Existence and multiplicity of solutions to 2mth-order ordinary differential equations

In this paper, the existence and multiplicity of solutions are obtained for the 2mth-order ordinary differential equation two-point boundary value problems (-1)(m)((2m))(u)(t) + Sigma(m)(i=1)(-1)(m-i) (ai) u((2(m-i)))(t) = f (t, u (t)) for all t is an element of [0, 1] subject to Dirichlet, Neumann, mixed and periodic boundary value conditions, respectively, where f is continuous, a(i) is an element of R for all i = 1, 2,..., m. Since these four boundary value problems have some common properties and they can be transformed into the integral equation of form u + Sigma(m)(i=1) a(i) T-i u = T-m fu, we firstly deal with this nonlinear integral equation. By using the strongly monotone operator principle and the critical point theory, we establish some conditions on f which are able to guarantee that the integral equation has a unique solution, at least one nonzero solution, and infinitely many solutions. Furthermore, we apply the abstract results on the integral equation to the above four 2mth-order two-point boundary problems and successfully resolve the existence and multiplicity of their solutions. (c) 2006 Elsevier Inc. All rights reserved.