A VARIATION OF PARAMETERS FORMULA FOR NONAUTONOMOUS LINEAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS OF GENERALIZED TYPE

The paper develops the fundamental matrix W(t, τ) for the linear impulsive IDEPCAG system and derives two equivalent variation-of-parameters formulas: a “long” piecewise-integral form and a compact Green-kernel form. The candidate solution reproduces the same construction on each interval Ik (where γ(t) ≡ ζk), uses E(t, τ) = Φ(t, τ)J(t, τ) with J(t, τ) = I + ∫_τ^t Φ(τ, s)B(s) ds and the identity ∂tE = A E + B, stitches across impulses via y(tk) = (I + Ck) y(tk−) + Dk, and concatenates the factors in left-ordered time to obtain the global W(t, τ), matching the paper’s definition (4.17) and solution formula (4.16) for the homogeneous case . It then derives the full inhomogeneous variation-of-parameters representation identical to (4.24), including the additive jump contributions Σr W(t, tr)Dr, and shows the equivalence to the Green-kernel form (4.28) by partitioning [τ, t], exactly as in the paper’s (4.25)–(4.27) and (4.28) . The only difference is a minor notational convention for J(t, τ) versus J(τ, t), which the model explicitly clarifies; logically, the arguments coincide. The paper’s hypotheses (H3) ensuring the invertibility of J and hence of E are invoked in the paper and implicitly assumed by the model; this is the only (minor) omission in the model’s write-up .