ be a symmetric $N_{b}$-FMS, where $o$ is product $t$ norm. Let $\{\zeta_{n}\}$ be a sequence in $X$. If $\{\zeta_{n}\}$ converges to $\zeta$ and $\{\zeta_{n}\}$ also converges to $\eta$ then $\zeta = \eta$. That is, if the limit of $\{\zeta_{n}\}$ exists, it is unique.

Proof. Let $\{\zeta_{n}\}$ converge to $\zeta$ and$\eta$. Then $N_{b}(\zeta, \zeta, \zeta_{n}, s)\rightarrow 1$ as $n\rightarrow \infty$ for each $s > 0$ and $N_{b}(\eta, \eta, \zeta_{n}, t-2s)\rightarrow 1$ as $n\rightarrow \infty$ for each $t-2s > 0$.

Example 3.17. Let $(X, N_{b}, o, k)$ be a symmetric $N_{b}$-FMS as defined in Example 2.3. Define a sequence $\{x_{n}\} = \{\frac {1}{n}\}$, $\forall n \in N$. Then $\{x_{n}\}$ converges to unique limit 0.

Definition 3.18. Let $(X, N_{b}, o, k)$ be a symmetric $N_{b}$-FMS and $\{\zeta_{n}\}$ be a sequence in $X$ is called Cauchy sequence, if for each $r > 0$ and $t > 0$, there exists $n_{0}\in N$ such that

for all $n, m\geq n_{0}.$

Definition 3.19. Let $(X, N_{b}, o, k)$ be a symmetric $N_{b}$-FMS. If every Cauchy sequence in $X$ is convergent in $X$, then $X$ is called a complete symmetric $N_{b}$-FMS.

Definition 3.20. Let $(X, N_{b}, o, k)$ be a symmetric $N_{b}$-FMS. A subset $A$ of $X$ is said to be $F$-bounded if there exist $t > 0$ and $0 < r < 1$ such that

Example 3.21. Let $X = R$ and $N_{b}$ be a function on $X^{3}\times (0, \infty)$ defined by

for all $\zeta, \eta, v \in X$ and $t > 0$. Then $(X, N_{b}, o, k)$ be a symmetric $N_{b}$-FMS. Let $A$ be a subset of $R$ defined by $A = \{x: 0 < x < 1\}$, then $A$ is $F$-bounded set. Particularly for $\zeta = 0$ and $\eta = 1$ the value of $r$ is $\frac{4}{t+4}$.

Definition 3.22. Let $(X, N_{b}, o, k)$ be a $N_{b}$-FMS. A self map $f:X\rightarrow X$ is a fuzzy $q$-contraction if for all $\zeta, \eta\in X$ and for some $q\in (0, 1),$ we have

Lemma 3.23. Let $(X, N_{b}, o, k)$ be a symmetric $N_{b}$-FMS, where $o$ is product $t$ norm and $\{\zeta_{n}\}$ be a sequence in $X$. If $\{\zeta_{n}\}$ converges to $\zeta$, then $\{\zeta_{n}\}$ is a Cauchy sequence.

Proof. For each $s, t > 0$ there is $p\in N$ such that

and

Hence, $\{\zeta_{n}\}$ is a Cauchy sequence.

Example 3.24. Let $(X, N_{b}, o, k)$ be a symmetric $N_{b}$-FMS as defined in Example 2.3. Define a sequence $\{x_{n}\} \in X$, by $\{1-\frac{1}{2^{n}}: n \in N\}$. Then $\{x_{n}\}$ is a convergence sequence and it converges to $1 \in X$ and it is also a Cauchy sequence.

Remark 3.25. The converse of Lemma 3.23 is not always true, i.e., every Cauchy sequence need not be convergent. For example if we take $X = [0, 1)$ in place of $R$ in the above example $\{x_{n}\}$ is not convergent in $X$ since 1 does not lie in $X$.

Remark 3.26. Symmetric $N_{b}$-FMS $(X, N_{b}, o, k)$ is complete iff the symmetric $S_{b}$-MS $(X, S_{b})$ is complete where

for all $\zeta, \eta, \upsilon \in X$ and $t\in(0, \infty)$.

Definition 3.27. Let $(X, N_{b}, o, k)$ and $(X^{'}, N_{b}^{'}, o^{'}, k^{'})$ be symmetric $N_{b}$-FMS. Then a function $f:X\rightarrow X^{'}$ is said to be continuous at a point $\zeta\in X$ iff it is sequentially continuous at $\zeta$, that is whenever $\{\zeta_{n}\}$ is convergent to $\zeta$ we have $\{f\zeta_{n}\}$ is convergent to $f(\zeta)$.

Proposition 3.28. Let $(X, N_{b}, o, k)$ be symmetric $N_{b}$-FMS and $f$ be a fuzzy $q$-contraction. If any fixed point $\zeta$ of $f$ satisfies

then

Proof. Let $\zeta\in X$ be a fixed point of $f$, as $f$ is a fuzzy $q$-contraction, so

and so

Lemma 3.29. Let $(X, N_{b}, o, k)$ be a symmetric $N_{b}$-FMS. Let $\{\zeta_{n}\}$ and $\{\eta_{n}\}$ be two sequences in $X$ and suppose $\zeta_{n} \rightarrow \zeta$, $\eta_{n} \rightarrow \eta$, as $n\rightarrow \infty$ and $N_{b}(\zeta, \zeta, \eta, t_{n}) \rightarrow N_{b}(\zeta, \zeta, \eta, t)$ as $n\rightarrow \infty$. Then $N_{b}(\zeta_{n}, \zeta_{n}, \eta_{n}, t_{n}) \rightarrow N_{b}(\zeta, \zeta, \zeta, t)$ as $n\rightarrow \infty$.

Proof. Since $\lim_{n\rightarrow \infty} \zeta_{n} = \zeta$, $\lim_{n\rightarrow \infty} \eta_{n} = \eta$ and $\lim_{n\rightarrow \infty} N_{b}(\zeta, \zeta, \eta, t_{n}) = N_{b}(\zeta, \zeta, \zeta, t)$ there is $n_{0}\in N$ such that $|t-t_{n}| < \delta$ for $n\geq n_{0}$ and $\delta < \frac{t}{2}$. We know that $N_{b}(\zeta, \zeta, \zeta, t)$ is nondecreasing with respect to $t$, so we have

and

In view of Definition 3.15 and combining the arbitrariness of $\delta$ and the continuity for $N_{b}(\zeta, \zeta, \eta, \cdot)$ with respect to $t$. For large enough $n$, we have

Consequently,

Lemma 3.30. Let $(X, N_{b}, o, k)$ be a symmetric $N_{b}$-FMS. If there exists $q\in (0, 1)$ such that $N_{b}(\zeta, \zeta, \eta, t) \geq N_{b}(\zeta, \zeta, \eta, \frac{t}{q})$ for all $\zeta, \eta\in X$, $t > 0$ and

Then $\zeta = \eta$.

Proof. Suppose that there exists $q\in (0, 1)$ such that $N_{b}(\zeta, \zeta, \eta, t) \geq N_{b}\Big(\zeta, \zeta, \eta, \frac{t}{q}\Big)$ for all $\zeta, \eta\in X$ and $t > 0$. Then,

and so

for positive integer $n$. Taking limit as $n\rightarrow \infty$, $N_{b}(\zeta, \zeta, \eta, t)\geq 1$ and hence $\zeta = \eta$.

4.   Application in fixed point theory

Fixed point theory is one of the most significant field of nonlinear functional analysis because of its wide applications in many disciplines such as studying the existence of solutions for algebraic equations, differential equations and integral equations, system of linear equations and convergence of many computational methods in economics, sports, medical science, computer science etc.

In this section as an application of fuzzy $q$-contraction in symmetric $N_{b}$-FMS, we establish an analogous of BCP in this space.

Theorem 4.1. Let $(X, N_{b}, o, k)$ be a complete symmetric $N_{b}$-FMS with

and $f$ be a fuzzy $q$-contraction. Then $f$ has a unique fixed point.

Proof. Let $\zeta_{0}\in X$ and generate a sequence $\{\zeta_{n}\}$ by the iterative process $\zeta_{n} = f^{n}(\zeta_{0}), n\in N$. Since $n, t > 0$. So by definition of fuzzy $q$-contraction, we get

Hence,

By the definition of fuzzy $q$-contraction (i.e., $q < 1$) together with condition (4.1) and letting $n\rightarrow \infty$, we get

Hence, $\{\zeta_{n}\}$ is Cauchy sequence. Since $(X, N_{b}, o, k)$ is a symmetric complete $N_{b}$-FMS, there exists $\zeta\in X$ such that

Now, we will show that $\zeta$ is a fixed point of $f$.

This shows that $f(\zeta) = \zeta,$ that is, $\zeta$ is fixed point of $f$. To prove the uniqueness, let $f(\eta) = \eta$ for some $\eta\in X$, then

Thus $\zeta = \eta$ and this completes the proof.

Example 4.2. Let $X = [0, 1]$ and $(X, N_{b}, o, k)$ be the complete symmetric $N_{b}$-FMS where $N_{b}$ is defined by

Let $f(\zeta) = \lambda \zeta, \lambda < \frac{\sqrt{2}}{2}, \zeta\in X, t > 0.$ Then, for $\frac{1}{2} > q$

where $\lambda^{2} = q$. Hence, all conditions of Theorem 4.1 are satisfied and so $f$ has a unique fixed point $0$ in $X$.

The first integral type of BCP was proved by Branciari ^[28] and see ^[19]. Let $\theta:(0, \infty)\rightarrow (0, \infty)$ as

be a nondecreasing and continuous function. Moreover for each $r > 0, \ \phi(r) > 0$. It also implies that $\phi(t) = 0$ iff $t = 0$.

In the following, we prove integral analogue of BCP in $N_{b}$-FMS.

Theorem 4.3. Let $(X, N_{b}, o, k)$ be a complete symmetric $N_{b}$-FMS and $f:X\rightarrow X$ be a map satisfying

for all $\zeta, \eta\in X, \; \phi\in\theta$, and $q\in (0, 1)$. Then $f$ has a unique fixed point.

Proof. By taking $\phi(1) = 1$ and applying Theorem 4.1, we obtain the result.

5.   Application of BCP in integral equations

Solution of any kind of equations is the most important and interesting tasks in mathematics. There are many techniques for solving many class of equations. Facing the problem of finding solutions and studying whether such solutions are unique or multiple. Fixed point theory is one of the important methodology that has great success in the field of integral equations, since its a iterative procedures has great variety of contexts.

In the study of existence of solution of differential and integral equation, fixed point theory plays a very important role, see ^[29,30].

In this section, we prove an application of Theorem 4.1 in particular nonlinear integral equation. The following theorem gives us an answer to the question "The solution of particular nonlinear equation (5.2) exists or not".

Consider $X = C[0, I]$ the class of all real valued continuous functions defined on $[0, I]$. Define a complete symmetric $N_{b}$-fuzzy metric $N_{b}:X^3\times (0, \infty)\rightarrow [0, 1]$ by

for $t > 0$ and for all $\zeta, \eta, \upsilon\in X$ and consider the integral equation.

where $I > 0$ and $g:[0, I]\rightarrow R$, $A:[0, I]^2\rightarrow R$ and $M:[0, I]^2\times R\rightarrow R$ are continuous functions.

Theorem 5.1. Let $(X, N_{b}, o, k)$ be a symmetric complete $N_{b}$-FMS defined by (5.1). Let $f:X\rightarrow X$ be the integral operator defined by

for all $\zeta\in X$ and $t, s\in [0, I]$. Suppose that the following conditions are satisfied:

(i) For all $t, s\in [0, 1]$ and $\zeta, \eta\in X$,

(ii) For all $t, s\in [0, I],$

Then the integral equation (5.2) has a unique solution $\zeta^{\star}\in X$.

Proof. For all $\zeta, \eta\in X$, we have

Since all conditions of Theorem 4.1 are satisfied in complete symmetric $N_{b}$-FMS. Hence, the integral equation (5.2) has a unique solution.

6.   Application of BCP in solving linear equation

In this section, we study a very general class of linear equation and apply BCP to find existence of solution in the setting of symmetric $N_{b}$-FMS. The main advantage of BCP is that it is a general contractive condition can be generalized in the wide range of ways depending on many parameters. Further more such a contractive condition involves many distinct terms that can be either adding or multiplying between terms. We demonstrate a application of BCP that guarantees the existence and in some cases, the uniqueness of fixed points that can be interpreted as solution of the mentioned linear equations.

In this section, we give an application of Theorem 4.1 for solving system of linear equations.

Let $X = R^{n}$ and define complete symmetric $N_{b}$-fuzzy metric on $X^{3}\times (0, \infty)$ by

for all $\zeta, \eta, \upsilon\in R^{n}$ and $k = 2$

Then the following system of linear equations has a unique solution.

Proof. Let $f:X\rightarrow X$ be defined by $f(\zeta) = c\zeta+d$ where $\zeta, d\in R^{n}$ and

For $\zeta, \eta\in R^{n},$ we get

Hence, $f$ is a fuzzy $q$-contraction and by Theorem 4.1, $f$ has a unique fixed point in complete symmetric $N_{b}$-FMS, that is, the system of linear equations (6.3) has a unique solution in $X$.

7.   Conclusions

In this article, we introduce the notions of $N_{b}$-FMS, quasi $N$-FMS, quasi $N_{b}$-FMS, pseudo $N_{b}$-FMS and proved decomposition theorem and BCP in the new setting with examples, counterexamples and applications.

We have built a fertile ground to study in further spaces like as extended $N_{b}$-FMS, partial $N_{b}$-FMS, extended partial $N_{b}$-FMS, intuitionistic $N_{b}$-FMS, partial intuitionistic $N_{b}$-FMS and many more generalized FMSs with fixed point theorems and their application in solution of different types of equations.

Conflict of interest

The authors declare to have no competing interests.