A general framework for level continuous fuzzy-valued functions

1. Problem Statement

The paper addresses the mathematical challenge of analyzing fuzzy-valued continuous functions under the level-convergence topology ($\tau_\ell$), which is weaker than the standard supremum metric topology ($d_\infty$).

• Context: Fuzzy analysis relies on fuzzy numbers ($E^1$). While the space of fuzzy numbers is well-studied under the supremum metric $d_\infty$ (making it a complete metric space), the level-convergence topology is distinct and weaker.
• The Gap: Previous research successfully embedded fuzzy-valued functions into Banach spaces (specifically products of $C([0,1])$) when dealing with continuous fuzzy numbers ($E^1_c$) or functions continuous under $d_\infty$. However, these embeddings fail or require modification when dealing with general fuzzy numbers (which may not be continuous) under the level-convergence topology.
• Specific Issue: Functions $f: K \to E^1$ that are continuous under $\tau_\ell$ do not necessarily yield jointly continuous real-valued representations. Specifically, the cross-section functions may lack continuity in the first coordinate (the $\lambda$-level), necessitating a new functional framework.

2. Methodology

The authors employ a parametric representation approach based on the Goetschel-Voxman characterization of fuzzy numbers.

1. Parametric Representation:
   Every fuzzy number $u \in E^1$ is characterized by its $\lambda$-level sets $[u]_\lambda = [u^-(\lambda), u^+(\lambda)]$. A fuzzy-valued function $f: K \to E^1$ is decomposed into two real-valued functions of two variables:
   $$f_1(\lambda, t) = [f(t)]^-(\lambda) \quad \text{and} \quad f_2(\lambda, t) = [f(t)]^+(\lambda)$$
   where $\lambda \in [0, 1]$ represents the level and $t \in K$ (a compact space) represents the domain.

2. Analysis of Continuity:
   The authors analyze the continuity properties of $f_1$ and $f_2$:

   • For $f \in C(K, E^1_\infty)$, the convergence is uniform in $\lambda$, leading to standard continuity properties.
   • For $f \in C(K, E^1_\ell)$, the convergence is pointwise in $\lambda$. This results in functions that are separately continuous in $t$ but only left-continuous (or right-continuous at 0) in $\lambda$.

3. Construction of a New Function Space:
   To accommodate the general case, the authors define a new space of functions, denoted as $C([0, 1] \times K)$, characterized by:

   • Joint Left-Continuity: Continuous in the product topology for $\lambda \in (0, 1]$.
   • Joint Right-Continuity: Continuous at $\lambda = 0$.
   • Monotonicity: For a fixed $t$, the function is monotone in $\lambda$ (non-decreasing for $f_1$, non-increasing for $f_2$).

4. Embedding Strategy:
   The paper constructs an isometric and isomorphic embedding of the space of level-continuous fuzzy-valued functions $C(K, E^1_\ell)$ into the product space $C([0, 1] \times K) \times C([0, 1] \times K)$.

3. Key Contributions

• Definition of $C([0, 1] \times K)$: The paper introduces a novel space of real-valued functions on a product domain that satisfies specific mixed continuity conditions (joint left/right continuity) and monotonicity.
  • Significance: This space is neither a vector space nor a cone because the sum of two functions with different monotonicity directions (one increasing, one decreasing) may not be monotone, violating the closure property required for vector spaces.
• Isometric Embedding Theorem (Theorem 4.7):
  The authors prove that $(C(K, E^1_\ell), D)$ can be embedded isomorphically and isometrically onto a closed convex cone within the product space $(C([0, 1] \times K) \times C([0, 1] \times K), D_\infty)$.
  • The metric $D$ is defined as $D(f, g) = \sup_{t \in K} d_\infty(f(t), g(t))$.
• Boundedness Proof (Theorem 4.4):
  It is proven that functions in $C([0, 1] \times K)$ are bounded, provided $K$ is compact. This relies crucially on the monotonicity condition; without it, such functions could be unbounded (demonstrated via Example 4.5).
• Differentiation of Topologies:
  The paper rigorously distinguishes between $C(K, E^1_\infty)$ and $C(K, E^1_\ell)$ by analyzing the behavior of right limits.
  • Functions in $C(K, E^1_\infty)$ satisfy a "jointly right limit-continuous" property.
  • Functions in $C(K, E^1_\ell)$ generally fail this property (demonstrated in Example 5.8), highlighting a fundamental structural difference between the two spaces.

4. Key Results

1. Representation: Every level-continuous fuzzy-valued function corresponds uniquely to a pair of functions $(f_1, f_2)$ in the newly defined space $C([0, 1] \times K)$.
2. Topological Distinction: While $C(K, E^1_c, \infty) = C(K, E^1_c, \ell)$ (they coincide for continuous fuzzy numbers), the spaces diverge for general fuzzy numbers. The key differentiator is the uniform convergence of the cross-sections in the first coordinate.
3. Limit Behavior:
   • Lemma 5.2 & 5.3: Establish that for functions in $C([0, 1] \times K)$, right limits exist and behave continuously under uniform convergence.
   • Counter-examples: The paper provides explicit examples (e.g., Example 3.1 and 5.8) showing that without uniform convergence (i.e., in the $\tau_\ell$ case), the interchange of limits (specifically regarding right limits) fails, breaking the joint right-limit continuity.

5. Significance and Impact

• Theoretical Advancement: This work provides the first general framework for handling level-continuous fuzzy-valued functions without restricting them to continuous fuzzy numbers. It bridges a gap in fuzzy analysis where previous methods relied heavily on the stronger $d_\infty$ metric.
• New Mathematical Objects: The introduction of the space $C([0, 1] \times K)$ expands the toolkit of classical analysis. It offers a setting for studying functions that are monotone and possess specific one-sided continuity properties, which appear naturally in fuzzy contexts but were previously unformalized.
• Applications: By embedding these functions into a product of real-valued function spaces, the authors enable the application of classical functional analysis tools (such as integration, differentiation, and optimization techniques adapted for cones) to fuzzy differential equations, fuzzy optimization, and decision-making problems involving uncertainty that is modeled via level convergence.
• Structural Insight: The paper clarifies that the algebraic structure of fuzzy-valued functions is more complex than a simple vector space, being a cone within a specific function space, and highlights the critical role of monotonicity in ensuring boundedness and continuity properties.

In summary, the paper successfully constructs a robust mathematical infrastructure to analyze fuzzy-valued functions under the weaker, more general level-convergence topology, distinguishing it clearly from the supremum metric case through the properties of right limits and monotonicity.