Qualitative Results of Conformable Linear Volterra Integro-Dynamic Equations on Time Scales

Source abstract

We study a class of conformable linear Volterra integro-dynamic equations (CVIDEs) on arbitrary time scales. Using the Anderson–Georgiev conformable dynamic calculus, we first derive an equivalent Volterra integral equation through the conformable variationof- constants formula. Explicit boundedness constants are then introduced on a time scale interval, and sufficient conditions for the existence of solutions are obtained by the Schauder and Krasnoselskii fixed point theorems. Under the same verifiable kernel smallness condition, the associated Volterra operator is a contraction, which yields uniqueness. We further establish Hyers–Ulam stability and Hyers–Ulam–Rassias stability with explicit stability constants. The latter result is formulated for positive nondecreasing Rassias weights, making all conditions used in the proof explicit. An example on the nonuniform time scale [Formula: see text] verifies the conformable regressivity, exponential factors, constants, contraction threshold, and exact solution values. Numerical plots illustrate the dependence of the solution and the stability bound on the Volterra-kernel amplitude.