The Method of Lower and Upper Solutions for Second, Third, Fourth, and Higher Order Boundary Value Problems

In this paper we develop the monotone method in the presence of lower and upper solutions for the problem u(n)(t)=ƒ(t, u(t));u(i)(a)−u(i)(b)=λi∈R; i=0, ..., n−1. Where ƒ is a Carathéodory function. We obtain necessary and sufficient conditions in ƒ to guarantee the existence of solutions between a lower solution α and an upper solution β for n=2 (if α≥β), n=3 (either α≤β or α≥β) and n=4 (if α≤β). Furthermore, we obtain sufficient conditions in ƒ for n=2k≥6 when α≤β.