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It is shown that the solution to a finite system of linear Volterra integral equations may be expressed in terms of the funda­ mental solution to a so-called associated system of ordinary differential equations. The only requirement on the kernel is that it be expandable in a sufficiently convergent series of continuous matrices. 1. Introduction. The only goal here will be to establish a representation formula for the solution to the finite system of Volterra integral equations of the form (I) u(x) = cp(x) + K(x, t)u(t) dt, x0 ^g a fg x < + oo, in terms of the solution to a certain infinite system of ordinary differential equations. The applicability of the resultant representation formula may be varied, but we will show in future work how it leads to stability theorems for general integral equations. It may be pointed out here that the nature of the representation theorem to be proved is such that an explicit, closed- form solution may be exhibited whenever such a solution can be found for, what will be called, the associated system of ordinary differential equations. Let K(x, t) be a continuous k x k matrix (k < +00) on x0 ^ t ^ x < + 00, and cp(x) be a continuous fc-vector on (x 0, + 00). In a previous work (1), we have shown that if U(x, s) satisfies the k x k matrix equation

A representation formula for linear Volterra integral equations