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) + ∫st 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) + ∫st B(t, u)Z(u, s)du, Z(s, s)=I. Furthermore, we prove that the solution of x′(t) = A(t)x(t) + ∫τt B(t, u)x(u)du + f(t), x(τ) = x0 is unique and given by the variation of parameters formula x(t) = Z(t, τ)x0 + ∫τt 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) - ∫st 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 ∂/∂t R(t, s) = -R(t, s)A(s) - ∫st 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. [ABSTRACT FROM AUTHOR]