The Method of Lower and Upper Solutions for Third-Order Periodic Boundary Value Problems

In this paper we develop the monotone method in the presence of lower and upper solutions for the problem Lnu(t) = ƒ(t, u(t)); u(i)(0) − u(i)(2π) = μi ∈ R; i = 0,..., n − 1, with Ln an nth-order linear operator and ƒ a Carathéodory function, We obtain sufficient conditions to guarantee the validity of the monotone method for this problem. For this, we study the sign of the Green function of the operator T−1n, with Tnu = Lnu + Mu, M > 0. Furthermore existence results are obtained. We show that the results obtained are optimal. Given a lower solution α and an upper solution β, we obtain the existence of solution between α and β when either α ≤ β or α ≥ β. This general study is applied to the third-order problem u‴(t) + Pu″(t) + Qu′(t) = ƒ(t,u(t)); u(i)(0) − u(i)(2π) = μi ∈ R; i = 0,1,2, with P, Q ∈ R. For this problem we obtain the best value on the constant M to guarantee that the Green function of the operator T−13 is positive.