Volterra integro-differential equations and infinite systems of ordinary differential equations

We establish a connection between finite-dimensional systems of integro-differential equations with the Hilbert-Schmidt kernel and ordinary differential equations in l2 (countable systems of differential equations). Such a reduction allows use of results obtained earlier for the countable systems of differential equations in study of integro-differential equations. In particular, it can be employed for study of stability and solutionsʹ constructions for integro-differential equations.