Journal of Applied Mathematics and Stochastic Analysis
Volume 7 (1994), Issue 1, Pages 33-47
doi:10.1155/S1048953394000043
The method of lower and upper solutions for n
th-order periodic boundary value problems
Universidade de Santiago de Compostela, Departamento de Análise Matemática, Facultade de Matemáticas, Spain
Received 1 August 1993; Revised 1 January 1994
Copyright © 1994 Alberto Cabada. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.