On Ulam–Hyers–Mittag-Leffler Stability of Fractional Integral Equations Containing Multiple Variable Delays

• Published in: Mathematics (Basel) Vol. 13; no. 4; p. 606
• Main Authors: Tunç, Osman, Tunç, Cemil
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.02.2025
• Subjects: Analysis; Banach’s fixed-point theorem; Chebyshev and Bielecki norms; Chebyshev approximation; Complex media; Differential equations; Electrodynamics; Fixed points (mathematics); Fractional calculus; fractional Volterra integral equation; Integers; Integral equations; Mathematical analysis; Mathematical models; Mathematical optimization; Nonlinear analysis; Norms; Ordinary differential equations; Rheological properties; Rheology; Stability; U-H-M-L stability; U-H-M-L-R stability; Volterra integral equations; New York; United States
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math13040606

In recent decades, many researchers have pointed out that derivatives and integrals of the non-integer order are well suited for describing various real-world materials, for example, polymers. It has also been shown that fractional-order mathematical models are more effective than integer-order mathematical models. Thereby, given these considerations, the investigation of qualitative properties, in particular, Ulam-type stabilities of fractional differential equations, fractional integral equations, etc., has now become a highly attractive subject for mathematicians, as this represents an important field of study due to their extensive applications in various branches of aerodynamics, biology, chemistry, the electrodynamics of complex media, polymer science, physics, rheology, and so on. Meanwhile, the qualitative concepts called Ulam–Hyers–Mittag-Leffler (U-H-M-L) stability and Ulam–Hyers–Mittag-Leffler–Rassias (U-H-M-L-R) stability are well-suited for describing the characteristics of fractional Ulam-type stabilities. The Banach contraction principle is a fundamental tool in nonlinear analysis, with numerous applications in operational equations, fractal theory, optimization theory, and various other fields. In this study, we consider a nonlinear fractional Volterra integral equation (FrVIE). The nonlinear terms in the FrVIE contain multiple variable delays. We prove the U-H-M-L stability and U-H-M-L-R stability of the FrVIE on a finite interval. Throughout this article, new sufficient conditions are obtained via six new results with regard to the U-H-M-L stability or the U-H-M-L-R stability of the FrVIE. The proofs depend on Banach’s fixed-point theorem, as well as the Chebyshev and Bielecki norms. In the particular case of the FrVIE, an example is delivered to illustrate U-H-M-L stability.