On some third order nonlinear boundary value problems: Existence, location and multiplicity results

We prove an Ambrosetti–Prodi type result for the third order fully nonlinear equation u (t) + f t,u(t),u (t ), u (t) = sp(t ) with f : [0, 1]×R3→R and p : [0, 1]→R+ continuous functions, s ∈ R, under several two-point separated boundary conditions. From a Nagumo-type growth condition, an a priori estimate on u is obtained. An existence and location result will be proved, by degree theory, for s ∈ R such that there exist lower and upper solutions. The location part can be used to prove the existence of positive solutions if a non-negative lower solution is considered. The existence, nonexistence and multiplicity of solutions will be discussed as s varies. © 2007 Elsevier Inc. All rights reserved