A Study of Fuzzy Fixed Points and Their Application to Fuzzy Fractional Differential Equations

Abstract

This study investigates fuzzy fixed points and fuzzy best proximity points for fuzzy mappings within the newly introduced framework $\theta$-fuzzy metric spaces that extends various existing fuzzy metric spaces. We establish novel fixed-point and best proximity-point theorems for both single-valued and multivalued mappings, thereby broadening the scope of fuzzy analysis. Furthermerefore, we have for aore, we apply one of our key results to derive conditions, ensuring the existence and uniqueness of a solution to Hadamard $\Psi$-Caputo tempered fuzzy fractional differential equations, particularly in the context of the SIR dynamics model. These theoretical advancements are expected to open new avenues for research in fuzzy fixed-point theory and its applications to hybrid models within $\theta$-fuzzy metric spaces.

1. Introduction

One of the fundamental challenges in the mathematical modeling of real-world phenomena is to address the uncertainty caused by the imprecision in categorizing events. Classical mathematics has historically faced difficulties in effectively managing imprecise or vague information. To address this limitation, in 1965, Zadeh [1] introduced the concept of fuzzy sets (FSs), providing a framework for modeling uncertainty that aligns with practical applications in fields such as engineering, life sciences, economics, medicine, and linguistics. Over the years, the foundational ideas of FSs have been significantly extended and developed. In particular, Heilpern [2] pioneered the concept of a fuzzy mapping and extended the fixed-point theorem for contraction mappings, making it applicable to fuzzy sets. Since then, various researchers have explored and applied fuzzy fixed-point (FFP) results in numerous contexts (see, for example, [3,4,5,6,7]).

It is worth noting that the fuzzy mappings involved in these studies are predominantly self-mappings. In a complete metric space $(\mathcal{X},d)$, the presence of the two nonempty subsets $\mathcal{U}$ and $\mathcal{V}$ does not necessarily imply that a contractive mapping $\mathcal{T}:\mathcal{U}\to \mathcal{V}$ will have a fixed point (FP). This lack of certainty has led researchers to explore points $\xi$ that achieve the minimum distance $d(\xi ,\mathcal{T}\xi )$. Specifically, the aim is to find a $\xi$ for which $d(\xi ,\mathcal{T}\xi )$ reaches the lowest possible value, which corresponds to the distance $d(\mathcal{U},\mathcal{V})$ separating the two subsets. This point $\xi$ is termed the best proximity point (BPP). As a result, a BPP theorem provides sufficient conditions that guarantee an approximate optimal solution $\xi$ satisfying $d(\xi ,\mathcal{T}\xi )=d(\mathcal{U},\mathcal{V})$, see [8,9,10,11].

Numerous authors in the literature have examined the existence and the convergence of FPs and BPPs under contractive conditions within distance metric spaces (see, for instance, [7,12,13,14]). However, these investigations have largely focused on mappings in classical or fuzzy metric spaces (FMSs) without considering the optimal proximity of fuzzy mappings. On the other hand, Amir et al. [15] defined the Hadamard $\Psi$-Caputo tempered fractional derivative ($\Psi$-CTFD), which is used as a mathematical tool in fuzzy calculus to measure the rate of change of a fuzzy function over time. It is considered a generalization of the classical derivative and can be applied to model systems with imprecise or uncertain data. For more studies, see [16,17]. In this research, we address a significant gap by exploring FFPs and fuzzy best proximity points (FBPPs) for fuzzy mappings within $\theta$-FMSs and by elucidating their interconnections. This comprehensive framework encompasses multiple spaces, such as FMSs and non-Archimedean FMSs, broadening the applicability of current findings in the field. Consequently, we derive pertinent theorems for FPs and BPPs, which apply to both multivalued and single-valued mappings. In addition, one of the derived results is utilized to examine the conditions for solving fuzzy fractional differential equation problems, especially concerning the Susceptible-Infectious-Removed (SIR) dynamics model. It is important to mention that these results could be further refined and expanded upon when examined within other generalized hybrid models in the larger field of fuzzy mathematics. The remainder of this paper is organized as follows: Section 2 provides fundamental definitions, lemmas, and theorems related to $\theta$-FMSs. Section 3 introduces the FFP theorem and its implications within $\theta$-FMSs. Furthermore, Section 4 focuses on FBPPs for fuzzy mappings and explores their consequences. Lastly, Section 5 presents an application that demonstrates the validity of the theoretical findings.

2. Preliminaries

This section gathers crucial definitions and findings related to the completion of $\theta$ fuzzy metrics, which are vital to the continuation of the article.

Definition 1

([18]). A binary operation $\ast :[0,1]\times [0,1]\to [0,1]$ is called a continuous t-norm (CtN) if ∗ is commutative, associative, $a\ast 1=a$ and for all $a,b,c,d\in [0,1],$ if $a\le c$ and $b\le d$ then $a\ast b\le c\ast d.$

Example 1.

• $\ast (a,b)=a·b$ ;
• $\ast (a,b)=min\left\{a,b\right\}$ ;
• $\ast (a,b)=max\left\{a+b-1,0\right\}$ .

Definition 2

([19]). Let $\mathcal{X}$ be a non-empty set and ∗ represents a CtN. Furthermore, let $\mathcal{M}:\mathcal{X}\times \mathcal{X}\times (0,+\infty )\to [0,1]$ be a fuzzy set. A triple $(\mathcal{X},\mathcal{M},\ast )$ is called a fuzzy metric space over $\mathcal{X}$ if the following conditions hold for any $\xi ,\eta ,\gamma \in \mathcal{X}$ and $t,\iota >0$:

(M1)

$\mathcal{M}(\xi ,\eta ,t)>0$;

(M2)

$\mathcal{M}(\xi ,\eta ,t)=1\mathit{if}\mathit{and}\mathit{only}\mathit{if}\xi =\eta$;

(M3)

$\mathcal{M}(\xi ,\eta ,t)=\mathcal{M}(\eta ,\xi ,t)$;

(M4)

$\mathcal{M}(\xi ,\eta ,t+\iota )\ge \mathcal{M}(\xi ,\gamma ,t)\ast \mathcal{M}(\gamma ,\eta ,\iota );$

(M5)

$\mathcal{M}(\xi ,\eta ,.):(0,+\infty )\to [0,1]$ is continuous and ${\displaystyle \underset{t\to +\infty }{lim}\mathcal{M}(\xi ,\eta ,t)=1.}$

Definition 3

([20]). Let $\theta :[0,+\infty )\times [0,+\infty )\to [0,+\infty )$ be a continuous mapping with respect to both variables. The image of θ is denoted by $Im\left(\theta \right)=\left\{\theta \right(\xi ,\eta ):\xi \ge 0,\eta \ge 0\}.$ The mapping θ is called an $\mathcal{B}$-action if and only if it satisfies the following conditions:

($\mathcal{B}$1)

$\theta (0,0)=0$ and $\theta (\xi ,\eta )=\theta (\eta ,\xi )$ for all $\xi ,\eta \ge 0$;

($\mathcal{B}$2)

$$\theta (\xi ,\eta )<\theta (u,v)\mathit{implies}\left\{\begin{array}{c}\mathit{either}\xi <u,\eta \le v,\hfill \\ \mathit{or}\xi \le u,\eta <v;\hfill \end{array}\right.$$

($\mathcal{B}$3)

For each $r\in Im\left(\theta \right)$ and for each $s\in [0,r],$ there exists $t\in [0,r]$ such that $\theta (t,s)=r;$

($\mathcal{B}$4)

$\theta (\xi ,0)\le \xi$ for all $\xi >0.$

The set of all $\mathcal{B}$-actions is denoted by $\Theta$.

Example 2

([20]). The following functions serve as examples of $\mathcal{B}$-actions on $[0,+\infty )\times [0,+\infty )$:

• ${\theta }_1(\xi ,\eta )=\xi +\eta ;$
• ${\theta }_2(\xi ,\eta )=k(\xi +\eta +\xi \eta );k\in [0,1)$
• ${\theta }_3(\xi ,\eta )=(\xi +\eta )(1+\xi \eta );$
• ${\theta }_4(\xi ,\eta )=\xi +\eta +\sqrt{\xi \eta };$
• ${\theta }_5(\xi ,\eta )=\sqrt{{\xi }^2+{\eta }^2}$;
• ${\theta }_6(\xi ,\eta )=max\{\xi ,\eta \}$.

Definition 4

([1,2]). In the set $\mathcal{X}$, a fuzzy set (FS) is characterized by a function $\mathcal{A}:\mathcal{X}\to [0,1]$, which assigns each element $\xi \in \mathcal{X}$ a membership value $\mathcal{A}\left(\xi \right)$ within the interval $[0,1]$. The collection of all fuzzy sets in $\mathcal{X}$ is denoted by $I^{\mathcal{X}}$. The α-level set of $\mathcal{A}$, indicated as ${\left[\mathcal{A}\right]}_{\alpha }$, is defined as follows:

$$\begin{array}{c}{\left[\mathcal{A}\right]}_{\alpha }=\left\{\xi \in \mathcal{X}:\mathcal{A}\left(\xi \right)\ge \alpha \right\},\mathit{for}\alpha \in (0,1],\hfill \\ {\left[\mathcal{A}\right]}_0=\overline{\left\{\xi \in \mathcal{X}:\mathcal{A}\left(\xi \right)\ge 0\right\}}.\hfill \end{array}$$

Definition 5

([21]). Let $\mathcal{X}$ be a non-empty set, and ∗ represents a CtN. Furthermore, let $\mathcal{N}:\mathcal{X}\times \mathcal{X}\times (0,+\infty )\to [0,1]$ be a fuzzy set. There exists $\theta \in \Theta$ such that a quadruple $(\mathcal{X},\mathcal{N},\ast ,\theta )$ is called a θ-fuzzy metric space (θ-FMS) over $\mathcal{X}$ if the following conditions hold for any $\xi ,\eta ,\gamma \in \mathcal{X}$ and $t,\iota >0$:

(N1)

$\mathcal{N}(\xi ,\eta ,t)>0$;

(N2)

$\mathcal{N}(\xi ,\eta ,t)=1\mathit{if}\mathit{and}\mathit{only}\mathit{if}\xi =\eta$;

(N3)

$\mathcal{N}(\xi ,\eta ,t)=\mathcal{N}(\eta ,\xi ,t)$;

(N4)

$\mathcal{N}(\xi ,\eta ,\theta (t,\iota \left)\right)\ge \mathcal{N}(\xi ,\gamma ,t)\ast \mathcal{N}(\gamma ,\eta ,\iota );$

(N5)

$\mathcal{N}(\xi ,\eta ,.):(0,+\infty )\to [0,1]$ is continuous and ${\displaystyle \underset{t\to +\infty }{lim}\mathcal{N}(\xi ,\eta ,t)=1.}$

Example 3.

Let $\mathcal{X}=\mathbb{R}.$ Define $a\ast b=a·b$, $\theta \in \Theta$ and $\mathcal{N}:\mathcal{X}\times \mathcal{X}\times (0,+\infty )\to [0,1]$ by

$$\mathcal{N}(\xi ,\eta ,t)=exp\left(-{\displaystyle \frac{|\xi -\eta |}{t}}\right).$$

(1)

Then $(\mathcal{X},\mathcal{N},\ast ,\theta )$ is a θ-FMS over $\mathcal{X}$. Our goal is to show condition (N4) from Definition 5, as the other assumptions, can be verified more straightforwardly.

$$\begin{array}{cc}\hfill \mathcal{N}(\xi ,\eta ,\theta (t,\iota ))=exp\left(-{\displaystyle \frac{|\xi -\eta |}{\theta (t,\iota )}}\right)& \ge exp\left(-{\displaystyle \frac{|\xi -\gamma |+|\gamma -\eta |}{\theta (t,\iota )}}\right)\hfill \\ & \ge exp\left(-{\displaystyle \frac{|\xi -\gamma |}{\theta (t,\iota )}}\right)·exp\left(-{\displaystyle \frac{|\gamma -\eta |}{\theta (t,\iota )}}\right)\hfill \\ & \ge exp\left(-{\displaystyle \frac{|\xi -\gamma |}{t}}\right)·exp\left(-{\displaystyle \frac{|\gamma -\eta |}{\iota }}\right)\hfill \\ & =\mathcal{N}(\xi ,\gamma ,t)\ast \mathcal{N}(\gamma ,\eta ,\iota ),\hfill \end{array}$$

for all $\xi ,\eta ,\gamma \in \mathcal{X}$ and $t,\iota >0.$

Example 4.

Let $\mathcal{X}=\mathbb{R}.$ Define $a\ast b=a·b$, $\theta \in \Theta$ and $\mathcal{N}:\mathcal{X}\times \mathcal{X}\times (0,+\infty )\to [0,1]$ by

$$\mathcal{N}(\xi ,\eta ,t)={\displaystyle \frac{t}{t+|\xi -\eta |}}.$$

(2)

Then $(\mathcal{X},\mathcal{N},\ast ,\theta )$ is a θ-FMS over $\mathcal{X}$. Our goal is to show that condition (N4) from Definition 5, as the other assumptions, can be verified more straightforwardly for all $\xi ,\eta ,\gamma \in \mathcal{X}$ and $t,\iota >0.$ Utilizing the characteristics of θ, we obtain

$$\begin{array}{cc}\hfill \mathcal{N}(\xi ,\gamma ,t)\ast \mathcal{N}(\gamma ,\eta ,\iota )& ={\displaystyle \frac{t}{t+|\xi -\gamma |}}·{\displaystyle \frac{\iota }{\iota +|\gamma -\eta |}}\hfill \\ & ={\displaystyle \frac{1}{1+\frac{|\xi -\gamma |}{t}}}·{\displaystyle \frac{1}{1+\frac{|\gamma -\eta |}{\iota }}}\hfill \\ & \le {\displaystyle \frac{1}{1+\frac{|\xi -\gamma |}{\theta (t,\iota )}}}·{\displaystyle \frac{1}{1+\frac{|\gamma -\eta |}{\theta (t,\iota )}}}\hfill \\ & \le {\displaystyle \frac{1}{1+\frac{|\xi -\gamma |+|\gamma -\eta |}{\theta (t,\iota )}}}\hfill \\ & \le {\displaystyle \frac{\theta (t,\iota )}{\theta (t,\iota )+|\xi -\gamma |+|\gamma -\eta |}}\hfill \\ & \le {\displaystyle \frac{\theta (t,\iota )}{\theta (t,\iota )+|\xi -\eta |}}\hfill \\ & =\mathcal{N}(\xi ,\eta ,\theta (t,\iota \left)\right).\hfill \end{array}$$

Hence, (N4) is satisfied.

Definition 6

([21]). Let $(\mathcal{X},\mathcal{N},\ast ,\theta )$ be a θ-FMS.

• A sequence $\left\{{\xi }_n\right\}\subset \mathcal{X}$ is considered to converge to a point $\xi \in \mathcal{X}$ if $\mathcal{N}({\xi }_n,\xi ,t)\to 1$ as $n\to +\infty$ for every $t>0$. The point ξ is called the limit of the sequence $\left\{{\xi }_n\right\}$.
• A sequence $\left\{{\xi }_n\right\}$⊆in $\mathcal{X}$ ⊆⊆ is called a Cauchy sequence if there exists $n_0\in \mathbb{N}$ such that $\mathcal{N}({\xi }_n,{\xi }_m,t)\to 1$ as $n,m\to +\infty$, for every $n,m\ge n_0$, $t>0$.
• A subset $\mathcal{Y}$ of $\mathcal{X}$ is said to be closed if the limit of a convergent sequence of $\mathcal{Y}$ always belongs to $\mathcal{Y}$.
• A subset $\mathcal{Y}$ of $\mathcal{X}$ is said to be complete if every Cauchy sequence in $\mathcal{Y}$ is a convergent and its limit is in $\mathcal{Y}.$
• The mapping $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ is called continuous at a point ${\xi }_0\in \mathcal{X}$ if for every sequence $\left\{{\xi }_n\right\}\subseteq \mathcal{X}$ with ${\xi }_n\to \xi$ as $n\to +\infty$ we have $\mathcal{T}\left({\xi }_n\right)\to \mathcal{T}\left(\xi \right)$ in $\mathcal{X}$ as $n\to +\infty$ .

Definition 7

([22]). Let $\mathcal{X}$ be an arbitrary set and $\mathcal{Y}$ a metric space. A mapping $\mathcal{T}$ from $\mathcal{X}$ to $𝒴$ is called a fuzzy mapping, which is a fuzzy subset of $\mathcal{X}\times \mathcal{Y}$ with the membership function $\mathcal{T}\left(\xi \right)\left(\eta \right)$ representing the degree of membership of η in $\mathcal{T}\left(\xi \right)$. For convenience, we denote the α-level set of $\mathcal{T}\left(\xi \right)$ by ${\left[\mathcal{T}\xi \right]}_{\alpha }$ instead of $\left[\mathcal{T}\right(\xi \left)\right]$.

3. Fuzzy Contractions

In what follows, we will use specific assumptions and definitions within the framework of $\theta$-FMS $(\mathcal{X},\mathcal{N},\ast ,\theta ).$ Let the set of all nonempty bounded proximal sets in $\mathcal{X}$ be denoted by $\mathcal{P}\left(\mathcal{X}\right),$ the set of all nonempty compact subsets of $\mathcal{X}$ be presented by $C\left(2^{\mathcal{X}}\right)$, and the set of all nonempty closed and bounded subsets of $\mathcal{X}$ be denoted by $CB\left(\mathcal{X}\right)$. Since every compact set is proximal and any proximal set is closed, the following are included:

$$C\left(2^{\mathcal{X}}\right)\subseteq \mathcal{P}\left(\mathcal{X}\right)\subseteq CB\left(\mathcal{X}\right).$$

(3)

For $\mathcal{U},\mathcal{V}\in C\left(2^{\mathcal{X}}\right)$ , we define the following:

•

$\mathcal{N}(\xi ,\mathcal{U},t)=sup\left\{\mathcal{N}\right(\xi ,\eta ,t):\eta \in \mathcal{U},t>0\}$.

•

$\mathcal{N}(\mathcal{U},\mathcal{V},t)=sup\left\{\mathcal{N}\right(\xi ,\eta ,t):\xi \in \mathcal{U},\eta \in \mathcal{V},t>0\}.$

•

We induce the Hausdorff fuzzy metric $\mathcal{H}$ on $C\left(2^{\mathcal{X}}\right)$ by the fuzzy $\theta$-metric $\mathcal{N}$, for all $t>0$ is defined as

$$\mathcal{H}(\mathcal{U},\mathcal{V},t)=\left\{\begin{array}{cc}min\left\{\underset{\xi \in \mathcal{U}}{inf}\mathcal{N}(\xi ,\mathcal{V},t),\underset{\eta \in \mathcal{V}}{inf}\mathcal{N}(\eta ,\mathcal{U},t)\right\},\hfill & \mathrm{if}\mathrm{it}\mathrm{exists},\hfill \\ 1,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(4)

Definition 8.

Let $(\mathcal{X},\mathcal{N},\ast ,\theta )$ be a θ-FMS. A subset $\mathcal{U}$ being a subset of $\mathcal{X}$ is called proximal, if for each $\xi \in \mathcal{X},$ there exists $\eta \in \mathcal{U}$ such that $\mathcal{N}(\xi ,\eta ,t)=\mathcal{N}(\xi ,\mathcal{U},t)$, for all $t>0.$

Definition 9.

Let $(\mathcal{X},\mathcal{N},\ast ,\theta )$ be a θ-FMS and $\mathcal{T}:\mathcal{X}\to I^{\mathcal{X}}$ be a fuzzy mapping. Then a point $\xi \in \mathcal{X}$ is called an FFP of $\mathcal{T}$ if there exists $\alpha \in (0,1]$ such that $\mathcal{N}(\xi ,{\left[\mathcal{T}\xi \right]}_{\alpha },t)=1$ for all $t>0$, i.e., $\xi \in {\left[\mathcal{T}\xi \right]}_{\alpha }$.

We will initially present a series of lemmas concerning $\theta$-FMSs.

Lemma 1.

If ${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}\in C\left(2^{\mathcal{X}}\right),{\alpha }_{\mathcal{T}}\left(\xi \right)\in (0,1],\xi \in \mathcal{X}$, then $\xi \in {\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}$ if and only if $\mathcal{N}({\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)},\xi ,t)=1$ for all $t>0.$

Proof. 

Assume $\mathcal{N}(\xi ,{\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)},t)=sup\left\{\mathcal{N}(\xi ,\eta ,t):\eta \in {\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)},t>0\right\}=1$ for all $t>0$. Then, there exists a sequence $\left\{{\eta }_n\right\}\in {\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}$ such that $\mathcal{N}(\xi ,{\eta }_n,t)\ge 1-{\displaystyle \frac{1}{n}}$. Since ${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}\in C\left(2^{\mathcal{X}}\right)$, ${\alpha }_{\mathcal{T}}\left(\xi \right)\in (0,1]$, and $\xi \in \mathcal{X}$, it follows that $\xi \in {\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}.$ Conversely, if $\xi \in {\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}$, we have for all $t>0$

$$\mathcal{N}({\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)},\xi ,t)=sup\left\{\mathcal{N}(\xi ,\eta ,t):\eta \in {\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}\right\}\ge \mathcal{N}(\xi ,\xi ,t)=1.$$

Thus, $\mathcal{N}({\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)},\xi ,t)=1$ for all $t>0.$ □

Lemma 2.

Let $(\mathcal{X},\mathcal{N},\ast ,\theta )$ be a complete θ-FMS , where $(C\left(2^{\mathcal{X}}\right),\mathcal{H},\ast )$ forms a Hausdorff FMS on $C\left(2^{\mathcal{X}}\right)$. Let $\mathcal{T}$ be a fuzzy mapping assuming, for every ${\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)}$ and ${\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)}$ in $C\left(2^{\mathcal{X}}\right)$, that for each $\xi \in {\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)}$, there exists an $\eta \in {\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)}$ satisfying $\mathcal{N}(\xi ,{\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},t)=\mathcal{N}(\xi ,\eta ,t)$, $t>0$; then the following inequality holds:

$$\mathcal{H}({\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t)\le \mathcal{N}(\xi ,\eta ,t).$$

Proof. 

Since

$$\begin{array}{cc}\hfill \mathcal{H}({\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t)=min\{& \underset{\xi \in {\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)}}{inf}\left\{\mathcal{N}(\xi ,{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t)\right\},\hfill \\ & \underset{\eta \in {\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)}}{inf}\left\{\mathcal{N}(\eta ,{\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},t)\}\right\},\hfill \end{array}$$

then we have two cases:

Case 1: If

$$\mathcal{H}({\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t)=\underset{\xi \in {\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)}}{inf}\{\mathcal{N}(\xi ,{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t)\},$$

implies that

$$\mathcal{H}({\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t)\le \mathcal{N}(\xi ,{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t),$$

(5)

then, by assumption, for each $\xi \in {\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)}$ and for $t>0,$ there exists $\eta \in {\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)}$, satisfying

$$\mathcal{N}(\xi ,\eta ,t)=\mathcal{N}(\xi ,{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t).$$

(6)

Therefore, based on (5) and (6), we can conclude that

$$\mathcal{H}({\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t)\le \mathcal{N}(\xi ,\eta ,t).$$

Case 2: If

$$\mathcal{H}({\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t)=\underset{\eta \in {\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)}}{inf}\{\mathcal{N}(\eta ,{\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},t)\},$$

then

$$\mathcal{H}({\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t)\le \mathcal{N}{(\eta ,\mathcal{T}{\xi }_1]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},t),$$

(7)

again, since there exists $\eta \in {\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)}$ satisfying

$$\mathcal{N}(\xi ,\eta ,t)=\mathcal{N}(\eta ,{\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},t).$$

(8)

Hence, from (7) and (8), we obtain

$$\mathcal{H}({\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t)\le \mathcal{N}(\xi ,\eta ,t).$$

□

Theorem 1.

Let $(\mathcal{X},\mathcal{N},\ast ,\theta )$ be a complete θ-FMS. Let $\mathcal{T}:\mathcal{X}\to I^{\mathcal{X}}$ be a fuzzy mapping. Assume that, for every $\xi \in \mathcal{X}$, there exists an ${\alpha }_{\mathcal{T}}\left(\xi \right)\in (0,1]$ such that ${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}\in C\left(2^{\mathcal{X}}\right)$. Additionally, suppose that the following condition holds:

$$\mathcal{H}({\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},kt)\ge \mathcal{N}({\xi }_1,{\xi }_2,t),$$

(9)

for all ${\xi }_1,{\xi }_2\in \mathcal{X},k\in (0,1)$ and $t>0.$ Then $\mathcal{T}$ has an FFP.

Proof. 

Let ${\xi }_0\in \mathcal{X}$ be arbitrary. We choose a sequence $\left\{{\xi }_n\right\}$ in $\mathcal{X}$ as follows: By hypothesis, there exists ${\alpha }_{\mathcal{T}}\left({\xi }_0\right)\in (0,1]$ such that ${\left[\mathcal{T}{\xi }_0\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_0\right)}\in C\left(2^{\mathcal{X}}\right)$. Since ${\left[\mathcal{T}{\xi }_0\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_0\right)}\in C\left(2^{\mathcal{X}}\right)$ is a nonempty compact subset of $\mathcal{X},$ there exists ${\xi }_1\in {\left[\mathcal{T}{\xi }_0\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_0\right)}$ such that $\mathcal{N}({\xi }_0,{\xi }_1,t)=\mathcal{N}({\xi }_0,{\left[\mathcal{T}{\xi }_0\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_0\right)},t)$. By Lemma 2, we can choose ${\xi }_2\in {\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)}$ such that

$$\mathcal{N}({\xi }_1,{\xi }_2,t)\ge \mathcal{H}({\left[\mathcal{T}{\xi }_0\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_0\right)},{\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},t)$$

for all $t>0.$ By induction, we have that ${\xi }_{n+1}\in \mathcal{T}{\xi }_n$, which satisfies the following inequality:

$$\mathcal{N}({\xi }_n,{\xi }_{n+1},\theta (t,s))\ge \mathcal{H}({\left[\mathcal{T}{\xi }_n\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_n\right)},{\left[\mathcal{T}{\xi }_{n+1}\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_{n+1}\right)},\theta (t,s)),$$

(10)

for all $t>0.$ Now, by (9) and (10) together with Lemma 2, we have,

$$\begin{array}{cc}\hfill \mathcal{N}({\xi }_n,{\xi }_{n+1},t)& \ge \mathcal{H}({\left[\mathcal{T}{\xi }_n\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_n\right)},{\left[\mathcal{T}{\xi }_{n+1}\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_{n+1}\right)},t)\hfill \\ & \ge \mathcal{N}({\xi }_n,{\xi }_{n+1},{\displaystyle \frac{t}{k}})\hfill \\ & \ge \mathcal{H}({\left[\mathcal{T}{\xi }_{n-1}\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_{n-1}\right)},{\left[\mathcal{T}{\xi }_n\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_n\right)},{\displaystyle \frac{t}{k}})\hfill \\ & \ge \mathcal{N}({\xi }_{n-1},{\xi }_n,{\displaystyle \frac{t}{k^2}})\hfill \\ & ⋮\hfill \\ & \ge \mathcal{H}({\left[\mathcal{T}{\xi }_0\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_0\right)},{\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},{\displaystyle \frac{t}{k^{n-1}}})\hfill \\ & \ge \mathcal{N}({\xi }_0,{\xi }_1,{\displaystyle \frac{t}{k^n}}).\hfill \end{array}$$

(11)

Let $m>n,$ then, by (N4) and (11). We have for all $t\in Im\left(\theta \right),$ which means by ($\mathcal{B}$3), there exists $s_i\le t$ for all $r_i\le t,i=1,2,\cdots ,m-n-1$ such that

$$\begin{array}{cc}\hfill \mathcal{N}({\xi }_n,{\xi }_m,t)& \ge \mathcal{N}({\xi }_n,{\xi }_n,s_1)\ast \mathcal{N}({\xi }_{n+1},{\xi }_m,r_1)\hfill \\ & \ge \mathcal{N}({\xi }_n,{\xi }_{n+1},s_1)\ast \mathcal{N}({\xi }_{n+1},{\xi }_{n+2},s_2)\ast \mathcal{N}({\xi }_{n+2},{\xi }_m,r_2)\hfill \\ & \ge \mathcal{N}({\xi }_n,{\xi }_{n+1},s_1)\ast \mathcal{N}({\xi }_{n+1},{\xi }_{n+2},s_2)\ast \cdots \ast \mathcal{N}({\xi }_{m-1},{\xi }_m,s_{m-n-1})\hfill \\ & \ge \mathcal{N}({\xi }_0,{\xi }_1,{\displaystyle \frac{s_1}{k^{n-1}}})\ast \cdots \ast \mathcal{N}({\xi }_0,{\xi }_1,{\displaystyle \frac{s_{m-n-1}}{k^{m-n-1}}}).\hfill \end{array}$$

By taking the limit as $n\to +\infty$, we obtain $\mathcal{N}({\xi }_n,{\xi }_m,t)=1.$ This shows that $\left\{{\xi }_n\right\}$ is a Cauchy sequence. Hence, the completeness of $(\mathcal{X},\mathcal{N},\ast ,\theta )$ implies that there exists $\eta \in \mathcal{X}$ such that ${\xi }_n\to \eta$ as $n\to +\infty$. Now, we have to prove $\eta \in {\left[\mathcal{T}\eta \right]}_{{\alpha }_{\mathcal{T}}\left(\eta \right)}$

$$\begin{array}{cc}\hfill \mathcal{N}(\eta ,{\left[\mathcal{T}\eta \right]}_{{\alpha }_{\mathcal{T}}\left(\eta \right)},t)& \ge \mathcal{N}(\eta ,{\xi }_{n+1},t)\hfill \\ & \ge \mathcal{H}({\left[\mathcal{T}\eta \right]}_{{\alpha }_{\mathcal{T}}\left(\eta \right)},{\left[\mathcal{T}{\xi }_n\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_n\right)},t)\hfill \\ & \ge \mathcal{N}(\eta ,{\xi }_n,{\displaystyle \frac{t}{k}})\hfill \\ & \to 1\mathrm{as}n\to +\infty .\hfill \end{array}$$

By Lemma 1, we have $\eta \in {\left[\mathcal{T}\eta \right]}_{{\alpha }_{\mathcal{T}}\left(\eta \right)}$. Hence, $\eta$ is an FFP for $\mathcal{T}$. □

Example 5.

Let $\mathcal{X}=[0,+\infty ).$ Define $\mathcal{N}:\mathcal{X}\times \mathcal{X}\times (0,+\infty )\to [0,1]$ as in Example 4, as follows:

$$\mathcal{N}(\xi ,\eta ,t)={\displaystyle \frac{t}{t+|\xi -\eta |}}.$$

Let $\alpha \in (0,1]$ and consider a fuzzy mapping $\mathcal{T}:\mathcal{X}\to I^{\mathcal{X}}$ defined as follows:

(i) If $\xi =0$

$$\mathcal{T}\left(\xi \right)\left(\eta \right)=\left\{\begin{array}{cc}1\hfill & ,\eta =0,\hfill \\ 0\hfill & ,\eta \ne 0.\hfill \end{array}\right.$$

(ii) If $0<\xi <\infty$

$$\mathcal{T}\left(\xi \right)\left(\eta \right)=\left\{\begin{array}{cc}\alpha \hfill & ,0\le \eta <{\displaystyle \frac{\xi }{3}},\hfill \\ {\displaystyle \frac{\alpha }{3}}\hfill & ,{\displaystyle \frac{\xi }{3}}<\eta \le {\displaystyle \frac{\xi }{2}},\hfill \\ {\displaystyle \frac{\alpha }{6}}\hfill & ,{\displaystyle \frac{\xi }{2}}\le \eta <{\displaystyle \frac{2\xi }{3}},\hfill \\ {\displaystyle \frac{\alpha }{18}}\hfill & ,{\displaystyle \frac{2\xi }{3}}\le \eta <\infty .\hfill \end{array}\right.$$

It is clear that, for ${\displaystyle \frac{\alpha }{3}},$ we have

$$\begin{array}{c}\hfill {\left[\mathcal{T}\xi \right]}_{{\displaystyle \frac{\alpha }{3}}}=\left\{\eta \in \mathcal{X}:\mathcal{T}\left(\xi \right)\left(\eta \right)\ge {\displaystyle \frac{\alpha }{3}}\right\}=[0,{\displaystyle \frac{\xi }{2}}].\end{array}$$

Thus, for every $\xi \in \mathcal{X},$ there exists ${{\displaystyle \frac{\alpha }{3}}}_{\mathcal{T}}\left(\xi \right)\in (0,1]$ such that ${\left[\mathcal{T}\xi \right]}_{{{\displaystyle \frac{\alpha }{3}}}_{\mathcal{T}}\left(\xi \right)}\in C\left(2^{\mathcal{X}}\right)$. Then,

$$\begin{array}{cc}\hfill \mathcal{H}({\left[\mathcal{T}{\xi }_1\right]}_{{{\displaystyle \frac{\alpha }{3}}}_{\mathcal{T}}\left({\xi }_1\right)},{\left[\mathcal{T}{\xi }_2\right]}_{{{\displaystyle \frac{\alpha }{3}}}_{\mathcal{T}}\left({\xi }_2\right)},kt)& =\mathcal{H}([0,{\displaystyle \frac{{\xi }_1}{2}}],[0,{\displaystyle \frac{{\xi }_2}{2}}],kt)\hfill \\ & ={\displaystyle \frac{kt}{kt+|\frac{{\xi }_1}{2}-\frac{{\xi }_2}{2}|}}\hfill \\ & \ge {\displaystyle \frac{t}{t+|{\xi }_1-{\xi }_2|}}\hfill \\ & =\mathcal{N}({\xi }_1,{\xi }_2,t),\hfill \end{array}$$

for all ${\xi }_1,{\xi }_2\in \mathcal{X},k={\displaystyle \frac{1}{2}}$ and $t>0.$ Consequently, all the conditions of Theorem 1 are satisfied to find $0\in {\left[\mathcal{T}0\right]}_{{{\displaystyle \frac{\alpha }{3}}}_{\mathcal{T}}\left(0\right)}.$

Corollary 1.

Let $(\mathcal{X},\mathcal{N},\ast ,\theta )$ be a complete θ-FMS. Let $\mathcal{S}:\mathcal{X}\to C\left(2^{\mathcal{X}}\right)\setminus \varphi$ be a multivalued mapping. Assume for every $\xi \in \mathcal{X}.$ Suppose that the following condition holds:

$$\mathcal{H}(\mathcal{S}{\xi }_1,\mathcal{S}{\xi }_2,kt)\ge \mathcal{N}({\xi }_1,{\xi }_2,t),$$

(12)

for all ${\xi }_1,{\xi }_2\in \mathcal{X},k\in (0,1)$ and $t>0.$ Then there exists $\xi \in \mathcal{X}$ such that $\xi \in \mathcal{S}\left(\xi \right)$.

Proof. 

Let ${\alpha }_{\mathcal{T}}:\mathcal{X}\to (0,1]$ be a mapping, and consider a fuzzy mapping $\mathcal{T}:\mathcal{X}\to {\mathcal{I}}^{\mathcal{X}}$ defined as follows:

$$\begin{array}{cc}\hfill & \mathcal{T}\left(\xi \right)\left(u\right)=\left\{\begin{array}{cc}{\alpha }_{\mathcal{T}}\left(\xi \right),\hfill & \mathrm{if}u\in \mathcal{T}\xi ,\hfill \\ 0,\hfill & \mathrm{if}u\notin \mathcal{T}\xi .\hfill \end{array}\right.\hfill \end{array}$$

(13)

Then, for all $\xi \in \mathcal{X},$ we have

$${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}}=\left\{u\in \mathcal{X}:\mathcal{T}\left(\xi \right)\left(u\right)\ge {\alpha }_{\mathcal{L}}\left(\xi \right)\right\}=\mathcal{S}\xi .$$

(14)

As a result,

$$\begin{array}{c}\hfill \mathcal{H}({\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},kt)=\mathcal{H}(\mathcal{S}{\xi }_1,\mathcal{S}{\xi }_2,kt)\ge \mathcal{N}({\xi }_1,{\xi }_2,t),\end{array}$$

(15)

for all ${\xi }_1,{\xi }_2\in \mathcal{X},k\in (0,1)$ and $t>0.$ Hence, Theorem 1 is applicable; then $\mathcal{S}$ has an FP. □

Corollary 2.

Let $(\mathcal{X},\mathcal{N},\ast ,\theta )$ be a complete θ-FMS. Let $\mathcal{S}:\mathcal{X}\to \mathcal{X}$ be a mapping. Assume that the following condition holds:

$$\mathcal{N}(\mathcal{S}{\xi }_1,\mathcal{S}{\xi }_2,kt)\ge \mathcal{N}({\xi }_1,{\xi }_2,t),$$

(16)

for all ${\xi }_1,{\xi }_2\in \mathcal{X}$ and $t>0,$ where $k\in (0,1).$ Then $\mathcal{S}$ has a unique FP.

Proof. 

Let ${\alpha }_{\mathcal{T}}:\mathcal{X}\to (0,1]$ be an arbitrary mapping, and consider a fuzzy mapping $\mathcal{T}:\mathcal{X}\to {\mathcal{I}}^{\mathcal{X}}$ defined as follows:

$$\begin{array}{cc}\hfill & \mathcal{T}\left(\xi \right)\left(u\right)=\left\{\begin{array}{cc}{\alpha }_{\mathcal{T}}\left(\xi \right),\hfill & \mathrm{if}u=\mathcal{S}\xi ,\hfill \\ 0,\hfill & \mathrm{if}u\ne \mathcal{S}\xi .\hfill \end{array}\right.\hfill \end{array}$$

(17)

Then, for all $\xi \in \mathcal{X},$ we have

$${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}}=\left\{u\in \mathcal{X}:\mathcal{T}\left(\xi \right)\left(u\right)\ge {\alpha }_{\mathcal{T}}\left(\xi \right)\right\}=\left\{\mathcal{S}\xi \right\}.$$

(18)

Notice that, in this case, for all ${\xi }_1,{\xi }_2\in \mathcal{X},k\in (0,1)$ and $t>0,$ we have

$$\begin{array}{c}\hfill \mathcal{H}({\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}},{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}},t)=\mathcal{N}(\mathcal{S}\left({\xi }_1\right),\mathcal{S}\left({\xi }_2\right),t)\ge \mathcal{N}({\xi }_1,{\xi }_2,t).\end{array}$$

(19)

Therefore, Theorem 1 can be applied to find $\xi \in \mathcal{X}$ such that $\xi \in \left\{\mathcal{S}\xi \right\}$; that is, $\xi$ is an FP.

Uniqueness: Suppose that there exist two fixed points $u,v\in \mathcal{X}$; then by the contraction condition, we obtain

$$\begin{array}{cc}\hfill \mathcal{N}(\mathcal{T}u,\mathcal{T}v,t)=\mathcal{N}(u,v,t)& \ge \mathcal{N}(u,v,{\displaystyle \frac{t}{k}})\hfill \\ & \ge \mathcal{N}(\mathcal{T}u,\mathcal{T}v,{\displaystyle \frac{t}{k}})\hfill \\ & \ge \mathcal{N}(u,v,{\displaystyle \frac{t}{k^2}})\hfill \\ & ⋮\hfill \\ & \ge \mathcal{N}(u,v,{\displaystyle \frac{t}{k^n}})\to 1,\mathrm{as}n\to +\infty ,\hfill \end{array}$$

which implies $u=v.$ □

4. Proximal Contractions

4.1. Proximal Fuzzy Contraction

This section introduces a new concept called k-proximal fuzzy contraction related to ${\mathcal{U}}_{\circ }$. For $\mathcal{U},\mathcal{V}\in C\left(2^{\mathcal{X}}\right)$ , we define the following:

•

${\mathcal{U}}_{\circ }=\left\{\xi \in \mathcal{U}:\mathcal{N}(\xi ,\eta ,t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),\mathrm{for}\mathrm{some}\eta \in \mathcal{V}\right\}.$

•

${\mathcal{V}}_{\circ }=\left\{\eta \in \mathcal{V}:\mathcal{N}(\xi ,\eta ,t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),\mathrm{for}\mathrm{some}\xi \in \mathcal{U}\right\}.$

Definition 10.

Let $\mathcal{U}$ and $\mathcal{V}$ be nonempty subsets of a θ-FMS $(\mathcal{X},\mathcal{N},\ast ,\theta )$ and $\mathcal{T}:\mathcal{U}\to I^{\mathcal{V}}$ be a fuzzy mapping. Then a point $\xi \in \mathcal{X}$ is called an FBPP of $\mathcal{T}$ if there exists $\alpha \in (0,1]$ such that $\mathcal{N}(\xi ,{\left[\mathcal{T}\xi \right]}_{\alpha },t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),$ for all $t>0$.

Definition 11.

Let $\mathcal{U}$ and $\mathcal{V}$ be non-empty subsets of a θ-FMS $(\mathcal{X},\mathcal{N},\ast ,\theta ).$ A fuzzy mapping $\mathcal{T}:\mathcal{U}\to I^{\mathcal{V}}$ is said to be a k-proximal fuzzy contraction with respect to ${\mathcal{U}}_{\circ }$ if there exists $k\in (0,1)$, such that, for each ${\xi }_1,{\xi }_2\in {\mathcal{U}}_{\circ },{\alpha }_{\mathcal{T}}\left({\xi }_1\right),{\alpha }_{\mathcal{T}}\left({\xi }_2\right)\in (0,1],$

$${\mathcal{V}}_{{\xi }_1}=\left\{\eta \in {\mathcal{U}}_0:\mathcal{N}(\eta ,{\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_1\right)},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),t>0\right\},$$

and

$${\mathcal{V}}_{{\xi }_2}=\left\{\eta \in {\mathcal{U}}_0:\mathcal{N}(\eta ,{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}\left({\xi }_2\right)},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),t>0\right\}$$

are nonempty, closed, and bounded sets and

$$\mathcal{H}({\mathcal{V}}_{{\xi }_1},{\mathcal{V}}_{{\xi }_2},kt)\ge \mathcal{N}({\xi }_1,{\xi }_2,t).$$

(20)

Lemma 3.

Let $(\mathcal{U},\mathcal{V})$ be a pair of nonempty subsets of a θ-FMS $(\mathcal{X},\mathcal{N},\ast ,\theta )$ with ${\mathcal{U}}_{\circ }\ne \varphi .$ Let $\mathcal{T}:\mathcal{U}\to I^{\mathcal{V}}$ be a fuzzy mapping such that, for every $\xi \in {\mathcal{U}}_{\circ }$, ${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}\cap {\mathcal{V}}_{\circ }$ is nonempty, and there exists ${\alpha }_{\mathcal{T}}\left(\xi \right)\in (0,1]$ with ${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}\in C\left(2^{\mathcal{X}}\right)$,then,

• For all $\xi \in {\mathcal{U}}_{\circ }$, the set ${\mathcal{V}}_{\xi }$ is nonempty.
• If ${\mathcal{U}}_{\circ }$ is closed and $\xi \in {\mathcal{U}}_{\circ }$, then ${\mathcal{V}}_{\xi }$ is closed.

Proof. 

(1) Let $\xi \in {\mathcal{U}}_{\circ }$; since ${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}\cap {\mathcal{V}}_{\circ }$ is nonempty, there exists $\eta \in {\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}\cap {\mathcal{V}}_{\circ }$, which implies that there exists $\zeta \in {\mathcal{U}}_{\circ }$ such that $\mathcal{N}(\zeta ,\eta ,t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),\mathrm{for}\mathrm{all}t>0.$ Therefore, $\mathcal{N}(\zeta ,{\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),$ proving that ${\mathcal{V}}_{\xi }$ is not empty.

(2) To prove that ${\mathcal{V}}_{\xi }$ is closed, consider a sequence $\left\{{\eta }_n\right\}$ in ${\mathcal{V}}_{\xi }$ that converges to a limit $\eta$. Since ${\eta }_n\in {\mathcal{U}}_{\circ }$ and satisfies

$$\mathcal{N}({\eta }_n,{\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),\forall t>0.$$

The continuity of $\mathcal{N}$ guarantees that

$$\mathcal{N}(\eta ,{\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),\forall t>0.$$

Since ${\mathcal{U}}_{\circ }$ is closed so $\eta \in {\mathcal{U}}_{\circ }$, it follows that $\eta \in {\mathcal{V}}_{\xi }$. Therefore, ${\mathcal{V}}_{\xi }$ must also be closed. □

Theorem 2.

Let $(\mathcal{U},\mathcal{V})$ be a pair of nonempty subsets of a complete θ-FMS $(\mathcal{X},\mathcal{N},\ast ,\theta )$ such that ${\mathcal{U}}_{\circ }$ is nonempty and closed. Assume that $\mathcal{T}:\mathcal{U}\to I^{\mathcal{V}}$ is a fuzzy mapping such that, for every $\xi \in \mathcal{U},$ there exists ${\alpha }_{\mathcal{T}}\left(\xi \right)\in (0,1]$ such that ${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}\in C\left(2^{\mathcal{V}}\right).$ Assume that the following conditions are also satisfied:

• $\mathcal{T}$ is an k-proximal fuzzy contraction with respect to ${\mathcal{U}}_{\circ };$
• for each $\xi \in {\mathcal{U}}_{\circ },{\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}\cap {\mathcal{V}}_{\circ }$ is nonempty.

Then there exist $\xi \in \mathcal{U}$ such that $\mathcal{N}(\xi ,{\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t).$

Proof. 

Let ${\xi }_0\in {\mathcal{U}}_{\circ }$. By Lemma 3 (1), we see that ${\mathcal{V}}_{\xi 0}$ is a nonempty set. Let ${\xi }_1\in {\mathcal{V}}_{{\xi }_0}.$ Then ${\xi }_1\in {\mathcal{U}}_{\circ }$, which implies that ${\mathcal{V}}_{{\xi }_1}$ is nonempty. ${\mathcal{U}}_{\circ }$ is closed, and by Lemma 3 (2), for each $\xi \in {\mathcal{U}}_{\circ }$, we get that ${\mathcal{V}}_{\xi }$ is closed and therefore is a compact subset of ${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}$, so we can choose ${\xi }_2\in {\mathcal{V}}_{{\xi }_1}$ such that

$$\begin{array}{c}\hfill \mathcal{N}({\xi }_1,{\xi }_2,t)\ge \mathcal{H}({\mathcal{V}}_{{\xi }_0},{\mathcal{V}}_{{\xi }_1},t).\end{array}$$

Continuing this process, we obtain a sequence $\left\{{\xi }_n\right\}$ in ${\mathcal{U}}_{\circ }$ such that $\mathcal{N}({\xi }_{n+1},{\xi }_n,t)=\mathcal{N}(\mathcal{U},\mathcal{V},t)$ and, by Lemma 2, we have

$$\begin{array}{c}\hfill \mathcal{N}({\xi }_{n+1},{\xi }_n,t)\ge \mathcal{H}({\mathcal{V}}_{{\xi }_n},{\mathcal{V}}_{{\xi }_{n-1}},t)\mathrm{for}\mathrm{all}n\in \mathbb{N}.\end{array}$$

Next, we show that $\left\{{\xi }_n\right\}$ is a Cauchy sequence in ${\mathcal{U}}_{\circ }$, and its limit is a BPP of $\mathcal{T}.$ Now, by (20) together with Lemma 2, for every $t>0,$ we find that

$$\begin{array}{cc}\hfill \mathcal{N}({\xi }_n,{\xi }_{n+1},t)& \ge \mathcal{H}({\mathcal{V}}_{{\xi }_n},{\mathcal{V}}_{{\xi }_{n-1}},t)\hfill \\ & \ge \mathcal{N}({\xi }_{n-1},{\xi }_n,{\displaystyle \frac{t}{k}})\hfill \\ & \ge \mathcal{H}({\mathcal{V}}_{{\xi }_{n-1}},{\mathcal{V}}_{{\xi }_{n-2}},{\displaystyle \frac{t}{k}})\hfill \\ & \ge \mathcal{N}({\xi }_{n-2},{\xi }_{n-1},{\displaystyle \frac{t}{k^2}})\hfill \\ & ⋮\hfill \\ & \ge \mathcal{H}({\mathcal{V}}_{{\xi }_0},{\mathcal{V}}_{{\xi }_1},{\displaystyle \frac{t}{k^{n-1}}})\hfill \\ & \ge \mathcal{N}({\xi }_0,{\xi }_1,{\displaystyle \frac{t}{k^n}}).\hfill \end{array}$$

(21)

Using (N4) and (21), let $m>n,$ for all $t\in Im\left(\theta \right),$ which means that there exists $s_i\le t$ for all $r_i\le t,i=1,2,\cdots ,m-n-1$ such that

$$\begin{array}{cc}\hfill \mathcal{N}({\xi }_n,{\xi }_m,t)& \ge \mathcal{N}({\xi }_n,{\xi }_{n+1},s_1)\ast \mathcal{N}({\xi }_{n+1},{\xi }_m,r_1)\hfill \\ & \ge \mathcal{N}({\xi }_n,{\xi }_{n+1},s_1)\ast \mathcal{N}({\xi }_{n+1},{\xi }_{n+2},s_2)\ast \mathcal{N}({\xi }_{n+2},{\xi }_m,r_2)\hfill \\ & \ge \mathcal{N}({\xi }_n,{\xi }_{n+1},s_1)\ast \mathcal{N}({\xi }_{n+1},{\xi }_{n+2},s_2)\ast \cdots \ast \mathcal{N}({\xi }_{m-1},{\xi }_m,s_{m-n-1})\hfill \\ & \ge \mathcal{N}({\xi }_0,{\xi }_1,{\displaystyle \frac{s_1}{k^{n-1}}})\ast \cdots \ast \mathcal{N}({\xi }_0,{\xi }_1,{\displaystyle \frac{s_{m-n-1}}{k^{m-n-1}}}).\hfill \end{array}$$

It follows that $\left\{{\xi }_n\right\}$ is a Cauchy sequence in ${\mathcal{U}}_{\circ }.$ Since ${\mathcal{U}}_{\circ }$ is closed, there exists $\xi \in {\mathcal{U}}_{\circ }$ such that $\left\{{\xi }_n\right\}$ converges to $\xi$ as $n\to +\infty$. By Lemma 3, it follows that ${\mathcal{V}}_{\xi }$ is nonempty and closed. Thus, there exists ${\xi }_n^{\prime }\in {\mathcal{V}}_{\xi }$ such that

$$\mathcal{N}({\xi }_{n+1},{\xi }_n^{\prime },t)\ge \mathcal{H}({\mathcal{V}}_{{\xi }_n},{\mathcal{V}}_{\xi },t)\ge \mathcal{N}({\xi }_n,\xi ,{\displaystyle \frac{t}{k}}),$$

which implies ${\displaystyle \underset{n\to +\infty }{lim}\mathcal{N}({\xi }_{n+1},{\xi }_n^{\prime },t)=1.}$ Therefore, $\left\{{\xi }_n^{\prime }\right\}$ converges to $\xi$, and since ${\mathcal{V}}_{\xi }$ is closed, it follows $\xi \in {\mathcal{V}}_{\xi }$, that is, $\mathcal{N}(\xi ,{\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t).$ □

Example 6.

Consider $\mathcal{X}={\mathbb{R}}^2.$ Define $a\ast b=a·b$, $\theta \in \Theta ,\theta (t,s)=t+s+ts,t,s\ge 0$ and $\mathcal{N}:\mathcal{X}\times \mathcal{X}\times (0,+\infty )\to (0,1]$ by

$$\mathcal{N}((a_1,a_2),(b_1,b_2),t)=exp\left(-{\displaystyle \frac{|a_1-b_1|+|a_2-b_2|}{t}}\right).$$

(22)

Suppose $\mathcal{U}=\left\{(1,\xi ):\xi \in [0,1]\right\}$ and $\mathcal{V}=\left\{\right(0,\eta ),\eta \in [0,1\left]\right\}.$ Let $\alpha \in (0,1]$; $\mathcal{T}:\mathcal{U}\to I^{\mathcal{V}}$ is defined by

(i) If $\xi =0$

$$\begin{array}{c}\hfill \mathcal{T}\left((1,0)\right)\left((0,u)\right)=\left\{\begin{array}{cc}1,\hfill & \mathit{if}u=0,\hfill \\ 0,\hfill & \mathit{if}u\ne 0.\hfill \end{array}\right.\end{array}$$

(ii) If $\xi \ne 0$

$$\begin{array}{c}\hfill \mathcal{T}\left(1,\xi \right)\left(0,u\right)=\left\{\begin{array}{cc}\alpha ,\hfill & \mathit{if}0<u<\frac{\xi }{4},\hfill \\ \frac{\alpha }{2},\hfill & \mathit{if}\frac{\xi }{4}\le u\le \frac{\xi }{2},\hfill \\ \frac{\alpha }{4},\hfill & \mathit{if}\frac{\xi }{2}<u\le \xi .\hfill \end{array}\right.\end{array}$$

$\mathcal{N}(\mathcal{U},\mathcal{V},t)=e^{{\displaystyle \frac{-1}{t}}},t>0$. As ${\mathcal{U}}_{\circ }=\mathcal{U},{\mathcal{V}}_{\circ }=\mathcal{V},$ for every $(1,\xi )\in \mathcal{U},$ there exists ${{\displaystyle \frac{\alpha }{2}}}_{\mathcal{T}}\left(\xi \right)\in (0,1];$ we obtain

$$\begin{array}{c}\hfill {\left[\mathcal{T}(1,\xi )\right]}_{{\displaystyle \frac{\alpha }{2}}}=\left\{(0,\eta )\in \mathcal{V}:\mathcal{T}\left((1,\xi )\right)\left((0,\eta )\right)\ge {\displaystyle \frac{\alpha }{2}}\right\}=\left\{0\right\}\times [0,{\displaystyle \frac{\xi }{2}}]\in C\left(2^{\mathcal{V}}\right).\end{array}$$

We can see for each $\xi \in {\mathcal{U}}_{\circ }$ we have that ${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\xi \right)}\cap {\mathcal{V}}_{\circ }$ is nonempty. Now, we show that the fuzzy mapping $\mathcal{T}:\mathcal{U}\to I^{\mathcal{V}}$ is a k-proximal fuzzy contraction with respect to ${\mathcal{U}}_{\circ }.$ Let ${\xi }_1,{\xi }_2\in {\mathcal{U}}_{\circ };$ then we have ${\mathcal{V}}_{{\xi }_1}=\left\{(1,{\displaystyle \frac{{\xi }_1}{2}})\right\},$ and ${\mathcal{V}}_{{\xi }_2}=\left\{(1,{\displaystyle \frac{{\xi }_2}{2}})\right\}$ are non-empty, closed, and bounded, and the condition

$$\mathcal{H}({\mathcal{V}}_{{\xi }_1},{\mathcal{V}}_{{\xi }_2},kt)\ge \mathcal{N}({\xi }_1,{\xi }_2,t)$$

(23)

holds with $k={\displaystyle \frac{1}{2}}\in (0,1).$ Consequently, all the conditions of Theorem 2 are satisfied to find an ${\displaystyle \frac{\alpha }{2}}\in (0,1]$ such that $\mathcal{N}((1,0),{\left[\mathcal{T}(1,0)\right]}_{{\displaystyle \frac{\alpha }{2}}},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),$ for all $t>0$.

4.2. Multivalued Proximal Mappings

Definition 12.

Let $\mathcal{U}$ and $\mathcal{V}$ be non-empty subsets of a θ-FMS $(\mathcal{X},\mathcal{N},\ast ,\theta ).$ The multivalued mapping $\mathcal{S}:\mathcal{U}\to 2^{\mathcal{V}}\setminus \varphi$ is said to be a k-proximal multivalued contraction with respect to ${\mathcal{U}}_{\circ }$ if there exists $k\in (0,1)$, such that, for each ${\xi }_1,{\xi }_2\in {\mathcal{U}}_{\circ },$

$${\mathcal{V}}_{{\xi }_1}=\left\{\eta \in {\mathcal{U}}_0:\mathcal{N}(\eta ,\mathcal{S}{\xi }_1,t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),t>0\right\},$$

and

$${\mathcal{V}}_{{\xi }_2}=\left\{\eta \in {\mathcal{U}}_0:\mathcal{N}(\eta ,\mathcal{S}{\xi }_2,t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),t>0\right\},$$

two sets are non-empty, closed, bounded, and

$$\mathcal{H}({\mathcal{V}}_{{\xi }_1},{\mathcal{V}}_{{\xi }_2},kt)\ge \mathcal{N}({\xi }_1,{\xi }_2,t).$$

(24)

Corollary 3.

Let $(\mathcal{U},\mathcal{V})$ be a pair of nonempty subsets of a complete θ-FMS $(\mathcal{X},\mathcal{N},\ast ,\theta )$ such that ${\mathcal{U}}_{\circ }$ is nonempty and closed. Assume that $\mathcal{S}:\mathcal{U}\to 2^{\mathcal{V}}\setminus \varnothing$ is a multivalued mapping satisfying the following conditions:

• $\mathcal{S}$ is a k-proximal fuzzy contraction with respect to ${\mathcal{U}}_{\circ }.$
• For each $\xi \in {\mathcal{U}}_{\circ },\mathcal{T}\xi \cap {\mathcal{V}}_{\circ }$ is nonempty.

Then there exists some $\xi \in \mathcal{U}$ such that $\mathcal{U}$ such that $\mathcal{N}(\xi ,\mathcal{T}\xi ,t)=\mathcal{N}(\mathcal{U},\mathcal{V},t).$

Proof. 

Let ${\alpha }_{\mathcal{T}}:\mathcal{X}\to (0,1]$ be an arbitrary mapping, and consider a fuzzy mapping $\mathcal{T}:\mathcal{U}\to {\mathcal{I}}^{\mathcal{V}}$ defined as follows:

$$\begin{array}{cc}\hfill & \mathcal{T}\left(\xi \right)\left(u\right)=\left\{\begin{array}{cc}{\alpha }_{\mathcal{T}}\left(\xi \right),\hfill & \mathrm{if}u\in \mathcal{S}\xi ,\hfill \\ 0,\hfill & \mathrm{if}u\notin \mathcal{S}\xi .\hfill \end{array}\right.\hfill \end{array}$$

(25)

Then, for all $\xi \in \mathcal{U},$ we have

$${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}}=\{u\in \mathcal{U}:\mathcal{T}\left(\xi \right)\left(u\right)\ge {\alpha }_{\mathcal{T}}\left(\xi \right)\}=\mathcal{S}\xi .$$

(26)

Thus, for each ${\xi }_1,{\xi }_2\in {\mathcal{U}}_{\circ },$

$${\mathcal{V}}_{{\xi }_1}=\left\{\eta \in {\mathcal{U}}_0:\mathcal{N}(\eta ,\mathcal{S}{\xi }_1,t)=\mathcal{N}(\eta ,{\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),t>0\right\},$$

and

$${\mathcal{V}}_{{\xi }_2}=\left\{\eta \in {\mathcal{U}}_0:\mathcal{N}(\eta ,\mathcal{S}{\xi }_2,t)=\mathcal{N}(\eta ,{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),t>0\right\},$$

two sets are non-empty, closed, bounded, and

$$\mathcal{H}({\mathcal{V}}_{{\xi }_1},{\mathcal{V}}_{{\xi }_2},kt)\ge \mathcal{N}({\xi }_1,{\xi }_2,t).$$

(27)

As a result, Theorem 2 is applicable. □

4.3. Single-Valued Proximal Contraction

Definition 13.

Let $\mathcal{U}$ and $\mathcal{V}$ be non-empty subsets of a θ-FMS $(\mathcal{X},\mathcal{N},\ast ,\theta ).$ A single-valued mapping $\mathcal{T}:\mathcal{U}\to \mathcal{V}$ is said to be a k-proximal contraction concerning ${\mathcal{U}}_{\circ }$ if there exists $k\in (0,1)$, such that, for each ${\xi }_1,{\xi }_2,{\eta }_1,{\eta }_2\in {\mathcal{U}}_{\circ },$

$$\left\{\begin{array}{c}\mathcal{N}({\eta }_1,\mathcal{T}{\xi }_1,t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),\hfill \\ \mathcal{N}({\eta }_2,\mathcal{T}{\xi }_2,t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),\hfill \end{array}\right.$$

implies that

$$\mathcal{N}({\eta }_1,{\eta }_2,kt)\ge \mathcal{N}({\xi }_1,{\xi }_2,t).$$

(28)

Corollary 4.

Let $(\mathcal{U},\mathcal{V})$ be a pair of nonempty subsets of a complete θ-FMS $(\mathcal{X},\mathcal{N},\ast ,\theta )$ such that ${\mathcal{U}}_{\circ }$ is nonempty and closed. Let $\mathcal{S}:\mathcal{U}\to \mathcal{V}$ be a single-valued mapping. Assume that the subsequent conditions are also met, as follows:

• $\mathcal{S}$ is a k-proximal contraction with respect to ${\mathcal{U}}_{\circ }.$
• For each $\xi \in {\mathcal{U}}_{\circ },\mathcal{S}\xi \in {\mathcal{V}}_{\circ }$.

Then there exists $\xi \in \mathcal{U}$ such that $\mathcal{N}(\xi ,\mathcal{S}\xi ,t)=\mathcal{N}(\mathcal{U},\mathcal{V},t).$

Proof. 

Let ${\alpha }_{\mathcal{S}}:\mathcal{X}\to (0,1]$ be an arbitrary mapping, and consider a fuzzy mapping $\mathcal{S}:\mathcal{U}\to {\mathcal{I}}^{\mathcal{V}}$ defined as follows:

$$\begin{array}{cc}\hfill & \mathcal{T}\left(\xi \right)\left(u\right)=\left\{\begin{array}{cc}{\alpha }_{\mathcal{T}}\left(\xi \right),\hfill & \mathrm{if}u=\mathcal{S}\xi ,\hfill \\ 0,\hfill & \mathrm{if}u\ne \mathcal{S}\xi .\hfill \end{array}\right.\hfill \end{array}$$

(29)

Then, for all $\xi \in \mathcal{U},$ we have

$${\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}}=\left\{u\in \mathcal{U}:\mathcal{T}\left(\xi \right)\left(u\right)\ge {\alpha }_{\mathcal{L}}\left(\xi \right)\}=\{\mathcal{S}\xi \right\}.$$

(30)

For each ${\xi }_1,{\xi }_2\in {\mathcal{U}}_{\circ },$ we have

$${\mathcal{V}}_{{\xi }_1}=\left\{{\eta }_1\in {\mathcal{U}}_0:\mathcal{N}({\eta }_1,\left\{\mathcal{S}{\xi }_1\right\},t)=\mathcal{N}({\eta }_1,{\left[\mathcal{T}{\xi }_1\right]}_{{\alpha }_{\mathcal{T}}},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),t>0\right\}=\left\{{\eta }_1\right\},$$

and

$${\mathcal{V}}_{{\xi }_2}=\left\{{\eta }_2\in {\mathcal{U}}_0:\mathcal{N}({\eta }_2,\left\{\mathcal{S}{\xi }_2\right\},t)=\mathcal{N}({\eta }_2,{\left[\mathcal{T}{\xi }_2\right]}_{{\alpha }_{\mathcal{T}}},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),t>0\right\}=\left\{{\eta }_2\right\},$$

are two sets non-empty, closed, bounded, and

$$\mathcal{H}({\mathcal{V}}_{{\xi }_1},{\mathcal{V}}_{{\xi }_2},kt)=\mathcal{N}({\eta }_1,{\eta }_2,kt)\ge \mathcal{N}({\xi }_1,{\xi }_2,t).$$

(31)

Therefore, Theorem 1 can be applied to find $\xi \in \mathcal{U}$ such that $\xi \in \left\{\mathcal{T}\right(\xi \left)\right\}$, which further implies $\mathcal{N}(\xi ,\mathcal{T}(\xi ),t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),$ for all $t>0$. □

Corollary 5.

Theorem 1 implies Theorem 2.

Proof. 

Define $\mathcal{G}:{\mathcal{U}}_{\circ }\to C\left(2^{{\mathcal{U}}_{\circ }}\right)$ by

$${\left[\mathcal{G}\xi \right]}_{{\alpha }_{\mathcal{G}}\left(\xi \right)}=\left\{\eta \in {\mathcal{U}}_{\circ }:\mathcal{N}(\eta ,{\left[\mathcal{T}\xi \right]}_{{\alpha }_{\mathcal{T}}\left(\zeta \right)},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t),t>0\right\},$$

for $\xi \in {\mathcal{U}}_{\circ },{\alpha }_{\mathcal{G}}\left(\xi \right)\in (0,1].$ It follows from Lemma 3 that ${\left[\mathcal{G}\xi \right]}_{{\alpha }_{\mathcal{G}}\left(\xi \right)}$ is a nonempty, closed, and bounded subset of ${\mathcal{U}}_{\circ }$ for each $\xi \in {\mathcal{U}}_{\circ }$ and so ${\left[\mathcal{G}\xi \right]}_{{\alpha }_{\mathcal{G}}\left(\xi \right)}$ is well defined. Since $\mathcal{T}$ is a k-proximal fuzzy contraction with respect to ${\mathcal{U}}_{\circ }$,

$$\mathcal{H}({\left[\mathcal{G}\xi \right]}_{{\alpha }_{\mathcal{G}}\left(\xi \right)},{\left[\mathcal{G}\eta \right]}_{{\alpha }_{\mathcal{G}}\left(\eta \right)},kt)=\mathcal{H}({\mathcal{V}}_{\xi },{\mathcal{V}}_{\eta },kt)\ge \mathcal{N}(\xi ,\eta ,t),$$

for all $\xi ,\eta \in {\mathcal{U}}_{\circ }$. It now follows from Theorem 1 that there exists $\zeta \in {\mathcal{U}}_{\circ }$ such that $\zeta \in {\left[\mathcal{G}\zeta \right]}_{{\alpha }_{\mathcal{T}}\left(\zeta \right)}$. By the definition of the mapping ${\left[\mathcal{G}\zeta \right]}_{{\alpha }_{\mathcal{G}}\left(\xi \right)}$, the point $\zeta$ satisfies $\mathcal{N}(\zeta ,{\left[\mathcal{T}\zeta \right]}_{{\alpha }_{\mathcal{T}}\left(\zeta \right)},t)=\mathcal{N}(\mathcal{U},\mathcal{V},t)$, and this completes the proof that Theorem 1 implies Theorem 2. □

5. Application to Fuzzy Fractional Differential Equations

The fuzzy Hadamard $\Psi$-CTFD was introduced by Abdou Amir et al. [15] as a comprehensive generalization, established through the integration of various fractional operators, including tempered Riemann–Liouville,$\Psi$-Riemann–Liouville–Hadamard, Riemann–Liouville, Caputo, and $\Psi$-Caputo. This unification provides a cohesive framework for understanding their applications across different mathematical settings, offering a systematic perspective on these operators and expanding their potential uses in various research fields and mathematical analysis.

Definition 14

([15]). Let ξ be a fuzzy number-valued function $n-1<\alpha <n,n\in \mathbb{N},\gamma ,p,q\ge 0$ and $\Psi \in C^n([a,b],\mathbb{R})$ such that ${\Psi }^{\prime }\left(t\right)>0,\forall t\in [a,b].$ The design of the generalized Hadamard Ψ-CTFD of level α of the function ξ is defined by

$$\begin{array}{cc}\hfill {}^{CH}\mathcal{D}_0^{\alpha ,\gamma ,p,q,\Psi }\xi \left(t\right)=& {\displaystyle \frac{E_{p,q}(-\gamma ln(\Psi (t\left)\right))}{\Gamma (n-\alpha )}}\odot {\int }_a^t{\displaystyle \frac{{\Psi }^{\prime }\left(s\right)}{\Psi \left(s\right)}}{\left[ln\left({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}}\right)\right]}^{n-\alpha -1}\hfill \\ & \odot {\left[{\displaystyle \frac{\Psi \left(s\right)}{{\Psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right]}^n(E_{p,q}(\gamma ln(\Psi (s\left)\right))\odot \xi \left(s\right)ds,\hfill \end{array}$$

(32)

where ${\displaystyle E_{p,q}\left(s\right)=\sum _{k=0}^{+\infty }\frac{s^k}{\Gamma (pk+q)},p,q>0,Re\left(s\right)>0.}$

First of all, we should consider the multiplication of a fuzzy number by a scalar in its level-wise form.

Suppose that $k\in \mathbb{R}$ is a scalar, and M is a fuzzy number. Then, in level-wise form, we have

$$k\odot M\left[r\right]=\left\{\begin{array}{cc}[k·M_l\left(r\right),k·M_u\left(r\right)]\hfill & \mathrm{if}k\ge 0,\hfill \\ ·M_u\left(r\right),k·M_l\left(r\right)]\hfill & \mathrm{if}k<0,\hfill \end{array}\right.\mathrm{for}\mathrm{all}r\in [0,1].$$

For any two arbitrary fuzzy numbers M and N, and any fixed $r\in [0,1]$, if $M\odot N=K$, then we have

$$K\left[r\right]=[K_l\left(r\right),K_u\left(r\right)]=M\left[r\right]\odot N\left[r\right]=[M_l\left(r\right),M_u\left(r\right)]\odot [N_l\left(r\right),N_u\left(r\right)].$$

Then,

$$K_l\left(r\right)=min\left\{\begin{array}{c}{M}_l\left(r\right)·N_l\left(r\right),\hfill \\ M_l\left(r\right)·N_u\left(r\right),\hfill \\ M_u\left(r\right)·N_l\left(r\right),\hfill \\ M_u\left(r\right)·N_u\left(r\right)\hfill \end{array}\right\},K_u\left(r\right)=max\left\{\begin{array}{c}{M}_l\left(r\right)·N_l\left(r\right),\hfill \\ M_l\left(r\right)·N_u\left(r\right),\hfill \\ M_u\left(r\right)·N_l\left(r\right),\hfill \\ M_u\left(r\right)·N_u\left(r\right)\hfill \end{array}\right\}.$$

As an application, we extend the SIR dynamics model investigated by Subramanian et al. [23] to include the fuzzy Hadamard $\Psi$-CTFD. Here, the susceptible population $S\left(t\right)$, the infected population $I\left(t\right)$, and the removed population $R\left(t\right)$ compose the overall population $N\left(t\right)$, structured as follows:

$$\left\{\begin{array}{c}{}^{CH}\mathcal{D}_0^{\alpha ,\gamma ,p,q,\Psi }S\left(t\right)=(1-p)\pi -\tilde{\beta }SI-\mu S,\hfill \\ {}^{CH}\mathcal{D}_0^{\alpha ,\gamma ,p,q,\Psi }I\left(t\right)=\tilde{\beta }SI-(\tilde{\gamma }+\mu )I,\hfill \\ {}^{CH}\mathcal{D}_0^{\alpha ,\gamma ,p,q,\Psi }R\left(t\right)=p\pi +\tilde{\gamma }I-\mu R,\hfill \end{array}\right.$$

(33)

where $\mu ,\pi ,p,\tilde{\beta },\tilde{\gamma }$ represent natural death rate, birth date, fraction of the vaccinated population at birth, contact rate of susceptible individuals, and infected individuals who recover at a rate, respectively. Now, the right-hand side of (33) becomes

$$\left\{\begin{array}{c}A(t,S\left(t\right))=(1-p)\pi -\tilde{\beta }SI-\mu S,\hfill \\ B(t,I\left(t\right))=\tilde{\beta }SI-(\tilde{\gamma }+\mu )I,\hfill \\ D(t,R\left(t\right))=p\pi +\tilde{\gamma }I-\mu R.\hfill \end{array}\right.$$

(34)

where $A,B,D$ are fuzzy functions. Then, for $r\in [0,1],$ the model in Equation (33) is expressed as

$$\left\{\begin{array}{c}{}^{CH}\mathcal{D}_0^{\alpha ,\gamma ,p,q,\Psi }S\left(t\right)=A(t,S\left(t\right)),\hfill \\ {}^{CH}\mathcal{D}_0^{\alpha ,\gamma ,p,q,\Psi }I\left(t\right)=B(t,I\left(t\right)),\hfill \\ {}^{CH}\mathcal{D}_0^{\alpha ,\gamma ,p,q,\Psi }R\left(t\right)=D(t,R\left(t\right)),\hfill \end{array}\right.$$

(35)

with fuzzy initial conditions

$$\begin{array}{cc}\hfill & \tilde{S}(0,r)=[\underline{S}(0,r),\overline{S}(0,r)],\hfill \\ & \tilde{I}(0,r)=[\underline{I}(0,r),\overline{I}(0,r)],\hfill \\ & \tilde{R}(0,r)=[\underline{R}(0,r),\overline{R}(0,r)].\hfill \end{array}$$

(36)

Let us put

$$\begin{array}{cc}\hfill & G\left(t\right)=\left\{\begin{array}{c}S\left(t\right),\hfill \\ I\left(t\right),\hfill \\ R\left(t\right),\hfill \end{array}\right.\hfill \end{array}$$

(37)

$$\begin{array}{cc}& G(0,r)=\left\{\begin{array}{c}\tilde{S}(0,r),\hfill \\ \tilde{I}(0,r),\hfill \\ R(0,r)\hfill \end{array}\right.\hfill \end{array}$$

(38)

$$\begin{array}{cc}& F(t,G\left(t\right))=\left\{\begin{array}{c}A(t,S(t\left)\right),\hfill \\ B(t,I(t\left)\right),\hfill \\ D(t,R(t\left)\right).\hfill \end{array}\right.\hfill \end{array}$$

(39)

Then, problem (33) can be reformulated as

$$\begin{array}{cc}\hfill {}^{CH}\mathcal{D}_0^{\alpha ,\gamma ,p,q,\Psi }& G\left(t\right)=F(t,G(t\left)\right),t\in [0,a],0<\alpha <1.\hfill \\ & G(0,r)=G_0\in E^1,\hfill \end{array}$$

(40)

where ${}^{CH}\mathcal{D}_0^{\alpha ,\gamma ,p,q,\Psi }$ design the generalized Hadamard $\Psi$-CTFD of level $\alpha$ and $F\in C([0,a]\times E^1,E^1)$, $\Psi$ is a continuously differentiable, increasing function on the interval $[0,\infty )$ with $\Psi \left(0\right)=0,{\Psi }^{\prime }\left(t\right)>0$ for all ${\displaystyle t\in (0,+\infty ),\underset{t\to +\infty }{lim}\Psi \left(t\right)=+\infty .}$

A complete fuzzy $\theta$-metric on $C([0,a]\times E^1,E^1)$ is defined as follows:

$$\mathcal{N}(G_1\left(t\right),G_2\left(t\right),\tau )=exp\left(-{\displaystyle \frac{\parallel G_1\left(t\right)-G_2\left(t\right)\parallel }{\tau }}\right),\tau >0,$$

(41)

where ${\displaystyle \parallel G_1\left(t\right)-G_2\left(t\right)\parallel =\underset{t\in [0,a]}{max}\left\{|S_1\left(t\right)-S_2\left(t\right)|+|I_1\left(t\right)-I_2\left(t\right)|+|R_1\left(t\right)-R_2\left(t\right)|\right\}}$ and $\theta (r,s)=\sqrt{r^2+s^2},a\ast b=a·b$.

Lemma 4

([15]). Let $G\left(t\right)$ represent the solution to Equation (40).

•

If $G\left(t\right)$ is Caputo $\left(i\right)$-gH differentiable,

$$\begin{array}{cc}\hfill G\left(t\right)=& E_{p,q}(-\gamma ln(\Psi (t\left)\right))\odot \xi \left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\hfill \\ & \odot {\int }_0^t{\displaystyle \frac{{\Psi }^{\prime }\left(s\right)}{\Psi \left(s\right)}}{\left[ln({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}})\right]}^{\alpha -1}{E}_{p,q}(-\gamma ln\left({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}}\right))\odot f(s,\xi \left(s\right))ds.\hfill \end{array}$$

(42)

•

If $G\left(t\right)$ is Caputo $\left(ii\right)$-gH differentiable,

$$\begin{array}{cc}\hfill G\left(t\right)=& E_{p,q}(-\gamma ln(\Psi (t\left)\right))\odot \xi \left(0\right)\ominus {\displaystyle \frac{-1}{\Gamma \left(\alpha \right)}}\hfill \\ & \odot {\int }_0^t{\displaystyle \frac{{\Psi }^{\prime }\left(s\right)}{\Psi \left(s\right)}}{\left[ln({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}})\right]}^{\alpha -1}{E}_{p,q}\left(-\gamma ln\left({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}}\right)\right)\odot f(s,\xi \left(s\right))ds.\hfill \end{array}$$

(43)

Theorem 3.

Assume that $F\in C([0,a]\times E^1,E^1)$ is bounded such that

$$\parallel F(t,G_1\left(t\right))-F(t,G_1\left(t\right))\parallel \le M\parallel G_1\left(t\right)-G_1\left(t\right)\parallel ,\mathrm{for}\mathrm{all}t\in [0,a]$$

(44)

such that ${\displaystyle \frac{M}{{\gamma }^{\alpha }}}<1$. Then by Theorem 1, Equation (35) has a unique solution for two cases in Lemma 4.

Proof. 

WOLG, assume that $G\left(t\right)$ is Caputo $\left(i\right)$-gH differentiable. Consider a closed convex subset $\mathcal{X}=\left\{G\in C([0,a]\times E^1,E^1):\parallel G\left(t\right)-E_{p,q}(-\gamma ln(\Psi (t\left)\right))\odot G(0)\parallel \le R\right\}$, where $R={\displaystyle \frac{N}{{\gamma }^{\alpha }}},N=\parallel f(t,G\left(t\right)\parallel .$ Additionally, consider a mapping ${\mathcal{P}}_G\left(t\right)$ over $\mathcal{X}$ such that

$$\begin{array}{cc}\hfill {\mathcal{P}}_G\left(t\right)=& E_{p,q}(-\gamma ln(\Psi (t\left)\right))\odot G\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\hfill \\ & \odot {\int }_0^t{\displaystyle \frac{{\Psi }^{\prime }\left(s\right)}{\Psi \left(s\right)}}{\left[ln({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}})\right]}^{\alpha -1}{E}_{p,q}(-\gamma ln\left({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}}\right))\odot f(s,G\left(s\right))ds.\hfill \end{array}$$

(45)

First, we show ${\mathcal{P}}_G\left(t\right)$ maps $\mathcal{X}$ into $\mathcal{X},asfollows:$

$$\begin{array}{cc}\hfill & \parallel {\mathcal{P}}_G\left(t\right)-E_{p,q}(-\gamma ln(\Psi (t\left)\right))\odot G(0)\parallel \hfill \\ & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{{\Psi }^{\prime }\left(s\right)}{\Psi \left(s\right)}}{\left[ln({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}})\right]}^{\alpha -1}{E}_{p,q}(-\gamma ln\left({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}}\right))\odot \parallel f(s,G\left(s\right))\parallel ds\hfill \\ & \le {\displaystyle \frac{N}{\Gamma \left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{{\Psi }^{\prime }\left(s\right)}{\Psi \left(s\right)}}{\left[ln({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}})\right]}^{\alpha -1}{E}_{p,q}(-\gamma ln\left({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}}\right))ds.\hfill \end{array}$$

Let us make a change to variables by putting $u=ln\left({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}}\right)$, which implies that $ds={\displaystyle \frac{-\Psi \left(s\right)}{{\Psi }^{\prime }\left(s\right)}}du$. Thus,

$$\begin{array}{cc}\hfill \parallel {\mathcal{P}}_G\left(t\right)-E_{p,q}(-\gamma ln(\Psi (t\left)\right))\odot G(0)\parallel & \le {\displaystyle \frac{N}{\Gamma \left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{{\Psi }^{\prime }\left(s\right)}{\Psi \left(s\right)}}{\left[ln({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}})\right]}^{\alpha -1}{E}_{p,q}(-\gamma ln\left({\displaystyle \frac{\Psi \left(t\right)}{\Psi \left(s\right)}}\right))ds\hfill \\ & \le {\displaystyle \frac{N}{\Gamma \left(\alpha \right)}}{\int }_0^{ln\Psi \left(t\right)}{\displaystyle \frac{{\Psi }^{\prime }\left(s\right)}{\Psi \left(s\right)}}{u}^{\alpha -1}{E}_{p,q}(-\gamma u\left)\right)du\hfill \\ & \le {\displaystyle \frac{N}{\Gamma \left(\alpha \right)}}{\int }_0^{ln\Psi \left(t\right)}{\displaystyle \frac{{\Psi }^{\prime }\left(s\right)}{\Psi \left(s\right)}}{u}^{\alpha -1}exp(-\gamma u))du\hfill \\ & \le {\displaystyle \frac{N}{\Gamma \left(\alpha \right)}}{\int }_0^{+\infty }{\displaystyle \frac{{\Psi }^{\prime }\left(s\right)}{\Psi \left(s\right)}}{u}^{\alpha -1}exp(-\gamma u))du\hfill \\ & ={\displaystyle \frac{N}{{\gamma }^{\alpha }}}=R.\hfill \end{array}$$

Let $\beta :C([0,a]\times E^1,E^1)\to [0,1]$ be a mapping. Consider a fuzzy mapping $\mathcal{T}:\mathcal{X}\to {\mathcal{I}}^{\mathcal{X}}$, defined by

$$\begin{array}{cc}\hfill & {\mu }_{\mathcal{T}\left(G\right)}r=\left\{\begin{array}{cc}\beta \left(G\right),\hfill & \mathrm{if}r\left(t\right)={\mathcal{P}}_G\left(t\right),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

(46)

Therefore, we have

$$\begin{array}{c}\hfill {\left[\mathcal{T}G\right]}_{\beta \left(G\right)}=\left\{r\left(t\right)\in \mathcal{X}:\left(\mathcal{T}\right(G\left)\right(t)\ge \beta (G)\right\}=\left\{{\mathcal{P}}_G\left(t\right)\right\}.\end{array}$$

(47)

Therefore, we have for all $\tau >0,$

$$\begin{array}{cc}\hfill & \mathcal{H}({\left[\mathcal{T}{G}_1\right]}_{\beta \left(G_1\right)},{\left[\mathcal{T}{G}_2\right]}_{\beta \left(G_2\right)},k\tau )\hfill \\ & =min\left\{\underset{G_1\in {\left[\mathcal{T}{G}_1\right]}_{\beta \left(G_1\right)}}{inf}\mathcal{N}(G_1,{\left[\mathcal{T}{G}_2\right]}_{\beta \left(G_2\right)},k\tau ),\underset{G_2\in {\left[\mathcal{T}{G}_2\right]}_{\beta \left(G_2\right)}}{inf}\mathcal{N}(G_2,{\left[\mathcal{T}{G}_1\right]}_{\beta \left(G_1\right)},k\tau )\right\}\hfill \\ & =\underset{t\in [0,a]}{inf}\mathcal{N}({\mathcal{P}}_{G_1}\left(t\right),{\mathcal{P}}_{G_2}\left(t\right),k\tau ) \hfill \\ & \ge \underset{t\in [0,a]}{inf}{exp}^{\left(\frac{-\frac{1}{\Gamma \left(\alpha \right))}\odot {\int }_0^t\frac{{\Psi }^{\prime }\left(s\right)}{\Psi \left(s\right)}{\left[ln(\frac{\Psi \left(t\right)}{\Psi \left(s\right)})\right]}^{\alpha -1}{E}_{p,q}(-\gamma ln\left(\frac{\Psi \left(t\right)}{\Psi \left(s\right)}\right))\odot \parallel F(s,G_1\left(s\right))-F(s,G_2\left(s\right))\parallel ds}{k\tau }\right)}\hfill \\ & \ge \underset{t\in [0,a]}{inf}{exp}^{\left(\frac{-\frac{M\parallel G_1-G_2\parallel }{\Gamma \left(\beta \right)}\odot {\int }_0^t\frac{{\Psi }^{\prime }\left(s\right)}{\Psi \left(s\right)}{\left[ln(\frac{\Psi \left(t\right)}{\Psi \left(s\right)})\right]}^{\alpha -1}{E}_{p,q}(-\gamma ln\left(\frac{\Psi \left(t\right)}{\Psi \left(s\right)}\right))ds}{k\tau }\right) }\hfill \\ & \ge \underset{t\in [0,a]}{inf}{exp}^{\left(\frac{-\frac{M\parallel G_1-G_2\parallel }{\Gamma \left(\alpha \right)}\odot {\int }_0^{ln\Psi \left(t\right)}{u}^{\alpha -1}{E}_{p,q}(-\gamma u)du}{k\tau }\right)}\hfill \\ & \ge exp\left(\frac{-\frac{M\parallel G_1-G_2\parallel }{\Gamma \left(\alpha \right)}\odot {\int }_0^{+\infty }{u}^{\alpha -1}{E}_{p,q}(-\gamma u)du}{k\tau }\right)\hfill \\ & \ge exp\left(\frac{-M\parallel G_1-G_2\parallel }{{\gamma }^{\alpha }\tau }\right)\hfill \\ & =exp\left(\frac{-M\parallel G_1-G_2\parallel }{{\gamma }^{\alpha }k\tau }\right)\hfill \\ & =\mathcal{N}(G_1,G_2,\tau ),k=\frac{M}{{\gamma }^{\alpha }}<1.\hfill \end{array}$$

Consequently, the requirements of Theorem 1 are satisfied, resulting in (35) possessing a unique type 1 solution; similar results are obtained when $G\left(t\right)$ is Caputo $\left(ii\right)$-gH differentiable. □

6. Conclusions and Future Works

This article addresses five key aspects. First, introducing $\theta$-FMSs provides a unifying framework that generalizes various existing spaces. Second, it establishes FFP and FBPP theorems within $\theta$-FMSs, deriving corresponding results for both single-valued and multivalued mappings. Third, it explores the intrinsic relationship between FFP and FBPP theorems, offering deeper insights into their interplay. From an application perspective, one of our main results is to establish existence conditions for solutions to the SIR dynamics model using the fuzzy Hadamard $\Psi$-Caputo tempered fractional derivative ($\Psi$-CTFD). To our knowledge, these findings are novel and fundamental in the study of $\theta$-FMSs and fuzzy set theory. In future studies, these ideas could be expanded to more extensive areas like L-fuzzy mappings, intuitionistic fuzzy mappings, soft set-valued maps, and other diverse hybrid models within fuzzy mathematics.