EXISTENCE PRINCIPLES FOR HIGHER-ORDER NONLOCAL BOUNDARY-VALUE PROBLEMS AND THEIR APPLICATIONS TO SINGULAR STURM-LIOUVILLE PROBLEMS

We present existence principles for the nonlocal boundary-value problem (phi(u((p- 1))))' = g(t, u, ... , u((p-1))), alpha(k)(u) = 0, 1 <= k <= p-1, where p >= 2, phi: R -> R is an increasing and odd homeomorphism, g is a Caratheodory function that is either regular or has singularities in its space variables, and alpha(k) : Cp-1[0, T] -> R is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (- 1)(n)(phi(u((2n-1))))' = f(t, u, ..., u((2n-1))), u((2k))(0) = 0, a(k)u((2k))(T) + b(k)u((2k+ 1))(T) = 0, 0 <= k <= n - 1, is given.