The method of lower and upper solutions for n
th-order periodic boundary value problems

Cabada, Alberto

doi:https://doi.org/10.1155/S1048953394000043

International Journal of Stochastic Analysis

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Volume 7 |Article ID 737583 | https://doi.org/10.1155/S1048953394000043

Alberto Cabada, "The method of lower and upper solutions for n th-order periodic boundary value problems", International Journal of Stochastic Analysis, vol. 7, Article ID 737583, 15 pages, 1994. https://doi.org/10.1155/S1048953394000043

The method of lower and upper solutions for n th-order periodic boundary value problems

Alberto Cabada

¹Universidade de Santiago de Compostela, Departamento de Análise Matemática, Facultade de Matemáticas, Spain

Received01 Aug 1993

Revised01 Jan 1994

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.

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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.

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