Principal Matrix Solutions and Variation of Parameters for Volterra Integro-Dynamic Equations on Time Scales

Abstract

We introduce the principal matrix solution Z(t, s) of the linear Volterra-type vector integro-dynamic equation x(Delta)(t) = A(t)x(t) + integral(t)(s) B(t, u)x(u)Delta u and prove that it is the unique matrix solution of Z(Delta t)(t, s) = A(t)Z(t, s) + integral(t)(s) B(t, u)Z(u, s)Delta u, Z(s, s) = I. We also show that the solution of x(Delta)(t) = A(t)x(t) + integral(t)(tau) B(t, u)x(u)Delta u + f(t), x(tau) = x(0) is unique and given by the variation of parameters formula x(t) = Z(t, tau)x(0) + integral(t)(tau) Z(t, sigma(s))f(s)Delta s.