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

Abstract

In this paper, the existence and multiplicity of solutions are obtained for the 2 mth-order ordinary differential equation two-point boundary value problems ( − 1 ) m u ( 2 m ) ( t ) + ∑ i = 1 m ( − 1 ) m − i a i 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, a i ∈ 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 + ∑ i = 1 m a i T i u = T m f u , 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 2 mth-order two-point boundary problems and successfully resolve the existence and multiplicity of their solutions.