Solutions of certain boundary value problems are shown to exist for the n th order differential equation y ( n ) = f ( t , y , y ′ , … , y ( n − 1 ) ) , where f is continuous on a slab ( a , b ) × R n and f satisfies a Lipschitz condition on the slab. Optimal length subintervals of ( a , b ) are determined, in terms of the Lipschitz coefficients, on which there exist unique solutions.