Principal matrix solutions and variation of parameters for a Volterra integro-differential equation and its adjoint

L. C. Becker, Christian Brothers University, Memphis, TN, U.S.A.

E. J. Qualitative Theory of Diff. Equ., No. 14. (2006), pp. 1-22.

Abstract: We define the principal matrix solution $Z(t,s)$ of the linear Volterra vector integro-differential equation
\[ x'(t) = A(t)x(t) + \int_s^t B(t,u)x(u)\,du \]
in the same way that it is defined for $x' = A(t)x$ and prove that it is the unique matrix solution of
\[ \frac{\partial}{\partial{t}}Z(t,s) = A(t)Z(t,s) + \int_{s}^t B(t,u)Z(u,s)\,du, \quad Z(s,s) = I. \]
Furthermore, we prove that the solution of
\[ x'(t) = A(t)x(t) + \int_{\tau}^t B(t,u)x(u)\,du + f(t), \quad x(\tau) = x_0\]
is unique and given by the variation of parameters formula
\[ x(t) = Z(t,\tau)x_0 + \int_{\tau}^t Z(t,s)f(s)\,ds.\]
We also define the principal matrix solution $R(t,s)$ of the adjoint equation
\[ r'(s) = -r(s)A(s) - \int_s^t r(u)B(u,s)\,du \]
and prove that it is identical to Grossman and Miller's resolvent, which is the unique matrix solution of
\[ \frac{\partial}{\partial{s}}R(t,s) = -R(t,s)A(s) - \int_{s}^t R(t,u)B(u,s)\,du, \quad R(t,t) = I. \]
Finally, we prove that despite the difference in their definitions $R(t,s)$ and $Z(t,s)$ are in fact identical.

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