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Journal of Applied Mathematics and Stochastic Analysis
Volume 7, Issue 1, Pages 33-47
http://dx.doi.org/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 Hindawi Publishing Corporation. 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,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.