The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense dx(t) = dA0(t) · x(t) + df0(t), x(t0) = c0 (t ∈ I) with a unique solution x0 is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems dx(t) = dAk(t) · x(t) + dfk(t), x(tk) = ck (k = 1, 2, . . .) to have a unique solution xk for any sufficiently large k such that xk(t) → x0(t) uniformly on I. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems. Moreover, efficient sufficient conditions for the problem of well-posedness are given.