Statistical Convergence for Grünwald–Letnikov Fractional Differences: Stability, Approximation, and Diagnostics in Fuzzy Normed Spaces

We present a unified framework for fuzzy statistical convergence of Grünwald–Letnikov (GL) fractional differences in Bag–Samanta fuzzy normed linear spaces, addressing memory effects and nonlocality inherent to fractional-order models. Theoretically, we establish the uniqueness, linearity, and invar...

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Vydané v:Axioms Ročník 14; číslo 10; s. 725
Hlavní autori: Öğünmez, Hasan, Türkmen, Muhammed Recai
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Basel MDPI AG 25.09.2025
Predmet:
Approximation
Approximation theory
Cesàro summability
Convergence
Convergence (Mathematics)
Fuzzy algorithms
Fuzzy logic
fuzzy normed spaces
Fuzzy systems
Grünwald–Letnikov fractional differences
Korovkin approximation
Linearity
Robustness (mathematics)
statistical convergence
Topology
Ulam–Hyers stability
Vector spaces
Turkey
ISSN:2075-1680, 2075-1680
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Shrnutí:We present a unified framework for fuzzy statistical convergence of Grünwald–Letnikov (GL) fractional differences in Bag–Samanta fuzzy normed linear spaces, addressing memory effects and nonlocality inherent to fractional-order models. Theoretically, we establish the uniqueness, linearity, and invariance of fuzzy statistical limits and prove a Cauchy characterization: fuzzy statistical convergence implies fuzzy statistical Cauchyness, while the converse holds in fuzzy-complete spaces (and in the completion, otherwise). We further develop an inclusion theory linking fuzzy strong Cesàro summability—including weighted means—to fuzzy statistical convergence. Via the discrete Q-operator, all statements transfer verbatim between nabla-left and delta-right GL forms, clarifying the binomial GL↔discrete Riemann–Liouville correspondence. Beyond structure, we propose density-based residual diagnostics for GL discretizations of fractional initial-value problems: when GL residuals are fuzzy statistically negligible, trajectories exhibit Ulam–Hyers-type robustness in the fuzzy topology. We also formulate a fuzzy Korovkin-type approximation principle under GL smoothing: Cesàro control on the test set {1,x,x2} propagates to arbitrary targets, yielding fuzzy statistical convergence for positive-operator sequences. Worked examples and an engineering-style case study (thermal balance with memory and bursty disturbances) illustrate how the diagnostics certify robustness of GL numerical schemes under sparse spikes and imprecise data.
Bibliografia:ObjectType-Article-1
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content type line 14
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms14100725