Solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions

We prove an existence and location result for the third order functional nonlinear boundary value problem u (t) = f t,u,u (t ), u (t) , for t ∈ [a, b], 0 = L0 u, u ,u(t0) , 0 = L1 u, u ,u (a), u (a) , 0 = L2 u, u ,u (b), u (b) , with t0 ∈ [a, b] given, f : I ×C(I )×R2→R is a L1-Carathéodory function and L0, L1, L2 are continuous functions depending functionally on u and u . The arguments make use of an a priori estimate on u , lower and upper solutions method and degree theory. Applications to a multipoint problem and to a beam equation will be presented. © 2005 Elsevier Inc. All rights reserved.