Comparison of eigenvalues for a fourth-order four-point boundary value problem

We establish the existence of a smallest eigenvalue for the fourth- order four-point boundary value problem ( p(u 00 (t))) 00 = h (t)u(t); u 0 (0) = 0; 0u( 0) = u(1); 0(u 00 (0)) = 0; 1 p(u 00 ( 1)) = p(u 00 (1)), p > 2, 0 < 1; 0 < 1; 0 < 1; 0 < 1, using the theory of u0-positive operators with respect to a cone in a Banach space. We then obtain a comparison theorem for the smallest positive eigenvalues, 1 and 2, for the dieren tial equations ( p(u 00 (t))) 00 = 1f(t)u(t) and ( p(u 00 (t))) 00 = 2g(t)u(t) where 0 f(t) g(t); t 2 (0; 1).