Impulsive Tempered Ψ-Fractional Differential Equations with Boundary and Integral Conditions

This paper studies mixed impulsive boundary value problems involving tempered Ψ-fractional derivatives of Caputo type. By introducing exponential tempering into the fractional framework, the proposed model effectively captures systems with fading memory—an improvement over conventional power-law kernels that assume long-range dependence. The generalized tempered Ψ-operator unifies several existing fractional derivatives, offering enhanced flexibility for modeling complex dynamical phenomena. Impulsive effects and integral boundary conditions are incorporated to describe processes subject to sudden changes and historical dependence. The problem is reformulated as a Volterra integral equation, and fixed-point theory is employed to establish analytical results. Existence and uniqueness of solutions are proven using the Banach Contraction Mapping Principle, while the Leray–Schauder nonlinear alternative ensures existence in non-contractive cases. The proposed framework provides a rigorous analytical basis for modeling phenomena characterized by both fading memory and sudden perturbations, with potential applications in physics, control theory, population dynamics, and epidemiology. A numerical example is presented to illustrate the validity and applicability of the main theoretical results.