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)mu(2m)(t) + m i=1(−1)m−iai u(2(m−i))(t) = f (t,u(t)) for all t ∈ [0, 1] subject to Dirichlet, Neumann, mixed and periodic boundary value conditions, respectively, where f is continuous, ai ∈ 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 + m i=1 aiT iu = T mfu, 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. © 2006 Elsevier Inc. All rights reserved