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

In recent years, fractional–order derivatives and integrals have proven tobe powerful tools for modeling complex systems with memory effects, mak-ing them well–suited for describing various real–world phenomena. Fractional–order models are often more effective than their integer–order counterpartsin capturing the dynamics of materials such as polymers, biological systems,and other physical processes. Given these considerations, the study of qual-itative 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–Hyersstability, Ulam–Hyers–Rassias stability, and σ–semi–Ulam–Hyers stability,for mild solutions of noninstantaneous impulsive fractional–order integro–differential equations with local conditions in a Banach space. By introduc-ing a novel set of suitable assumptions, we establish stability results andobtain five new findings concerning these stability concepts. Our analysisconsiders both finite and infinite intervals. As an application, we presentthree illustrative examples that support the theoretical findings. The anal-ysis in this work relies on the Banach fixed–point theorem and the Bieleckimetric, which are fundamental tools in nonlinear analysis with applicationsin functional equations, fractal theory, and optimization. Our findings con-tribute to the ongoing development of stability theories for fractional–ordermodels and their applications in various scientific fields.