In this paper we develop the monotone method in the presence of lower and upper solutions for the problem u(n)(t)=f(t,u(t));u(i)(a)u(i)(b)=λi,i=0,,n1 where f is a Carathéodory function. We obtain sufficient conditions for f to guarantee the existence and approximation of solutions between a lower solution α and an upper solution β for n3 with either αβ or αβ.For this, we study some maximum principles for the operator Luu(n)+Mu. Furthermore, we obtain a generalization of the method of mixed monotonicity considering f and u as vectorial functions.