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

Abstract

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 = L 0 ( u , u ′ , u ( t 0 ) ) , 0 = L 1 ( u , u ′ , u ′ ( a ) , u ″ ( a ) ) , 0 = L 2 ( u , u ′ , u ′ ( b ) , u ″ ( b ) ) , with t 0 ∈ [ a , b ] given, f : I × C ( I ) × R 2 → R is a L 1 -Carathéodory function and L 0 , L 1 , L 2 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.