Boundary Value Problems
Volume 2010 (2010), Article ID 410986, 26 pages
doi:10.1155/2010/410986
Research Article

Optimal Conditions for Maximum and Antimaximum Principles of the Periodic Solution Problem

Meirong Zhang1,2

1Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
2Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China

Received 18 September 2009; Accepted 11 April 2010

Academic Editor: Pavel Drábek

Copyright © 2010 Meirong Zhang. 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

Given a periodic, integrable potential q(t), we will study conditions on q(t) so that the operator Lqx=x″+qx admits the maximum principle or the antimaximum principle with respect to the periodic boundary condition. By exploiting Green functions, eigenvalues, rotation numbers, and their estimates, we will give several optimal conditions.