Brouwer topological degree theory, the shooting type method, the disconjugacy theory of Hamiltonian systems and the Liapunov-Razumikhin technique of Volterra integrodifferential equations are employed to establish some solvability results for the 2n-dimensional differential system
subject to one of the following boundary conditions:
(i) x(0) = Qx(l), Qg(l x(1), qy(1)= g(0, x(0), y(0)),
(ii) Blx(O) = B2g(O, x(O), y(O)), C1x(l) = −C2g(l, x(l), y(l)),
where Q, Bi, Ci, i = 1, 2, are n x n real matrices. An application is given to the second order equation xn = h(t, x, x') subject to certain nonlinear boundary conditions.