Principal Matrix Solutions and Variation of Parameters for a Volterra Integro-differential Equation and Its Adjoint

We define the principal matrix solution Z(t, s) of the linear Volterra vector integro-differential equation x (t) = A(t)x(t) + t s B(t, u)x(u) du in the same way that it is defined for x = A(t)x and prove that it is the unique matrix solution of ∂ ∂t Z(t, s) = A(t)Z(t, s) + t s B(t, u)Z(u, s) du, Z(s, s) = I. Furthermore, we prove that the solution of x (t) = A(t)x(t) + t τ B(t, u)x(u) du + f (t), x(τ) = x 0 is unique and given by the variation of parameters formula x(t) = Z(t, τ)x 0 + t τ Z(t, s)f (s) ds. We also define the principal matrix solution R(t, s) of the adjoint equation r (s) = −r(s)A(s) − t s r(u)B(u, s) du and prove that it is identical to Grossman and Miller's resolvent, which is the unique matrix solution of ∂ ∂s R(t, s) = −R(t, s)A(s) − t s R(t, u)B(u, s) du, R(t, t) = I. Finally, we prove that despite the difference in their definitions R(t, s) and Z(t, s) are in fact identical.