Jackson-Type Approximation via Hybrid Neural Operators under Levelwise Continuity for Fuzzy-Valued Functions

Abstract This paper introduces a novel class of fuzzy neural network interpolation operators designed for the approximation andsynchronization of fuzzy-valued functions under weak hybrid continuity. To address structural irregularities and topologicalimprecision in fuzzy data, we propose operators that act level-wise on α-cuts and are compatible with Kuratowski uppersemicontinuity and generalized Hausdorff-type metrics. The proposed framework generalizes classical models, such as theCardaliaguet-Euvrard approach, by establishing a Jackson-type fuzzy approximation theorem under hybrid moduli of continuity.To the best of our knowledge, this is the first framework that rigorously integrates hybrid fuzzy interpolation into fuzzy-valued reservoir computing (FVRC) for the synchronization of chaotic systems under fuzzy uncertainty. Extensive numericalsimulations—including fuzzy versions of the Kuramoto–Sivashinsky equation, the complex Ginzburg–Landau system, and afuzzy chemo-fluidic insulin oscillator—demonstrate improved accuracy, robustness to noise, and preservation of fuzzy structuralproperties. These results provide a solid theoretical and computational foundation for next-generation fuzzy inference systemsin biomedical and uncertain environments.The proposed framework unifies classical and modern approaches by rigorously establishing a Jackson-type theorem underhybrid fuzzy metrics. The proposed operators are therefore suitable for intelligent systems and biomedical applications thatrequire structurally faithful fuzzy approximations. Notably, the operator guarantees convergence even in the presence ofstructural discontinuities that are not addressed by classical fuzzy interpolation schemes. Mathematics Subject Classification: 41A25; 46S40; 68T05; 93C41; 35Q93