Calculus for Functions with Fuzzy Inputs and Outputs: Applications to Fuzzy Differential Equations

TL;DR

The paper addresses calculus for functions whose inputs and outputs are fuzzy numbers, framing mappings $f:\mathbb{R}_{\mathcal{F}(A)}\to\mathbb{R}_{\mathcal{F}(A)}$ within a complex-analytic-inspired structure. It constructs the space $\mathbb{R}_{\mathcal{F}(A)}$ via the bijection $\Phi:\mathbb{C}\to \mathbb{R}_{\mathcal{F}(A)}$ with an asymmetric $A$ and defines a complex-derived multiplication $\odot$ (and division) to develop limits, continuity, derivatives, and integrals for fuzzy mappings and curves, including a chain rule for fuzzy arguments. Key contributions include proving the isomorphism between $\mathbb{R}_{\mathcal{F}(A)}$ and $\mathbb{C}$, providing explicit differential and integral rules, and applying the framework to two-dimensional fuzzy ODEs and a fuzzy Lotka-Volterra model with phase portraits. This work offers rigorous tools for uncertainty-aware differential equations, enabling analysis of systems with both fuzzy inputs and outputs and suggesting extensions to higher-dimensional fuzzy dynamics.

Abstract

This article presents a theory of differential and integral calculus for mapping between Banach spaces formed by subsets of fuzzy numbers called A-linearly correlated fuzzy numbers, where both the domain and codomain are spaces composed of fuzzy numbers. This is one of the main contributions of this study from a theoretical point of view, as well-known approaches to fuzzy calculus in the literature typically deal with fuzzy number-valued functions defined on intervals of real numbers. Notions of differentiability and integrability based on complex functions are proposed. Moreover, we introduce the study of ordinary differential equations for which the solutions are functions from A-linearly correlated fuzzy numbers to A-linearly correlated fuzzy numbers or from real functions to A-linearly correlated fuzzy numbers. For the latter case, we present an initial study of the solution and its phase portrait for two-dimensional differential equation systems. In particular, for the former case, we examine the Lotka Volterra model and analyze its phase portrait.

Figures (15)

• Figure 1: Vector and Polar representation of a fuzzy number $B=(r+qA) \in \mathbb{R}_{\mathcal{F}(A)}$.
• Figure 2: Graphical representation of the fuzzy curve $w$ given by Eq. \ref{['soljuntas']}, which is the solution of linear fuzzy differential equation \ref{['lineqfuzz']} with $A=(-0.5;0;0.51)$, $\lambda_1=-0.5, \lambda_2=0.8$, and $x_0 = y_0=2$. Gray lines, from white to black, represent the endpoints of the $\alpha$-level of $w$ for $\alpha$ varying from 0 to 1, respectively.
• Figure 3: Graphical representation of the fuzzy curve $w$ given by Eq. \ref{['psisol']}, which is the solution of linear fuzzy differential equation \ref{['crescimento']} with $A=(-0.5;0;0.51)$, $\lambda_1=-0.5, \lambda_2=0.8$, and $x_0 = y_0=2$. Gray lines, from white to black, represent the endpoints of the $\alpha$-level of $w$ for $\alpha$ varying from 0 to 1, respectively.
• Figure 4: Graphical representation of the fuzzy curve $w$ given by Eq. \ref{['soljuntas']}, which is the solution of linear fuzzy differential equation \ref{['lineqfuzz']} with $A=(-0.5;0;0.51)$, $\lambda_1=0.5, \lambda_2=1$, and $x_0 = y_0=2$. Gray lines, from white to black, represent the endpoints of the $\alpha$-level of $w$ for $\alpha$ varying from 0 to 1, respectively.
• Figure 5: Graphical representation of the fuzzy curve $w$ given by Eq. \ref{['psisol']}, which is the solution of linear fuzzy differential equation \ref{['crescimento']} with $A=(-0.5;0;0.51)$, $\lambda_1=0.5, \lambda_2=1$, and $x_0 = y_0=2$.. Gray lines, from white to black, represent the endpoints of the $\alpha$-level of $w$ for $\alpha$ varying from 0 to 1, respectively.

Theorems & Definitions (37)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Remark 1
• Theorem 3.3
• proof
• Definition 3.1
• Example 1
• Example 2