Certain new aspects in fuzzy fixed point theory

1.   Introduction

The symbols in Table 1 are used throughout this study.

FSs were introduced by Zadeh ^[1] as a useful tool for situations where data is ambiguous and FS theory contains the concept of degree of membership. The terms "fuzziness" and "probability" are not interchangeable. The term "probability" refers to the objective uncertainty resulting from a large number of observations. The term "fuzziness" describes a perception of ambiguity. Fuzzy notions are used to describe the degrees of possession of a specific property. The ability of FS theory to tackle issues that fixed point theory finds problematic is what makes it valuable in dealing with control challenges. FSs are used to control ill-defined, complex, or non-linear systems.

Metric FP theory has been extensively investigated due to its vast range of applications in mathematics, science and economics. Harnadi ^[14] explained MLSs and demonstrated FP results. For an extended multi-valued F-contraction in MLSs, Hammad et al. ^[13] proposed a modified dynamic process. Alghamdi ^[10] developed the concept of b-MLSs and provided several couple FP techniques for contraction mappings. Mlaiki et al. ^[15] introduced the concept of rectangular MLSs and used contraction mappings to demonstrate FP results. Rectangular b-metric spaces were introduced by Georgea et al. ^[16].

CTNs were proposed by Schweizer and Sklar ^[8]. FMSs were proposed by Kramosil and Michalek ^[2], who combined the concepts of FSs with metric spaces. Garbiec ^[5] gave a fuzzy interpretation of the Banach contraction principle in FMSs, while Kaleva and Seikkala ^[3] defined a distance between two points in FMSs as a non-negative fuzzy number. Hausdorff topology was defined on FMSs by George and Veermani ^[4]. In the development of control FMSs, Uddin et al. ^[7] developed different Banach FP findings. Saleem et al. ^[17] defined fuzzy double controlled metric spaces and established a number of FP theorems. Uddin et al. ^[18] used fuzzy contractions of the Suzuki type to solve problems. In fuzzy b-metric spaces, K. Javed et al. ^[19] showed ordered-theoretic FP findings. In the scenario of orthogonal partial b-metric spaces, K. Javed et al. ^[20] developed various FP findings. For generalised contractions, Ali et al. ^[21] demonstrated a number of FP findings. Several FP findings were reported in fuzzy b-metric spaces by Rakic et al. ^[22]. Rakic et al. ^[23] proved novel FPs in FMSs for the Ciric type. Debnath et al. ^[24] demonstrated some incredible FP results.

The concept of fuzzy MLSs was proposed by Shukla and Abbas ^[11] using the principles of MLSs and FSs. The fuzzy MLSs approach was established by Shukla and Gopal ^[12], who also demonstrated numerous FP solutions. Javed et al. ^[6] proposed the concept of FBBMLSs and demonstrated a number of FP results. The concept of FRBMSs was developed by Mehmood et al. ^[9], and the Banach contraction principle was shown in this context. In this study, we elaborated on the ideas offered in ^[6,9]. The manuscript's main goals are as follows:

(a) Introduce the concepts of FRMLSs and FRBMLSs;

(b) To establish several FP results;

(c) To enhance existing literature of FMSs and fuzzy FP theory.

In this manuscript, we aim to establish several fixed point results in new introduced spaces in this manuscript known as fuzzy rectangular metric-like spaces and rectangular b-metric-like spaces. Few non-trivial examples and an application also verify the uniqueness of solution.

2.   Preliminaries

This section includes some basic definitions that will aid in the comprehension of the main material.

Definition 2.1. ^[8] A binary operation $\mathit{*}$ : $E\times E\to E$ is known as CTN if

C1. $\kappa *ɴ = ɴ*\kappa, \left(\forall \right)\kappa, ɴ\in E;$

C2. $*$ is continuous;

C3. $\kappa *1 = \kappa, \left(\forall \right)\kappa \in E;$

C4. $\left(\kappa *ɴ\right)*ũ = \kappa *\left(ɴ*ũ\right), \left(\forall \right)\kappa, ɴ, ũ\in E;$

C5. If $\kappa \le ũ$ and $ɴ\le \sigma,$ with $\kappa, ɴ, ũ, \sigma \in E,$ then $\kappa *ɴ\le ũ*\sigma.$

Definition 2.2. ^[6] Suppose $\mathfrak{R}\ne \mathfrak{\varnothing }.$ A triplet $\left(\mathfrak{R}, {F}_{ɴ}, *\right)$ is known as FBMLS if $*$ is a CTN, ${F}_{ɴ}$ is a FS on $\mathfrak{R}\times \mathfrak{R}\times (0, +\infty)$ if for all $\sigma, \varkappa, \mathit{g}\in \mathfrak{R}$ and $\mathfrak{є}, s > 0,$

R1. ${F}_{ɴ}\left(\sigma, \varkappa, є\right) > 0;$

R2. ${F}_{ɴ}\left(\sigma, \varkappa, є\right) = 1$ then $\sigma = \varkappa;$

R3. ${F}_{ɴ}\left(\sigma, \varkappa, є\right) = {F}_{ɴ}\left(\varkappa, \sigma, є\right);$

R4. ${F}_{ɴ}\left(\sigma, \mathit{g}, ɴ\left(є+s\right)\right)\ge {F}_{ɴ}\left(\sigma, \varkappa, є\right)*{F}_{ɴ}\left(\varkappa, \mathit{g}, s\right);$

R5. ${F}_{ɴ}\left(\sigma, \varkappa, .\right):\left(0, +\infty \right)\to E$ is continuous and $\underset{є\to +\infty }{\mathrm{lim}}{F}_{ɴ}\left(\sigma, \varkappa, є\right) = 1.$

Definition 2.3. ^[9] Let $\mathfrak{R}\ne \mathfrak{\varnothing }$. A triplet $\left(\mathfrak{R}, {\mathfrak{{\rm{ \mathsf{ δ} }} }}_{v}, *\right)$ is known as FRMS if $*$ is a CTN, ${{\rm{ \mathsf{ δ} }} }_{v}$ is a FS on $\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)$ if for all $\sigma, \varkappa, \mathit{g}\in \mathfrak{R}$ and $\mathfrak{є}, s, w > 0,$

F1. ${{\rm{ \mathsf{ δ} }} }_{v}\left(\sigma, \varkappa, 0\right) = 0;$

F2. ${{\rm{ \mathsf{ δ} }} }_{v}\left(\sigma, \varkappa, є\right) = 1$ if and only if $\sigma = \varkappa;$

F3. ${{\rm{ \mathsf{ δ} }} }_{v}\left(\sigma, \varkappa, є\right) = {{\rm{ \mathsf{ δ} }} }_{v}\left(\varkappa, \sigma, є\right);$

F4. ${{\rm{ \mathsf{ δ} }} }_{v}\left(\sigma, \mathit{g}, є+s+w\right)\ge {{\rm{ \mathsf{ δ} }} }_{v}\left(\sigma, \varkappa, є\right)*{{\rm{ \mathsf{ δ} }} }_{v}\left(\varkappa, u, s\right)*{{\rm{ \mathsf{ δ} }} }_{v}\left(u, \mathit{g}, w\right)$ for all distinct $\varkappa, u\in \mathfrak{R}\backslash \{\sigma, \mathit{g}\};$

F5. ${{\rm{ \mathsf{ δ} }} }_{v}\left(\sigma, \varkappa, .\right):\left(0, +\infty \right)\to E$ is left continuous and $\underset{є\to +\infty }{\mathrm{lim}}{{\rm{ \mathsf{ δ} }} }_{v}\left(\sigma, \varkappa, є\right) = 1.$

Definition 2.4. ^[9] Let $\mathfrak{R}\ne \mathfrak{\varnothing }$. A triplet $\left(\mathfrak{R}, {\mathfrak{{\rm{ \mathsf{ δ} }} }}_{\mathfrak{ɴ}}, \mathfrak{*}\right)$ is known as FRBMS if $\mathfrak{ɴ}\ge 1$, $*$ is a CTN and ${{\rm{ \mathsf{ δ} }} }_{ɴ}$ is a FS on $\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)$ if for all $\sigma, \varkappa, \mathit{g}\in \mathfrak{R}$ and $\mathfrak{є}, s, w > 0,$

L1. ${{\rm{ \mathsf{ δ} }} }_{ɴ}\left(\sigma, \varkappa, 0\right) = 0;$

L2. ${{\rm{ \mathsf{ δ} }} }_{ɴ}\left(\sigma, \varkappa, є\right) = 1$ if and only if $\sigma = \varkappa;$

L3. ${{\rm{ \mathsf{ δ} }} }_{ɴ}\left(\sigma, \varkappa, є\right) = {{\rm{ \mathsf{ δ} }} }_{ɴ}\left(\varkappa, \sigma, є\right);$

L4. ${{\rm{ \mathsf{ δ} }} }_{ɴ}\left(\sigma, \mathit{g}, ɴ\left(є+s+w\right)\right)\ge {{\rm{ \mathsf{ δ} }} }_{ɴ}\left(\sigma, \varkappa, є\right)*{{\rm{ \mathsf{ δ} }} }_{ɴ}\left(\varkappa, u, s\right)*{{\rm{ \mathsf{ δ} }} }_{ɴ}\left(u, \mathit{g}, w\right)$ for all distinct $\varkappa, u\in \mathfrak{R}\backslash \{\sigma, \mathit{g}\};$

L5. ${{\rm{ \mathsf{ δ} }} }_{ɴ}\left(\sigma, \varkappa, .\right):\left(0, +\infty \right)\to E$ is left continuous and $\underset{є\to +\infty }{\mathrm{lim}}{{\rm{ \mathsf{ δ} }} }_{ɴ}\left(\sigma, \varkappa, є\right) = 1.$

3.   Main results

In this section, we provide numerous new concepts as generalizations of FRMSs and FRBMSs, as well as several FP results.

Definition 3.1. Suppose $\mathfrak{R}\ne \mathfrak{\varnothing }$. A triplet $\left(\mathfrak{R}, {L}_{v}, *\right)$ is known as FRMLS if $*$ is a CTN, ${L}_{v}$ is a FS on $\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)$ if for all $\sigma, \varkappa, \mathit{g}\in \mathfrak{R}$ and $\mathfrak{є}, s, w > 0,$

S1. ${L}_{v}\left(\sigma, \varkappa, 0\right) = 0;$

S2. ${L}_{v}\left(\sigma, \varkappa, є\right) = 1$ implies $\sigma = \varkappa;$

S3. ${L}_{v}\left(\sigma, \varkappa, є\right) = {L}_{v}\left(\varkappa, \sigma, є\right);$

S4. ${L}_{v}\left(\sigma, \mathit{g}, є+s+w\right)\ge {L}_{v}\left(\sigma, \varkappa, є\right)*{L}_{v}\left(\varkappa, u, s\right)*{L}_{v}\left(u, \mathit{g}, w\right)$ for all distinct $\varkappa, u\in \mathfrak{R}\backslash \{\sigma, \mathit{g}\};$

S5. ${L}_{v}\left(\sigma, \varkappa, .\right):\left(0, +\infty \right)\to E$ is left continuous and $\underset{є\to +\infty }{\mathrm{lim}}{L}_{v}\left(\sigma, \varkappa, є\right) = 1.$

Example 3.1. Suppose $\left(\mathfrak{R}, d\right)$ be a rectangular MLS, define ${L}_{v}:\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)\to E$ by

with $*$ be a CTN on $\mathfrak{R}.$ Then it is easy to see that $\left(\mathfrak{R}, {L}_{v}, *\right)$ is a FRMLS.

Example 3.2. Define ${L}_{v}:\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)\to E$ by

CTN is given by $\kappa *ɴ = \kappa · ɴ,$ then it is obvious that $\left(\mathfrak{R}, {L}_{v}, *\right)$ is a FRMLS.

Remark 3.1. In the preceding case, the self-distance is not equal to 1, i.e.,

In the case of FRMS, however, the self-distance must be equal to one. As a result, every FRMS is a FRMLS, but the opposite may not be true.

Definition 3.2. Let $\mathfrak{R}\ne \mathfrak{\varnothing }$ and a triplet $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is known as FRBMLS if $\mathfrak{ɴ}\ge 1$, $*$ is a CTN and ${\rm{ \mathsf{ δ} }}$ is a FS on $\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)$ if for all $\sigma, \varkappa, \mathit{g}\in \mathfrak{R}$ and $\mathfrak{є}, s, w > 0,$

(a) ${\rm{ \mathsf{ δ} }} \left(\sigma, \varkappa, 0\right) = 0;$

(b) ${\rm{ \mathsf{ δ} }} \left(\sigma, \varkappa, є\right) = 1$ implies $\sigma = \varkappa;$

(c) ${\rm{ \mathsf{ δ} }} \left(\sigma, \varkappa, є\right) = {\rm{ \mathsf{ δ} }} \left(\varkappa, \sigma, є\right);$

(d) ${\rm{ \mathsf{ δ} }} \left(\sigma, \mathit{g}, ɴ\left(є+s+w\right)\right)\ge {\rm{ \mathsf{ δ} }} \left(\sigma, \varkappa, є\right)*{\rm{ \mathsf{ δ} }} \left(\varkappa, u, s\right)*{\rm{ \mathsf{ δ} }} \left(u, \mathit{g}, w\right)$ for all distinct $\varkappa, u\in \mathfrak{R}\backslash \{\sigma, \mathit{g}\};$

(e) ${\rm{ \mathsf{ δ} }} \left(\sigma, \varkappa, .\right):\left(0, +\infty \right)\to E$ is left continuous and $\underset{є\to +\infty }{\mathrm{lim}}{\rm{ \mathsf{ δ} }} \left(\sigma, \varkappa, є\right) = 1.$

Example 3.3. Suppose $\left(\mathfrak{R}, d\right)$ be a rectangular b-MLS (RBMLS), define ${\rm{ \mathsf{ δ} }} :\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)\to E$ by

with CTN $\text{'}*\text{'}$. Therefore, it is clear that $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is a FRBMLS.

Example 3.4. Define $\mathfrak{{\rm{ \mathsf{ δ} }} }:\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)\to E$ by

CTN is defined by $\kappa *ɴ = \kappa · ɴ~~\text{and}~~ p\ge 1,$ then it is obvious that $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is a FRBMLS.

Example 3.5. Assume $\left(\mathfrak{R}, d\right)$ be a RBMLS, define ${\rm{ \mathsf{ δ} }} :\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)\to E$ by

with CTN $\kappa *ɴ = \mathrm{min}\left\{\kappa, ɴ\right\}$. Then it is obvious that $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is a FRBMLS.

Example 3.6. Assume $\left(\mathfrak{R}, d\right)$ be a RBMLS, define ${\rm{ \mathsf{ δ} }} :\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)\to E$ by

with $p\ge 1$ and CTN $\kappa *ɴ = \mathrm{min}\left\{\kappa, ɴ\right\}$. Then it obvious that $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is a FRBMLS.

Remark 3.2. If CTN given by $\kappa *ɴ = \kappa · ɴ$, then Example 3.6 is also a FRBMLS.

Remark 3.3. The self distance in FRBMLS may be not equal to 1.

Pick Example 3.6 with $p = 2$, then it yields

Remark 3.4. The preceding statement demonstrates that every FRBMLS is not a FRBMS, because in order to be a FRBMS, self distance must equal 1.

Remark 3.5. FRBMLS may be not continuous.

Example 3.7. Suppose $\mathfrak{R} = \left[0, +\infty \right),$ ${\rm{ \mathsf{ δ} }} \left(\sigma, \varkappa, є\right) = \frac{є}{є+d\left(\sigma, \varkappa \right)}$ for all $\sigma, \varkappa \in \mathfrak{R}, \mathfrak{ }\mathfrak{є} > 0$ and

If we define CTN by $\kappa *ɴ = \kappa · ɴ,$ then $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is an FRBMLS. Now, to illustrate continuity, we have

However,

Hence, $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is not continuous.

Definition 3.3. Let $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ be a FRBMLS and assume $\left\{{\sigma }_{n}\right\}$ is a sequence in $\mathfrak{R}.$ Then

(a) $\left\{{\sigma }_{n}\right\}$ is named to be a convergent sequence if there exists $\sigma \in \mathfrak{R}$ such that

(b) $\left\{{\sigma }_{n}\right\}$ is named to be Cauchy sequence if $\underset{n\to +\infty }{\mathrm{lim}}{\rm{ \mathsf{ δ} }} \left({\sigma }_{n}, {\sigma }_{n+q}, є\right)$ is exists and is finite $\text{for}~~\text{all}~~є > 0.$

(c) If every Cauchy sequence is convergent in $\mathfrak{R}$ then $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is said to be a complete FRBMLS such that

for all $є > 0~~\text{and}~~ q\ge 1.$

Remark 3.6. A convergent sequence's limit may not be unique in a FRBMLS.

Consider the FRBMLS in Example 3.4, and describe a sequence as ${\sigma }_{n} = 1-\frac{1}{n}\text{for}~~\mathrm{ }\mathrm{ }\mathrm{ }\text{all}~~n\ge 1.$ If $\sigma \ge 1,$ $\text{for}~~\text{all}~~є > 0,$ then

That is, the sequence $\left\{{\sigma }_{n}\right\}$ converges to all $\sigma \ge 1.$

Remark 3.7. It is not necessary for convergent sequence to become Cauchy in a FRBMLS.

Consider the example given in the preceding remark and describe a sequence as ${\sigma }_{n} = 1+{(-1)}^{n}$ for all $n\ge 1.$ If $\sigma \ge 2,~~ \text{for}~~\text{all}~~є > 0,$ then

That is, the sequence $\left\{{\sigma }_{n}\right\}$ converges to all $\sigma \ge 2$ but it is not Cauchy as $\underset{n\to +\infty }{\mathrm{lim}}{\rm{ \mathsf{ δ} }} \left({\sigma }_{n}, {\sigma }_{n+q}, є\right)$ does not exist.

Definition 3.4. Let $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ be a FRBMLS. For $\sigma \in \mathfrak{R},$ $\theta \in \left(\mathrm{0, 1}\right), є > 0,$ we define the open ball as $B\left(\sigma, \theta, є\right) = \left\{\varkappa \in \mathfrak{R}:\mathfrak{{\rm{ \mathsf{ δ} }} }\left(\sigma, \varkappa, є\right) > 1-\theta \right\}$ $\left(\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}~~\sigma, \mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{s}~~\theta ~~\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{ }~~\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}~~єoє\right)$.

Remark 3.8. FRBMLS may not have to be Hausdorff.

Example 3.8. Let $\mathfrak{R} = \left\{\mathrm{1, 2}, \mathrm{3, 4}\right\}.$ Define ${\rm{ \mathsf{ δ} }} :\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)\to \left[\mathrm{0, 1}\right]$ by

CTN is defined by $\kappa *ɴ = \kappa · ɴ,$ then $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is a FRBMLS.

Now, take $\sigma = 1, є = 20~~\text{and}~~ \varkappa \in \mathfrak{R},$ then

Now, if we take $\theta = 0.4,$ then

Hence, $B\left(\mathrm{1, 0.4, 20}\right) = \left\{\mathrm{2, 3}\right\}$ is an open ball. Now, take $\sigma = 2, є = 10~~\text{and}~~ \varkappa \in \mathfrak{R},$ then

Now, if we take $\theta = 0.5,$ then

Hence, $B\left(\mathrm{2, 0.5, 10}\right) = \left\{\mathrm{1, 3}\right\}$ is an open ball. But $B\left(\mathrm{1, 0.4, 20}\right)\cap B\left(\mathrm{2, 0.5, 10}\right) = \left\{\mathrm{2, 3}\right\}\cap \left\{\mathrm{1, 3}\right\}\ne \varnothing.$ This implies that FRBMLS $\left(\mathfrak{R}, {\mathfrak{{\rm{ \mathsf{ δ} }} }}_{\mathfrak{ɴ}}, \mathfrak{*}\right)$ is not Hausdorff.

Lemma 3.1. Let $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ be a FRBMLS and

for all $\sigma, \varkappa \in \mathfrak{R}, 0 < \varsigma < 1~~\text{and}~~є > 0, \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\sigma = \varkappa.$

Proof. From (d) of Definition 3.2, it is immediate.

Theorem 3.1. (Banach contraction theorem in fuzzy rectangular b-metric-like spaces)

Suppose $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ be a complete FRBMLS with $\mathfrak{ɴ}\ge 1$ such that

Let $\xi :\mathfrak{R}\to \mathfrak{R}$ be a mapping satisfying

for all $\sigma, \varkappa \in \mathfrak{R}, \varsigma \in \left[0, \frac{1}{ɴ}\right).$ Then $\xi$ has a unique fixed point $u\in \mathfrak{R}$ and $\mathfrak{{\rm{ \mathsf{ δ} }} }\left(u, u, є\right) = 1.$

Proof. Fix an arbitrary point ${\kappa }_{0}\in \mathfrak{R}$ and for $n = \mathrm{0, 1}, 2, \dots,$ start an iterative process ${\kappa }_{n+1} = \xi {\kappa }_{n}.$ Successively applying inequality (3.1), we get for all $n, є > 0,$

Since $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is a FRBMLS. For the sequence $\left\{{\kappa }_{n}\right\}$, writing $є = \frac{є}{3}+\frac{є}{3}+\frac{є}{3}$ and using the rectangular inequality given in (d) of Definition 3.2 on ${\rm{ \mathsf{ δ} }} \left({\kappa }_{n}, {\kappa }_{n+p}, є\right),$ we have the following cases.

Case 1. If $p$ is odd, then said $p = 2m+1$ where $m\in \left\{\mathrm{1, 2}, 3, \dots \right\},$ we have

Using (d) of Definition 3.2 in the above inequalities, we deduce

Case 2. If $p$ is even, then said $p = 2m, m\in \left\{\mathrm{1, 2}, 3, \cdots \right\},$ then we have

Using (3.4) in the above inequalities, we deduce

Therefore, from $\underset{є\to +\infty }{\mathrm{lim}}{\rm{ \mathsf{ δ} }} \left(\sigma, \varkappa, є\right) = 1,$ Cases 1, 2 and (3.2) conclude that for all $p\in \left\{\mathrm{1, 2}, 3, \cdots \right\},$ we have

Hence, $\left\{{\kappa }_{n}\right\}$ is a Cauchy sequence. Since $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is a complete FRBMLS, so there exists $u\in \mathfrak{R}$ such that

Now, we examine that $u$ is a fixed point of $\xi.$

Hence, $u$ is a fixed point of $\xi.$

Uniqueness: Let $v$ is another fixed point of $\xi$ for some $v\in \mathfrak{R},$ then

and by using the fact $\underset{є\to +\infty }{\mathrm{lim}}{\rm{ \mathsf{ δ} }} \left(\sigma, \varkappa, є\right) = 1.$ Thus, $u = v.$ Hence, the fixed point is unique.

Theorem 3.2. (Banach contraction theorem in fuzzy rectangular metric-like spaces)

Suppose $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ be a FRMLS such that

Let $\xi :\mathfrak{R}\to \mathfrak{R}$ be a mapping satisfying

for all $\sigma, \varkappa \in \mathfrak{R}, \varsigma \in \left[\mathrm{0, 1}\right).$ Then $\xi$ has a unique fixed point $u\in \mathfrak{R}$ and $\mathfrak{{\rm{ \mathsf{ δ} }} }\left(u, u, є\right) = 1.$

Proof. It is immediate if we take $ɴ = 1$ in the above theorem.

Example 3.10. Let $\mathfrak{R} = \left[\mathrm{0, 1}\right]$, define ${\rm{ \mathsf{ δ} }} :\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)\to \left[\mathrm{0, 1}\right]$ by

$\text{for}~~\text{all}~~\sigma, \varkappa \in \mathfrak{R}~~\text{and}~~є > 0,$ with CTN $\kappa *ɴ = \kappa.ɴ.$ Then it is obvious that $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is a complete FRBMLS.

Define $\xi :\mathfrak{R}\to \mathfrak{R}~~\mathrm{b}\mathrm{y}~~\xi \left(\sigma \right) = \frac{1-{2}^{-\sigma }}{3}.$ Then

for all $\sigma, \varkappa \in \mathfrak{R},$ where $\varsigma \in \left[\frac{1}{2}, 1\right).$ Thus, all the conditions of Theorem 3.1 satisfied and hence, 0 is the unique fixed point of $\xi.$

Theorem 3.3. Let $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ be a complete FRBMLS with $\mathfrak{ɴ}\ge 1$ such that

Let $\xi :\mathfrak{R}\to \mathfrak{R}$ be a mapping satisfying

for all $\sigma, \varkappa \in \mathfrak{R}, \varsigma \in \left[0, \frac{1}{ɴ}\right).$ Then $\xi$ has a unique fixed point $u\in \mathfrak{R}$ and $\mathfrak{{\rm{ \mathsf{ δ} }} }\left(u, u, є\right) = 1.$

Proof. Let $(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{ }\mathfrak{*})$ be a complete FRBMLS. For arbitrary ${\kappa }_{0}\in \mathfrak{R}$, define a sequence $\left\{{\kappa }_{n}\right\}$ in $\mathfrak{R}$ by

${\kappa }_{1} = \xi {\kappa }_{0}$, ${\kappa }_{2} = {\xi }^{2}{\kappa }_{0} = {\xi \kappa }_{1}, \dots, {\kappa }_{n} = {\xi }^{n}{\kappa }_{0} = {\xi \kappa }_{n-1}$ for all $n\in \mathbb{N}.$

If ${\kappa }_{n} = {\kappa }_{n-1}$ for some $n\in \mathbb{N}$, then ${\kappa }_{n}$ is a fixed point of $\xi$. We assume that ${\kappa }_{n}\ne {\kappa }_{n-1}$ for all $n\in \mathbb{N}$. For $є > 0$ and $n\in \mathbb{N}$, we get from (3.6) that

We have

Continuing this way, we get

We have

Since $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ be a FRBMLS for the sequence $\left\{{\kappa }_{n}\right\}$, writing $є = \frac{є}{3}+\frac{є}{3}+\frac{є}{3}$ and using the rectangular inequality given in (d) of Definition 3.2 on ${\rm{ \mathsf{ δ} }} \left({\kappa }_{n}, {\kappa }_{n+p}, є\right),$ in the following cases.

Case 1. If $p$ is odd, then said $p = 2m+1$ where $m\in \left\{\mathrm{1, 2}, 3, \dots \right\},$ we have

By using (3.7) in the above inequality, we have

Case 2. If $p$ is even, then said $p = 2m, m\in \left\{\mathrm{1, 2}, 3, \cdots \right\},$ then we have

We deduce from the cases 1 and 2 that

Therefore, $\left\{{\kappa }_{n}\right\}$ is a Cauchy sequence in $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{ }\mathfrak{*}\right).$ By the completeness of $(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{ }\mathfrak{*})$, there exists $u\in \mathfrak{R}$ such that

Now we prove that $u$ is a fixed point for $\xi$. For this we obtain from (3.6) that

and

Using the above inequalities, we deduce

Taking limit as $n\to +\infty$ and utilizing (3.8) in the preceding inequality, we examine that ${\rm{ \mathsf{ δ} }} \left(u, \xi u, є\right) = 1,$ that is, $\xi u = u.$ Hence, $u$ is a fixed point of $\xi$ and ${\rm{ \mathsf{ δ} }} \left(u, u, є\right) = 1$ for all $є > 0.$

Now we prove the uniqueness of $u$ of $\xi$. Let $v$ be another fixed point of $\xi$, such that ${\rm{ \mathsf{ δ} }} (u, v$, t) $< 1$ for some $є > 0$, and follows from (3.6) that

a contradiction. Therefore, we must have ${\rm{ \mathsf{ δ} }} \left(u, v, є\right) = 1,$ for all $є > 0,$ and hence $u = v.$

Corollary 3.1. Let $(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{ }\mathfrak{*})$ be a complete FRBMLS and a mapping $\xi :\mathfrak{R}\to \mathfrak{R}$ satisfying

For some $n\in \mathbb{N}, \forall \sigma, \varkappa \in \mathfrak{R}, \mathfrak{ }\mathfrak{є} > 0$, where $\varsigma \in \left[0, \frac{1}{ɴ}\right)$. Then $\xi$ has a unique fixed point $u\in \mathfrak{R}$ and $\mathfrak{{\rm{ \mathsf{ δ} }} }\left(u, u, є\right) = 1, \forall є > 0$.

Proof. $u\in \mathfrak{R}$ is a unique fixed point of ${\xi }^{n}$ by using Theorem 3.3, and ${\rm{ \mathsf{ δ} }} \left(u, u, є\right) = 1, \forall є > 0.$ $\xi u$ is also a fixed point of ${\xi }^{n}$ as ${\xi }^{n}\left(\xi u\right) = \xi u$ and from Theorem 3.3, $\xi u = u$, $u$ is a unique fixed point, since the unique fixed point of $\xi$ is also a unique fixed point of ${\xi }^{n}.$

Example 3.10. Let $\mathfrak{R} = \left[\mathrm{0, 1}\right]$, define ${\rm{ \mathsf{ δ} }} :\mathfrak{R}\times \mathfrak{R}\times \left[0, +\infty \right)\to \left[\mathrm{0, 1}\right]$ by

$\text{for}~~\text{all}~~\sigma, \varkappa \in \mathfrak{R}~~\text{and}~~є > 0,$ with CTN $\kappa *ɴ = \kappa.ɴ.$ Then it is obvious that $\left(\mathfrak{R}, \mathfrak{{\rm{ \mathsf{ δ} }} }, \mathfrak{*}\right)$ is a complete FRBMLS. Define $\xi :\mathfrak{R}\to \mathfrak{R}\mathrm{b}\mathrm{y}$

Then $\xi$ verifies the contractive form in Theorem 3.3, where $\varsigma \in \left[\frac{1}{2}, 1\right),$ with unique fixed point 0 and ${\rm{ \mathsf{ δ} }} \left(\mathrm{0, 0}, є\right) = 1\text{for}~~\text{all}~~є > 0.$ Hence, all conditions of Theorem 3.3 are satisfied.

4.   Application

An application of Theorem 3.1's integral equation is presented in this section. We show that an integral equation of the type

for all $j\in \left[0, l\right]$ where $l > 0$, has a solution. Let $C\left(\left[0, l\right], \mathbb{R}\right)$ be the space of all continuous functions defined on $[0, l]$ with CTN $\kappa *ɴ = \kappa.ɴ$ for all $\kappa, ɴ\in \left[\mathrm{0, 1}\right]$ and define a complete FRBMLS by

Theorem 4.1. Let $\xi :C\left(\left[0, l\right], \mathbb{R}\right)\to C\left(\left[0, l\right], \mathbb{R}\right)$ be the integral operator given by

where $F\in C\left(\left[0, l\right]\times \left[0, l\right]\times \mathbb{R}, \mathbb{R}\right)$ satisfies the following conditions:

There exists $f:\left[0, l\right]\times \left[0, l\right]\to \left[0, ++\infty \right]$ such that for all $r, j\in \left[0, l\right],$ $f\left(j, r\right)\in {L}^{1}\left(\left[0, l\right], \mathbb{R}\right)$ and for all $\sigma, \varkappa \in C\left(\left[0, l\right], \mathbb{R}\right),$ we have

and

Then the integral equation has the solution ${\sigma }_{*}\in C\left(\left[0, l\right], \mathbb{R}\right).$

Proof. For all $\sigma, \varkappa \in C\left(\left[0, l\right], \mathbb{R}\right),$ we have

Hence, ${\sigma }_{*}$ is a fixed point of $\xi,$ which is the solution of integral equation (4.1).

Remark 4.1. In the above theorem, if we take $p = 1,$ then application holds for FRMLS.

5.   Conclusions

In this manuscript, we established several fixed point results in new introduced spaces in this manuscript known as fuzzy rectangular metric-like spaces and rectangular b-metric-like spaces. Few non-trivial examples and an application also verify the uniqueness of solution. Fixed point theory receives a lot of attention since it has so many applications in mathematics, science, and economics. Using the ideas presented in the paper, several types of fixed point solutions for single and multi-valued mappings can be established. Intuitionistic fuzzy rectangular metric-like spaces, intuitionistic fuzzy rectangular b-metric-like spaces, Fuzzy controlled rectangular metric-like spaces, and other mathematical structures can be used to further extend the principles provided.

Acknowledgments

The authors are very thankful to DSR for providing necessary facilities.

Conflict of interest

The authors declare that they have no competing interests.