On the Ulam-Hyers-Rassias stability of noninstantaneous impulsive fractional integro-differential equations

Abstract

In recent years, fractional-order derivatives and integrals have proven to be powerful tools for modeling complex systems with memory effects, making them well-suited for describing various real-world phenomena. Fractional-order models are often more effective than their integer-order counterparts in capturing the dynamics of materials such as polymers, biological systems, and other physical processes. Given these considerations, the study of qualitative properties, particularly Ulam-type stabilities, of fractional integro-differential equations has gained significant attention in mathematical research due to their wide-ranging applications in aerodynamics, biology, physics, electrodynamics, polymer science, and other disciplines. Motivated by this, we investigate various types of Ulam-type stabilities, including Ulam-Hyers stability, Ulam-Hyers-Rassias stability, and \(\sigma\)-semi-Ulam-Hyers stability, for mild solutions of noninstantaneous impulsive fractional-order integro-differential equations with local conditions in a Banach space. By introducing a novel set of suitable assumptions, we establish stability results and obtain five new findings concerning these stability concepts. Our analysis considers both finite and infinite intervals. As an application, we present three illustrative examples that support the theoretical findings. The analysis in this work relies on the Banach fixed-point theorem and the Bielecki metric, which are fundamental tools in nonlinear analysis with applications in functional equations, fractal theory, and optimization. Our findings contribute to the ongoing development of stability theories for fractional-order models and their applications in various scientific fields.

Keywords

• fractional integro–differential equations
• semigroup
• noninstantaneous impulses
• Banach fixed–point theorem
• Bielecki metric
• Ulam–Hyers–Rassias stability

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