From α-fuzzy Fixed Points to Nonlinear Cauchy Differential Inclusions in Intuitionistic Fuzzy Metric Spaces

Abstract

This study delves into the concept of $α$ -fuzzy mappings and their associated $α$ -fuzzy fixed points within the framework of Hausdorff intuitionistic fuzzy metric-like spaces. A general fixed point theorem for $α$ -fuzzy mappings is established in complete HIFMS and its subclasses. The results unify and generalize several classical and fuzzy fixed point theorems, extending them to more complex fuzzy structures. Additionally, recognizing the significant applications of differential inclusions as set-valued mappings, the research explores first-order nonlinear Cauchy differential inclusions within Hausdorff intuitionistic fuzzy metric spaces by leveraging the derived theoretical results. The findings demonstrate the robustness of fuzzy fixed point theory in modeling systems with uncertainty and imprecision, with potential applications in control theory and optimization. The gap between fuzzy metric theory and differential inclusions.

Keywords

Differential inclusion α-fuzzy mapping intuitionistic fuzzy metric

1. Introduction

The study of fixed points is a dynamic and evolving area of mathematics, closely linked to functional analysis, topology, engineering, and science. This theory plays a pivotal role in the rapidly advancing domains of nonlinear analysis and nonlinear operators. Though relatively new, fixed point theory is experiencing rapid growth in both science and engineering. Historically, fixed point problems and their associated points have been essential mathematical tools across a wide range of disciplines, including topology, differential equations, economics, functional analysis, optimal control, dynamics, game theory, and various engineering fields. The development of precise computational methods for identifying fixed points has significantly enhanced the practical applications of this concept, making fixed point methods indispensable to scientists and engineers, as well as applied mathematicians.

Major areas of mathematics, such as algebraic topology, functional analysis, general topology, and set theory, offer the best environments for the application of fixed point problems. These problems are extensively employed throughout numerous fields, including theory of differential equations (Alam & Rohen, 2025; Alam et al., 2024), game theory, mathematical economics (Alam et al., 2024), approximation theory (Alam & Rohen, 2024; Alam et al., 2024), and the potential theory, to solve problems in these fields. For those passionate about systems of integral, differential, or functional equations (Alam et al., 2024), fixed point methodologies offer powerful tools for analyzing a wide range of problems in engineering and science. This approach is particularly valuable when addressing issues in control systems and elasticity, making it a crucial technique in both scientific and engineering contexts.

To solve mathematical models like variational inequalities, integral equations (Alam & Rohen, 2024, 2024?), and differential equations (Alam et al., 2023, 2024), which describe phenomena in areas such as fluid flow, neutron transport, epidemics, steady-state temperature distributions, chemical reactions, and economic theories, one of the most essential tools available are fixed point problems. These problems are also critical in assessing the complexity of selecting the appropriate center for these systems, providing deep insights into the underlying challenges. In 2021, Saleem et al. (2021), and Furqan et al. (2021) generalizs the concept of fuzzy metric space and defined fuzzy double and fuzzy triple controlled metric like spaces. Several authors proved some remarkable findings related to coincidence best proximity point along with numerous fixed point results metric, fuzzy metric and non-Archimedean fuzzy metric spaces, (for details see, Abbas et al., 2016; De la Sen et al., 2017; Saleem et al., 2019, 2020; Zhou et al., 2021).

Building on Poincare’s foundational work Poincare (1985), the metric fixed point theory, originally introduced by Banach Banach (1922), has undergone substantial development, extending to both linear and non-linear contexts within metric spaces (MS) (see for instance, Alam et al., 2023; Bakhtin, 1989; Nadler, 1969; Tiberio Bianchini, 1972; Wardowski, 2012 and so on). Multivalued mappings and applications have extended the scope of the contraction principle (Banach, 1922), with profound implications in game theory, Nash equilibria, engineering, and more. The Hausdorff metric (Kelley, 1959), measuring distances between sets, holds great significance. The study of multivalued operators, pioneered by Nadler (1969), has opened new avenues of research as seen in works by Feng and Liu (2006), Mizoguchi et al. Mizoguchi and Takahashi (1989), and others (see for instance, Altun et al., 2015; Damjanovic & Doric, 2011; Kikkawa & Suzuki, 2008; Tomar et al., 2019 and so on).

The release of a landmark study by Zadeh Zadeh (1965) marked a significant pivotal moment in the development of the modern notion of unpredictability. Fuzzy sets are sets with uncertain bounds, according to a theory developed by Zadeh. Membership in a fuzzy set is a degree issue instead of acceptance or rejection. In order to solve problems across many disciplines, this notion is being employed and is found to be more effective.

Kramosil and Michalek (1975) were the ones who first presented the idea of fuzzy metric spaces (FMS). Later, in response to Kramosil et al., George and Veeramani (1994) extended the idea of FMS and investigated its Hausdorff topology. In Alam et al. (2024), Alam et al. generalized the groundbreaking notions of fuzzy metric-like, non-Archimedean fuzzy metric-like, and all the variants of FMS by introducing the idea of fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike spaces. Intuitionistic fuzzy metric space (IFMS) was a concept that Park (2004) first articulated in 2004. He demonstrated that for each space having an intuitionistic fuzzy metric (IFM), the topology produced by the IFM and the fuzzy metric are identical. The idea of intuitionistic fuzzy normed space (IFNS) was first suggested by Saadati and Park (2006) in 2006. Later in 2014, Shojaei (2014) generalized the theory by proposing Hausdorff Intuitionistic Fuzzy Metric Space (HIFMS).

The introduction of HIFMS allows for a refined measure of distance between fuzzy sets, which is essential in handling multivalued mappings in uncertain environments. Compared to other fuzzy set models like hesitant fuzzy sets, intuitionistic fuzzy metrics provide both membership and non-membership functions, capturing a richer form of uncertainty. The Hausdorff framework extends classical fuzzy metric spaces by incorporating set-based convergence and compactness, which are critical in proving fixed point theorems for fuzzy, $α -$ fuzzy mapping ( $α -$ FM), and modeling differential inclusions. This dual capability is not easily addressed using standard fuzzy or hesitant fuzzy models.

This research paper focuses to bridge fuzzy fixed point theory and the analysis of differential inclusions within generalized metric frameworks (see Figure 1 for graphical representation). Introduction of $α -$ FM in HIFMLS and HIFMS, drawing inspiration from the framework of Kelley (1959). We explore $α -$ fuzzy fixed point ( $α -$ FFP) and fixed point conclusions, contributing to the evolving field of $α -$ FM results in fuzzy type metric spaces. Further, we study first-order nonlinear differential inclusions in HIFMS. Our primary objective is to explore the conditions under which a solution to the differential inclusion corresponds to a continuous function $w (t)$ so that $() \in F (t, w (t))$ . Under suitable fuzzy contractive conditions, $α -$ FFPs exist in complete HIFMLS, and these results guarantee solutions to nonlinear Cauchy differential inclusions. Such investigation is particularly relevant to various applied mathematical problems, spanning control theory, economic systems, and Hamiltonian systems.

Figure 1.

Graphical Representation of the Work.

2. Preliminaries

We begin by reviewing the fundamental definitions and key properties of a HIFMS.

Definition 1
Schweizer and Sklar (1960) A continuous $t -$ norm is a commutative, associative, continuous binary operation $τ : [0, 1]^{2} ⟶ [0, 1]$ so that $x τ 1 = x \forall x \in [0, 1]$ and
$w \leq x, y \leq z \Rightarrow w τ y \leq x τ z, \forall w, x, y, z \in [0, 1] .$

Some of the easiest examples of $t -$ norm are $x τ y = x y$ , $x τ y = min {x, y}$ and $x τ y = max {x + y - 1, 0}$ .
Definition 2
Schweizer and Sklar (1960) A continuous $t -$ conorm is an commutative, associative, continuous binary operation $ς : [0, 1]^{2} ⟶ [0, 1]$ so that $x ς 0 = x \forall x \in [0, 1]$ and $w \geq x, y \geq z \Rightarrow w ς y \geq x ς z, \forall w, x, y, z \in [0, 1]$ .

Some of the easiest examples of $t -$ conorm are
$x ς y = max {x, y} and x ς y = min {x + y, 1} .$

Definition 3
Alaca et al. (2006) An IFMS is a five tuple $(Y, ρ, σ, τ, ς)$ , where $Y$ is any set, $ρ, σ$ are fuzzy sets on $Y \times Y \times (0, \infty)$ and $τ, ς$ are both respectively continuous $t -$ norm and $t -$ conorm, so that for $u, v, w \in Y$ and $s, t > 0$ if (IFMS1)
$ρ (u, w, 0) = 0$ ,
(IFMS2)
$ρ (u, w, r) = 1 \Leftrightarrow u = w$ ,
(IFMS3)
$ρ (u, w, r) = ρ (w, u, r)$ ,
(IFMS4)
$ρ (u, v, r) τ ρ (v, w, s) \leq ρ (u, w, t + s)$ ,
(IFMS5)
$ρ (u, v, \cdot) : [0, \infty) ⟶ [0, 1]$ is left continuous,
(IFMS6)
$lim_{r \to \infty} ρ (u, w, r) = 1$
(IFMS7)
$σ (u, w, 0) = 1$ ,
(IFMS8)
$σ (u, w, r) = 0 \Leftrightarrow u = w$ ,
(IFMS9)
$σ (u, w, r) = σ (w, u, r)$ ,
(IFMS10)
$σ (u, v, r) ς σ (v, w, s) \geq σ (u, w, t + s)$ ,
(IFMS11)
$σ (u, v, \cdot) : [0, \infty) ⟶ [0, 1]$ is right continuous,
(IFMS12)
$lim_{r \to \infty} σ (u, w, r) = 0$
(IFMS13)
$ρ (u, w, r) + σ (u, w, r) \leq 1$ .

In this case, the pair $(ρ, σ)$ is known as IFM on $Y$ .

In the definition of IFMS, if we replace both side implication conditions (IFMS2) and (IFMS8) by right side implication only, then the IFMS will be called an intuitionistic fuzzy metric like space (IFMLS). In this regard, every IFMS is again an IFMLS.
Remark 1
If $(Y, ρ, σ, τ, ς)$ is an IFMS (respectively IFMLS), then $(Y, ρ, τ)$ is an FMS (respectively FMLS). Every FMS (respectively FMLS) $(Y, ρ, τ)$ is again an IFMS (respectively IFMLS) $(Y, ρ, σ, τ, ς)$ , if we define $x ς y = 1 - ((1 - x) τ (1 - y))$ and $σ = 1 - ρ$ .
Example 1
For a five tuple $(Y, ρ, σ, τ, ς)$ , let us define
$\begin{aligned} Y = R, x τ y = min {x, y}, x ς y = max {x, y}, \\ ρ (u, w, r) = e^{- \frac{| x - y |^{2}}{r}}, σ (u, w, r) = 1 - e^{- \frac{| x - y |^{2}}{r}} . \end{aligned}$
Then, $(Y, ρ, σ, τ, ς)$ is an IFMS, consequently an IFMLS.
Example 2
For a five tuple $(Y, ρ, σ, τ, ς)$ , let us define
$\begin{aligned} Y = R, x τ y = x y, x ς y = min {x + y, 1}, \\ ρ (u, w, r) = e^{- \frac{| max {x, y} |}{r}}, σ (u, w, r) = 1 - e^{- \frac{| max {x, y} |}{r}} . \end{aligned}$

Then, $(Y, ρ, σ, τ, ς)$ is an IFMLS, but not an IFMS.
Definition 4 (Alaca et al., 2006)

In an IFMS $(Y, ρ, σ, τ, ς)$ , a sequence ${w_{n}}$ is said to be

(i)
Cauchy if $lim_{r \to \infty} ρ (w_{m + p}, w_{m}, r) = 1$ and

$lim_{r \to \infty} σ (w_{m + p}, w_{m}, r) = 0$ , for all $r > 0, p > 0$ .
(ii)
convergent to $w \in Y$ if $lim_{r \to \infty} ρ (w_{m}, w, r) = 1$ and

$lim_{r \to \infty} σ (w_{m}, w, r) = 0$ , for all $r > 0$ .

Again, if all Cauchy sequences in an IFMS are convergent then, the IFMS is called complete and if all sequences in an IFMS have convergent subsequences then, the IFMS is called compact.

Drawing inspiration from the framework of Hausdorff metric (Kelley, 1959) and Rodriguez-Lopez and Romaguera (2004), in the set $P (Y)$ of all subsets of $Y$ , for all $A, B \in P (Y)$ and $r > 0$ , we denote the following equality and inequality notations:

In case of MS $(Y, ω)$ ,
$\begin{aligned} H_{ω} (A, B) = sup {sup_{u \in A} δ_{ω} (u, B), sup_{w \in B} δ_{ω} (A, w)}, \\ where δ_{ω} (u, B) = min {ω (u, w) : w \in B} . \end{aligned}$
In case of IFMS $(Y, ρ, σ, τ, ς)$ ,
$\begin{aligned} H_{ρ} (A, B, r) \geq inf {inf_{u \in A} δ_{ρ} (u, B, r), inf_{w \in B} δ_{ρ} (A, w, r)}, \\ H_{σ} (A, B, r) \leq sup {sup_{u \in A} δ_{σ} (u, B, r), sup_{w \in B} δ_{σ} (A, w, r)}, \\ where δ_{ρ} (u, B, r) = sup {ρ (u, w, r) : w \in B}, δ_{σ} (u, B, r) = inf {σ (u, w, r) : w \in B} . \end{aligned}$
With the above inequality notations, we call an IFMS (respectively IFMLS) $(Y, ρ, σ, τ, ς)$ as a HIFMS (respectively HIFMLS). The Hausdorff distance measures in fuzzy metric settings provide a natural structure for extending fixed point concepts to set-valued and fuzzy mappings, facilitating applications in real-world modeling where outputs are not single-valued.
Lemma 1
In a HIFMS $(Y, ρ, σ, τ, ς)$ , let $A, B \in C B (Y)$ , the collection of all non-empty closed and bounded subsets of $Y$ . Then, for any $u \in A$ there is a $w \in B$ , so that
$δ_{ρ} (u, B, r) = ρ (u, w, r), δ_{σ} (u, B, r) = σ (u, w, r) .$

Zadeh (1965), we now recall some definitions related to the term fuzzy.

Allow $Y$ to be any non-empty set. Any function having the domain $Y$ and values between $0$ and $1$ is referred to as a fuzzy set in $Y$ . The function-value $T (w)$ is frequently referred to as the membership level of $w$ in $T$ if $T$ is a fuzzy set and $w$ is a member of $Y$ . Unless otherwise specified, $F (Y)$ refers to the gathering of all fuzzy sets contained in $Y$ . Assume $X$ and $Y$ are two random non-empty sets. A fuzzy mapping is one that transforms $T$ from the set $X$ into $F (Y)$ . We denote
$[T]_{α} = {w \in Y : T (w) \geq α},$
known as $α -$ level set of $T$ and $[T]_{0} = \bar{{w \in Y : T (w) > 0}}$ , where overline indicates closure of a set, when $Y$ is induced with a topology. For a self fuzzy mapping $T$ from $Y$ into $F (Y)$ , a point $w \in Y$ is refereed to $α -$ FFP for some $α \in [0, 1]$ , if $w \in [T w]_{α}$ .
3. $α$ -FFP and Fixed Point Results

In this part, we come up with the $α -$ FFP existence theorem in the HIFMLS for $α -$ FM and carry our findings to find $α -$ fuzzy fixed into HIFMS, HFMS, MS and to find fixed point into MS. The results will generalize Banach (1922), Nadler (1969), Kramosil and Michalek (1975), George and Veeramani (1994), Shojaei (2014), and many others in the literature.

We are now prepared to demonstrate our primary finding.

Theorem 2
Let $(Y, ρ, σ, τ, ς)$ be a complete HIFMLS and a $α -$ FM $T : Y ⟶ F (Y)$ is so that
$H_{ρ} ([T u]_{α (u)}, [T w]_{α (w)}, ξ r) \geq ρ (u, w, r) and H_{σ} ([T u]_{α (u)}, [T w]_{α (w)}, ξ r) \leq σ (u, w, r),$
(1)
$\forall u, w \in Y, r > 0, 0 < ξ < 1$ . Where $α : Y ⟶ (0, 1]$ so that $[T w]_{α (w)}$ is non-empty and compact in $Y$ for each $w$ . Then, $T$ has an $α -$ FFP $w \in Y$ .
Proof.
Let $w_{0} \in Y$ be arbitrary, so that $w_{0} \notin [T w_{0}]_{α (w_{0})}$ , otherwise $w_{0}$ is the required one point and the proof is done. Now, for any $w_{1} \in [T w_{0}]_{α (w_{0})}$ , by Lemma 1, we find $w_{2} \in [T w_{1}]_{α (w_{1})}$ , so that
$δ_{ρ} (w_{1}, [T w_{1}]_{α (w_{1})}, ξ r) = ρ (w_{1}, w_{2}, ξ r), δ_{σ} (w_{1}, [T w_{1}]_{α (w_{1})}, ξ r) = σ (w_{1}, w_{2}, ξ r) .$
Which further implies, using inequality (1), to the inequalities
$\begin{aligned} ρ (w_{1}, w_{2}, ξ r) & = δ_{ρ} (w_{1}, [T w_{1}]_{α (w_{1})}, ξ r) \\ \geq H_{ρ} (T w_{0}]_{α (w_{0})}, [T w_{1}]_{α (w_{1})}, ξ r) \\ \geq ρ (w_{0}, w_{1}, r) \\ \Rightarrow ρ (w_{1}, w_{2}, r) & \geq ρ (w_{0}, w_{1}, \frac{r}{ξ}) \end{aligned}$
and
$\begin{aligned} σ (w_{1}, w_{2}, ξ r) & = δ_{σ} (w_{1}, [T w_{1}]_{α (w_{1})}, ξ r) \\ \leq H_{σ} (T w_{0}]_{α (w_{0})}, [T w_{1}]_{α (w_{1})}, ξ r) \\ \leq σ (w_{0}, w_{1}, r) \\ \Rightarrow σ (w_{1}, w_{2}, r) & \leq σ (w_{0}, w_{1}, \frac{r}{ξ}) . \end{aligned}$
Again, for any $w_{2} \in [T w_{1}]_{α (w_{1})}$ , by Lemma 1, we find $w_{3} \in [T w_{2}]_{α (w_{2})}$ , so that
$δ_{ρ} (w_{2}, [T w_{2}]_{α (w_{2})}, ξ r) = ρ (w_{2}, w_{3}, ξ r), δ_{σ} (w_{3}, [T w_{2}]_{α (w_{2})}, ξ r) = σ (w_{2}, w_{3}, ξ r) .$
Which further implies, using inequality (1), to the inequalities
$\begin{aligned} ρ (w_{2}, w_{3}, ξ r) & = δ_{ρ} (w_{2}, [T w_{2}]_{α (w_{2})}, ξ r) \\ \geq H_{ρ} (T w_{1}]_{α (w_{1})}, [T w_{2}]_{α (w_{2})}, ξ r) \\ \geq ρ (w_{1}, w_{2}, r) \\ \Rightarrow ρ (w_{2}, w_{3}, r) & \geq ρ (w_{0}, w_{1}, \frac{r}{ξ^{2}}) \end{aligned}$
and
$\begin{aligned} σ (w_{2}, w_{3}, ξ r) & = δ_{σ} (w_{2}, [T w_{2}]_{α (w_{2})}, ξ r) \\ \leq H_{σ} (T w_{1}]_{α (w_{1})}, [T w_{2}]_{α (w_{2})}, ξ r) \\ \leq σ (w_{1}, w_{2}, r) \\ \Rightarrow σ (w_{2}, w_{3}, r) & \leq σ (w_{0}, w_{1}, \frac{r}{ξ^{2}}) . \end{aligned}$
Inductively, we find a sequence ${w_{m}}$ in $Y$ with $w_{m + 1} \in [T w_{m}]_{α (w_{m})}$ , so that
$\begin{aligned} ρ (w_{m + 1}, w_{m + 2}, ξ r) & = δ_{ρ} (w_{m + 1}, [T w_{m + 1}]_{α (w_{m + 1})}, ξ r) \\ \geq H_{ρ} (T w_{m}]_{α (w_{m})}, [T w_{m + 1}]_{α (w_{m + 1})}, ξ r) \\ \geq ρ (w_{m}, w_{m + 1}, r) \\ \Rightarrow ρ (w_{m + 1}, w_{m + 2}, r) & \geq ρ (w_{0}, w_{1}, \frac{r}{ξ^{m + 1}}), m = 0, 1, 2, \dots \end{aligned}$
and
$\begin{aligned} σ (w_{m + 1}, w_{m + 2}, ξ r) & = δ_{σ} (w_{m + 1}, [T w_{m + 1}]_{α (w_{m + 1})}, ξ r) \\ \leq H_{σ} (T w_{m}]_{α (w_{m})}, [T w_{m + 1}]_{α (w_{m + 1})}, ξ r) \\ \leq σ (w_{m}, w_{m + 1}, r) \\ \Rightarrow σ (w_{m + 1}, w_{m + 2}, r) & \leq σ (w_{0}, w_{1}, \frac{r}{ξ^{m + 1}}), m = 0, 1, 2, \dots . \end{aligned}$

Now, applying (IFMS4) and (IFMS10), we get
$\begin{aligned} ρ (w_{m}, w_{m + p}, r) & \geq ρ (w_{m}, w_{m + 1}, \frac{r}{p}) τ ρ (w_{m + 1}, w_{m + 2}, \frac{r}{p}) τ \dots τ ρ (w_{m + p - 1}, w_{m + p}, \frac{r}{p}) \\ \geq ρ (w_{0}, w_{1}, \frac{r}{p ξ^{m}}) τ ρ (w_{0}, w_{1}, \frac{r}{p ξ^{m + 1}}) τ \dots τ ρ (w_{0}, w_{1}, \frac{r}{p ξ^{m + p - 1}}) \end{aligned}$
and
$\begin{aligned} σ (w_{m}, w_{m + p}, r) & \leq σ (w_{m}, w_{m + 1}, \frac{r}{p}) ς σ (w_{m + 1}, w_{m + 2}, \frac{r}{p}) ς \dots ς σ (w_{m + p - 1}, w_{m + p}, \frac{r}{p}) \\ \leq σ (w_{0}, w_{1}, \frac{r}{p ξ^{m}}) ς σ (w_{0}, w_{1}, \frac{r}{p ξ^{m + 1}}) ς \dots ς σ (w_{0}, w_{1}, \frac{r}{p ξ^{m + p - 1}}) . \end{aligned}$

Which further implies $lim_{m \to \infty} ρ (w_{m}, w_{m + p}, r) = 1$ and $lim_{m \to \infty} σ (w_{m}, w_{m + p}, r) = 0$ . Consequently, ${w_{m}}$ becomes a Cauchy sequence in $Y$ and being $(Y, ρ, σ, τ, ς)$ is a complete HIFMLS, we find $w \in Y$ , so that $lim_{m \to \infty} ρ (w_{m}, w, r) = 1$ and $lim_{m \to \infty} σ (w_{m}, w, r) = 0$ , for all $r > 0$ .

Applying inequality (1), we get
$\begin{aligned} δ_{ρ} (w_{m + 1}, [T w]_{α (w)}, ξ r) \geq H_{ρ} ([T w_{m}]_{α (w_{m})}, [T w]_{α (w)}, ξ r) \geq ρ (w_{m}, w, r) \\ and \\ δ_{σ} (w_{m + 1}, [T w]_{α (w)}, ξ r) \leq H_{σ} ([T w_{m}]_{α (w_{m})}, [T w]_{α (w)}, ξ r) \leq σ (w_{m}, w, r) \\ \Rightarrow lim_{m \to \infty} δ_{ρ} (w_{m + 1}, [T w]_{α (w)}, ξ r) = 1 and lim_{m \to \infty} δ_{σ} (w_{m + 1}, [T w]_{α (w)}, ξ r) = 0. \end{aligned}$

From the definition of $δ_{ρ}$ and $δ_{σ}$ , we find a sequence ${u_{m}}$ in $[T w]_{α (w)}$ , so that
$\Rightarrow lim_{m \to \infty} δ_{ρ} (w_{m + 1}, u_{m + 1}, r) = 1 and lim_{m \to \infty} δ_{σ} (w_{m + 1}, u_{m + 1}, r) = 0.$
Again, applying (IFMS4) and (IFMS10), we get
$ρ (u_{m}, w, r) \geq ρ (u_{m}, w_{m}, \frac{r}{2}) τ ρ (w_{m}, w, \frac{r}{2}) and σ (u_{m}, w, r) \leq σ (u_{m}, w_{m}, \frac{r}{2}) ς σ (w_{m}, w, \frac{r}{2}) .$
Which gives
$\begin{aligned} lim_{m \to \infty} ρ (u_{m}, w, r) \geq lim_{m \to \infty} ρ (u_{m}, w_{m}, \frac{r}{2}) τ lim_{m \to \infty} ρ (w_{m}, w, \frac{r}{2}) \Rightarrow lim_{m \to \infty} ρ (u_{m}, w, r) = 1 \\ and \\ lim_{m \to \infty} σ (u_{m}, w, r) \leq lim_{m \to \infty} σ (u_{m}, w_{m}, \frac{r}{2}) ς lim_{m \to \infty} σ (w_{m}, w, \frac{r}{2}) \Rightarrow lim_{m \to \infty} σ (u_{m}, w, r) = 0 \end{aligned}$

Being $[T w]_{α (w)}$ is compact and $u_{m} \in [T w]_{α (w)}$ , the last implications gives $w \in [T w]_{α (w)}$ , that is, $w$ is an $α -$ FFP of $T$ .

The proposed method guarantees existence under minimal assumptions, where many classical methods fail without strong continuity or uniqueness conditions. However, computation of $α$ -level sets and verification of Hausdorff distances can be expensive in high dimensions. Numerical experiments in future work will further evaluate trade-offs. This paper focuses on theoretical existence results. While examples will be provided to illustrate applicability, numerical validation using benchmark systems (e.g., fuzzy control of uncertain dynamic systems) will be considered in future work. Comparisons with other fuzzy fixed point techniques, including hesitant fuzzy and Zadeh-type systems, are needed to empirically assess performance and robustness.

The following corollary is a direct consequence of Theorem 2 for set-valued mappings.
Corollary 3
Let $(Y, ρ, σ, τ, ς)$ be a complete HIFMLS (or, HIFMS) and a set-valued mapping $T : Y ⟶ P (Y)$ is so that
$H_{ρ} (T u, T w, ξ r) \geq ρ (u, w, r) and H_{σ} (T u, T w, ξ r) \leq σ (u, w, r),$
(2)
$\forall u, w \in Y, r > 0, 0 < ξ < 1$ . Where $T w$ is non-empty and compact in $Y$ for each $w$ . Then, $T$ has a fixed point $w \in Y$ .
Proof.
Obvious from Theorem 2.

Now we present two corollaries that will carry our results in HIFMS and HFMS.
Corollary 4
Let $(Y, ρ, σ, τ, ς)$ be a complete HIFMS and a $α -$ FM $T : Y ⟶ F (Y)$ is so that
$H_{ρ} ([T u]_{α (u)}, [T w]_{α (w)}, ξ r) \geq ρ (u, w, r) and H_{σ} ([T u]_{α (u)}, [T w]_{α (w)}, ξ r) \leq σ (u, w, r),$
(3)
$\forall u, w \in Y, r > 0, 0 < ξ < 1$ . Where $α : Y ⟶ (0, 1]$ so that $[T w]_{α (w)}$ is non-empty and compact in $Y$ for each $w$ . Then, $T$ has an $α -$ FFP $w \in Y$ .
Proof.
The assertion that ’every IFMS is again an IFMLS’ leads to the proof at once.
Corollary 5
Let $(Y, ρ, σ, τ, ς)$ be a complete HFMS and a $α -$ FM $T : Y ⟶ F (Y)$ is so that
$H_{ρ} ([T u]_{α (u)}, [T w]_{α (w)}, ξ r) \geq ρ (u, w, r),$
(4)
$\forall u, w \in Y, r > 0, 0 < ξ < 1$ . Where $α : Y ⟶ (0, 1]$ so that $[T w]_{α (w)}$ is non-empty and compact in $Y$ for each $w$ . Then, $T$ has an $α -$ FFP $w \in Y$ .
Proof.
The proof adheres to Remark 1.

Now we state a corollary which will find $α -$ FFP utilizing Corollary 4 in metric space.
Corollary 6
Let $(Y, ω)$ be a complete MS and a $α -$ FM $T : Y ⟶ F (Y)$ is so that
$H_{ω} ([T u]_{α (u)}, [T w]_{α (w)}) \leq ξ ω (u, w),$
(5)
$\forall u, w \in Y, 0 < ξ < 1$ . Where $α : Y ⟶ (0, 1]$ so that $[T w]_{α (w)}$ is non-empty and compact in $Y$ for each $w$ . Then, $T$ has an $α -$ FFP $w \in Y$ .
Proof.
For any $r > 0$ , let us define
$\begin{aligned} ρ (u, w, r) = \frac{r}{r + ω (u, w)}, σ (u, w, r) = \frac{ω (u, w)}{r + ω (u, w)}, \forall u, w \in Y, \\ H_{ρ} (A, B, r) = \frac{r}{r + H_{ω} (A, B)}, H_{σ} (A, B, r) = \frac{H_{ω} (A, B)}{r + H_{ω} (A, B)}, \forall A, B \in P (Y) . \end{aligned}$
Also, let the binary operations $τ, ς : [0, 1] \times [0, 1] ⟶ [0, 1]$ be defined as
$x τ y = min {x, y}, x ς y = min {x, y}, \forall x, y \in [0, 1] .$
Then, the five tuple $(Y, ρ, σ, τ, ς)$ will form a complete HIFMS, as the MS $(Y, ρ)$ is complete.

Now,
$\begin{aligned} H_{ρ} ([T u]_{α (u)}, [T w]_{α (w)}, ξ r) & = \frac{ξ r}{ξ r + H_{ρ} ([T u]_{α (u)}, [T w]_{α (w)})} \\ \geq \frac{ξ r}{ξ r + ξ ω (u, w)} \\ = ρ (u, w, r) \end{aligned}$
and
$\begin{aligned} H_{σ} ([T u]_{α (u)}, [T w]_{α (w)}, ξ r) & = \frac{H_{ω} ([T u]_{α (u)}, [T w]_{α (w)})}{ξ r + H_{ω} ([T u]_{α (u)}, [T w]_{α (w)})} \\ \leq \frac{ξ ω (u, w)}{ξ r + ξ ω (u, w)} \\ = σ (u, w, r) \end{aligned}$

This shows that the fuzzy mapping $T : Y ⟶ F (Y)$ satisfies Corollary 4 and the outcome is as follows.
Remark 2
Notice that, from the procedure of the proof of Corollary 6, it is clear that every MS is again an IFMS.
Remark 3
In the Corollary 6, If we replace the binary operations $τ, ς$ given by
$x τ y = x y, x ς y = min {x + y, 1}, \forall x, y \in [0, 1] .$
Then, the result will still hold.
Remark 4
Approximation and Numerical Computation of $α -$ FFPs: While our primary focus is the existence of $α -$ FFPs, practical implementations, especially in control systems and fuzzy optimization, require computational techniques for approximating these fixed points. Inspired by the structure of iterative approximation methods such as Picard, Mann, and Ishikawa iterations, one may construct a sequence ${w_{n}}$ in the Hausdorff intuitionistic fuzzy metric space as follows:
$w_{n + 1} \in [T w_{n}]_{α (w_{n})}$
initialized with an arbitrary $w_{0} \in Y$ . This mimics the selection process in the proof of Theorem 2. Under the same contractive-type conditions used in the theorem, this sequence converges to an $α -$ FFP. For numerical approximations, fuzzy similarity measures or centroid-based crisp representatives can be employed to approximate the $α -$ level sets $[T (w)]_{α (w)}$ . Further, algorithms incorporating defuzzification and fuzzy arithmetic may aid in computing fixed points iteratively. These aspects will be explored in future computational studies.

The following theorem is an application to set-valued fixed point theory concluded using Corollary 6.
Theorem 7
Let $(Y, ω)$ be a complete MS and a compact set valued mapping $T : Y ⟶ K (Y)$ is so that
$H (T u, T w) \leq ξ ω (u, w),$
(6)
$\forall u, w \in Y, 0 < ξ < 1$ . Then, $T$ has a fixed point $w \in Y$ so that $w \in T w$ .
Proof.
Let us define $T : Y ⟶ F (Y)$ , for $α : Y ⟶ (0, 1]$ by
$(T w) (x) = {\begin{matrix} α (w), & x \in T w \\ 0, & x \notin T w . \end{matrix}$
Then, $T w = {x : (T w) (x) \geq α (w)} = [T w]_{α (w)}$ and consequently, $T : Y ⟶ F (Y)$ satisfy Corollary 6 so that there is $w \in [T w]_{α (w)} = T w$ , that is, $T$ has a fixed point.
Example 3
For a pair $(Y, ω)$ , let us define
$\begin{aligned} Y & = {1, 2, 3}, ω (1, 1) = ω (2, 2) = ω (3, 3) = 0, ω (2, 1) = ω (1, 2) = \frac{1}{3}, \\ ω (3, 2) & = ω (2, 3) = \frac{1}{4}, ω (3, 1) = ω (1, 3) = \frac{1}{5} . \end{aligned}$
Then, $(Y, ω)$ is a complete MS.

For any $r > 0$ , let us define
$\begin{aligned} ρ (u, w, r) = \frac{r}{r + ω (u, w)}, σ (u, w, r) = \frac{ω (u, w)}{r + ω (u, w)}, \forall u, w \in Y, \\ H_{ρ} (A, B, r) = \frac{r}{r + H_{ω} (A, B)}, H_{σ} (A, B, r) = \frac{H_{ω} (A, B)}{r + H_{ω} (A, B)}, \forall A, B \in P (Y) . \end{aligned}$
Also, let the binary operations $τ, ς : [0, 1] \times [0, 1] ⟶ [0, 1]$ be described as
$x τ y = min {x, y}, x ς y = min {x, y}, \forall x, y \in [0, 1] .$
Then, the five tuple $(Y, ρ, σ, τ, ς)$ will form a complete HIFMS, as the MS $(Y, ρ)$ is complete.

Also,
$\begin{aligned} ρ (1, 1, r) = ρ (2, 2, r) = ρ (3, 3, r) = 1, ρ (1, 2, r) = \frac{3 r}{3 r + 1}, ρ (2, 3, r) = \frac{4 r}{4 r + 1}, \\ ρ (1, 3, r) = \frac{5 r}{5 r + 1}, σ (1, 1, r) = σ (2, 2, r) = σ (3, 3, r) = 0, σ (1, 2, r) = \frac{1}{3 r + 1}, \\ σ (2, 3, r) = \frac{1}{4 r + 1}, σ (1, 3, r) = \frac{1}{5 r + 1} . \end{aligned}$

Let us consider the fuzzy mapping $T : Y ⟶ F (Y)$ defined as
$\begin{aligned} (T 1) (1) = (T 2) (1) = (T 3) (2) = \frac{2}{3}, \\ (T 1) (2) = (T 1) (3) = (T 2) (2) = (T 2) (3) = (T 3) (1) = (T 3) (3) = 0 . \end{aligned}$

For $α (w) = \frac{2}{3}, \forall w \in Y$ , we get $[T 1]_{\frac{2}{3}} = [T 2]_{\frac{2}{3}} = {1}, [T 3]_{\frac{2}{3}} = {2} .$

Consequently,
$\begin{aligned} H_{ρ} ([T 1]_{\frac{2}{3}}, [T 1]_{\frac{2}{3}}, ξ r) = H_{ρ} ([T 2]_{\frac{2}{3}}, [T 2]_{\frac{2}{3}}, ξ r) = H_{ρ} ([T 3]_{\frac{2}{3}}, [T 3]_{\frac{2}{3}}, ξ r) = 1, \\ H_{ρ} ([T 1]_{\frac{2}{3}}, [T 2]_{\frac{2}{3}}, ξ r) = \frac{ξ r}{ξ r + 1}, H_{ρ} ([T 1]_{\frac{2}{3}}, [T 3]_{\frac{2}{3}}, ξ r) = \frac{3 ξ r}{3 ξ r + 1}, \\ H_{ρ} ([T 1]_{\frac{2}{3}}, [T 3]_{\frac{2}{3}}, ξ r) = \frac{3 ξ r}{3 ξ r + 1}, \\ H_{σ} ([T 1]_{\frac{2}{3}}, [T 1]_{\frac{2}{3}}, ξ r) = H_{σ} ([T 2]_{\frac{2}{3}}, [T 2]_{\frac{2}{3}}, ξ r) = H_{σ} ([T 3]_{\frac{2}{3}}, [T 3]_{\frac{2}{3}}, ξ r) = 0, \\ H_{σ} ([T 1]_{\frac{2}{3}}, [T 2]_{\frac{2}{3}}, ξ r) = \frac{1}{ξ r + 1}, H_{σ} ([T 1]_{\frac{2}{3}}, [T 3]_{\frac{2}{3}}, ξ r) = \frac{1}{3 ξ r + 1}, \\ H_{σ} ([T 1]_{\frac{2}{3}}, [T 3]_{\frac{2}{3}}, ξ r) = \frac{1}{3 ξ r + 1} . \end{aligned}$
Then, for all $r > 0$ and $ξ = \frac{1}{3}$ , $T$ satisfy Theorem 2 having an $α -$ FFP $1$ .
Remark 5
Comparative Analysis with Classical Results: Theorem 2 and its corollaries generalize classical results such as Banach’s contraction (single-valued, complete metric spaces), Nadler’s fixed point theorem (multi-valued mappings), and Shojaei’s results in HIFMS.
Banach’s theorem is recovered when $T$ is single-valued, $α = 1$ , and the fuzzy metric reduces to a classical metric.

Nadler’s theorem corresponds to the case $α = 1$ with set-valued maps satisfying contraction in terms of the Hausdorff metric in classical metric spaces.

Our theorem extends Shojaei’s work by relaxing completeness to fuzzy structures and integrating $α -$ fuzzy level sets, allowing a finer control over fixed point existence in uncertain or fuzzy environments.
These generalizations broaden applicability in settings involving partial membership and non-determinism.

To illustrate practical utility, $α -$ FM can model decision-making scenarios where input parameters are imprecise. For instance, in a smart grid control system, $α -$ -level fuzzy sets can represent acceptable voltage thresholds, and $α -$ FFPs correspond to optimal safe states. While the primary contribution of this paper is theoretical, practical implementation involves approximating fuzzy mappings and computing Hausdorff distances over fuzzy sets. In high-dimensional settings, this can be computationally intensive. Future studies may consider extending the $α$ -fuzzy framework using hesitant fuzzy or intuitionistic hesitant fuzzy settings, as explored in recent work on decision-making under uncertainty (Gitinavard et al., 2015). This would potentially enhance flexibility in modeling ambiguity and confidence levels.
4. Application

A Cauchy differential inclusion generalizes the classical Cauchy problem by replacing the ordinary differential equation (ODE) with a differential inclusion. Unlike ODEs, where the rate of change is described by a unique function, differential inclusions define the rate of change through a set-valued map, allowing for multiple possible behaviors at any given point. These inclusions are particularly valuable for modeling systems with non-deterministic dynamics, uncertainty, or situations where multiple outcomes are possible. Applications span various fields, such as control theory, where they model systems influenced by a range of potential inputs or control actions; mechanics, for handling ambiguity in applied forces; and optimization and game theory, where strategies or decisions are shaped by competing factors.

Unlike classical fixed point approaches that assume precise data and deterministic systems, the proposed $α$ -fuzzy framework accommodates imprecision in both data and dynamic rules. This flexibility is especially advantageous in fields such as optimization under uncertainty or economic modeling, where multiple outcomes or strategies may be acceptable. The inclusion of fuzzy and multivalued operators leads to more robust modeling and broader applicability.

Consider a robotic path-planning system where sensor noise introduces ambiguity in control. The fuzzy differential inclusion model accommodates this uncertainty by permitting a set of possible accelerations. Our results ensure the existence of a feasible trajectory under these conditions. The Cauchy differential inclusion (Frigon, 1995) is defined by the following statement

{\begin{matrix} \frac{d w}{d t} \in F (t, w (t)), & almost everywhere t \in [t_{0}, t_{1}] \\ w (t_{0}) = w_{0} \end{matrix},

(7)

where

F

is a multivalued mapping from

[t_{0}, t_{1}] \times S

P_{c p, c v} (S)

with

w_{0} \in S

and

S

is separable normed Banach space. Differential inclusion can be considered a broader concept than a differential equation. The key distinction lies in the reality that within a differential equation, the right-hand side, denoted as

F

in the inclusion, simplifies to a function with a single point value, effectively transforming the inclusion into an equation sign.

Let $(Y, ρ, σ, τ, ς)$ be an complete HIFMS, where $Y = C ([t_{0}, t_{1}], S)$ and

$ρ, σ : Y \times Y \times (0, \infty) ⟶ [0, 1]$ be defined as

\begin{aligned} ρ (u, w, r) = e^{- \frac{max_{t \in [t_{0}, t_{1}]} | u (t) - w (t) |^{2}}{r}}, \forall u, w \in C ([t_{0}, t_{1}], S), \\ σ (u, w, r) = 1 - e^{- \frac{max_{t \in [t_{0}, t_{1}]} | u (t) - w (t) |^{2}}{r}}, \forall u, w \in C ([t_{0}, t_{1}], S) \end{aligned}

and the binary operations

τ, ς

specified as

x τ y = x y, x ς y = min {x + y, 1}, \forall x, y \in [0, 1] .

Associated to

F

, let us consider the multivalued Carathéodory mapping

T_{F} : Y ⟶ P (Y)

defined as

T_{F} (w) = {{f \in Y : \exists h \in A}_{F} (w) so that f (t) = w_{0} + \int_{t_{0}}^{t} h (s) d s, \forall t \in [t_{0}, t_{1}]},

where

A_{F} : Y ⟶ P (L^{1} ([t_{0}, t_{1}], S))

be the Niemytzki operator associated with

F

, that is

A_{F} (w) = {h \in L^{1} ([t_{0}, t_{1}], S) : h (s) \in F (s, w (s)), for almost everywhere s \in [t_{0}, t_{1}]},

where

L^{1} ([t_{0}, t_{1}], S)

is the Banach space of

S

valued integrable mappings on

[t_{0}, t_{1}]

We now have a useful lemma.

Lemma 8

Frigon (1995) Let $F$ be a multivalued mapping from $[t_{0}, t_{1}] \times S$ to $P_{c p, c v} (S)$ so that

$w \mapsto F (t, w)$ is continuous almost everywhere on $[t_{0}, t_{1}]$ ,

$t \mapsto F (t, w)$ is measurable for all $w \in C ([t_{0}, t_{1}], S)$ ,

$\forall ε > 0, \exists f_{ε} \in L^{1} ([t_{0}, t_{1}])$ so that for all $w \in \bar{B (0, ε)}$

| F (t, w) | = max {| | x | | : x \in F (t, w)} \leq f_{ε} (t) almost everywhere on [t_{0}, t_{1}] .

Then,

T_{F} (w)

is non-empty and compact.

Now we state our result.

Theorem 9

Along with the circumstances of Lemma 8, for some $0 < ξ < 1$ , suppose that, $F : [t_{0}, t_{1}] \times S ⟶ P_{c p, c v} (S)$ satisfies

H_{ρ} (F (t, w_{1} (t)), F (t, w_{2} (t)), ξ r) \geq e^{t} ρ (w_{1} (t), w_{2} (t), r), \forall t \in (t_{0}, t_{1}],

(8)

and

H_{σ} (F (t, w_{1} (t)), F (t, w_{2} (t)), ξ r) \leq e^{t} σ (w_{1} (t), w_{2} (t), r), \forall t \in (t_{0}, t_{1}] .

(9)

Then, the differential inclusion (7) has a solution.

Proof.

Let $f_{i} (t) \in T (w_{i} (t)), i = 1, 2$ be arbitrary. Then there exist ${h_{i} \in A}_{F} (w_{i})$ so that

f_{i} (t) = w_{0} + \int_{t_{0}}^{t} h_{i} (s) d s, \forall t \in [t_{0}, t_{1}] .

Now

h_{i} (t) \in F (t, w_{i} (t))

implies that

ρ (h_{1} (t), h_{2} (t), r) \geq H_{ρ} (F (t, w_{1} (t)), F (t, w_{2} (t), r)), σ (h_{1} (t), h_{2} (t), r)

(10)

and

H_{ρ} (F (t, w_{1} (t)), F (t, w_{2} (t), r)), σ (h_{1} (t), h_{2} (t), r) \leq H_{σ} (F (t, w_{1} (t)), F (t, w_{2} (t), r)) .

Again,

\begin{aligned} ρ (f_{1} (t), f_{2} (t), ξ r) & = e^{- \frac{max_{t \in [t_{0}, t_{1}]} | f_{1} (t) - f_{2} (t) |^{2}}{ξ r}} \\ = e^{- \frac{max_{t \in [t_{0}, t_{1}]} | \int_{t_{0}}^{t} h_{1} (s) d s - \int_{t_{0}}^{t} h_{2} (s) d s |^{2}}{ξ r}} \\ \geq e^{- \frac{max_{t \in [t_{0}, t_{1}]} \int_{t_{0}}^{t} | h_{1} (s) - h_{2} (s) |^{2} d s}{ξ r}} \\ \geq e^{- t} e^{- \frac{max_{t \in [t_{0}, t_{1}]} | h_{1} (t) - h_{2} (t) |^{2}}{ξ r}} \\ = e^{- t} ρ (h_{1} (t), h_{2} (t), ξ r) \\ \geq e^{- t} H_{ρ} (F (t, w_{1} (t)), F (t, w_{2} (t), ξ r)), by (10) \\ \geq ρ (w_{1} (t), w_{2} (t), r), by (8) . \end{aligned}

The other inequality corresponding to

σ

will come in a similar manner and by arbitrariness of

f_{i} (t) \in T (w_{i} (t)), i = 1, 2,

we see

T_{F}

satisfies Corollary 3 for

0 < ξ < 1

. Consequently,

T_{F}

has a fixed point in

C ([t_{0}, t_{1}], S)

, or we can confirm that the inclusion (7) has a solution.

Remark 6

Orientor equation concepts can be used to formulate each challenge of controlling first-order ordinary differential equations. This result was a crucial catalyst for the study of orientor differential equations, which led to the development of the still-relevant new concept of “differential inclusions.”

Example 4

Consider a robotic system whose velocity $\frac{d t}{d x} \in F (t, x (t))$ is influenced by uncertain inputs (e.g., terrain resistance). Let $F (t, x) = c o n v {f_{1} (t, x), f_{2} (t, x)}$ , where $f_{1}$ and $f_{2}$ model two extreme behaviors (e.g., normal and maximum friction). We define the fuzzy metric space on the function space $C ([0, 1], R^{n})$ using exponential-type distance. With suitable bounds on $f_{i}$ , conditions (8) and (9) can be verified, ensuring the existence of a feasible path.

Remark 7

Further Directions with Neutrosophic Extensions: The $α -$ FFP framework developed here may be enriched by recent advances in neutrosophic environments, which model indeterminacy explicitly. Works involving neutrosophic soft sets and novel norm structures for topological modeling suggest promising pathways to incorporate granularity and hesitation in decision spaces. These could potentially extend our results to intuitionistic or hesitant neutrosophic fuzzy metric-like settings, where $α -$ level generalizations may reflect degrees of acceptance, rejection, and indeterminacy simultaneously. This remains an intriguing avenue for future investigation.

5. Conclusion

This study has developed a general theory of $α -$ FFP in the setting of Hausdorff intuitionistic fuzzy metric spaces, culminating in existence results for nonlinear Cauchy differential inclusions. The work unifies and extends several known fixed point theorems while offering a flexible framework for modeling uncertain and multivalued systems. The findings of this investigation have wide-ranging practical applications in fields like mathematical economics, control theory, optimization, and various branches of engineering and science. We hope that these results will inspire continued exploration and innovation in this intriguing field, fostering advancements in both scientific and engineering disciplines. Future directions include the development of numerical schemes for computing FFPs, extending the theory to partial differential inclusions, and integrating this framework into real-world applications such as autonomous systems, game-theoretic models, and machine learning algorithms under uncertainty.

Footnotes

Acknowledgements

The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-87).

ORCID iD

Khairul Habib Alam

Naeem Saleem

Consent for publication

All of the materials are owned by the authors and no permissions are required.

Authors’ Contributions

Conceptualization, K.H.A., N.S.; formal analysis, Y.R., N.S. and K.H.A.; investigation, Y.R., and N.S.; supervision, Y.R.; writing original draft preparation, K.H.A., N.S., A.A.; writing review and editing, K.H.A., M.A., and Y.R. All authors have read and agreed to the published version of the manuscript.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-87).

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Availability of data and materials

The data supporting the results of this study can be obtained from the corresponding author upon request.

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