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 equationu‴(t)+f(t,u(t),u′(t),u″(t))=sp(t)u‴(t)+f(t,u(t),u′(t),u″(t))=sp(t) with f:[0,1]×R3→Rf:[0,1]×R3→R and p:[0,1]→R+p:[0,1]→R+ continuous functions, s∈Rs∈R, under several two-point separated boundary conditions. From a Nagumo-type growth condition, an a priori   estimate on u″u″ is obtained. An existence and location result will be proved, by degree theory, for s∈Rs∈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.