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

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.