Abstract
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∈ℝ,i=0,…,n−1
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 n≥3 with either α≤β or
α≥β.
For this, we study some maximum principles for the operator
Lu≡u(n)+Mu. Furthermore, we obtain a generalization of the method of mixed monotonicity considering f and u as vectorial functions.