Existence of solution for discontinuous third order boundary value problems

In this paper, we obtain existence results for the problem u″′=q(u″) f(t,u) with boundary conditions u(a)=A, u(b)=B, u″(a)=C and u(a)=u(b), u′(a)=u′(b), u″(a)=C. We assume f a Carathéodory function, q∈L∞(R,(0,∞)) such that 1/q∈Lloc∞(R,(0,∞)) and suppose the existence of lower and upper solutions. The existence of solution for the first considered conditions is obtained as a consequence of the fixed-points theorems. We obtain the solution of the second problem as a limit of solutions of the first case. For the first problem, the monotone method is developed.