Optimal conditions for the maximum principle for second-order periodic problems

Abstract

We provide alternate necessary and/or sufficient conditions on the  sign-changing coefficient p(t) for the maximum principle for the  second-order periodic problem \(u''=p(t) u + q(t) \) to hold, i.e., for nonnegative \(q\) to yield a nonpositive periodic solution \(u\). See also https://ejde.math.txstate.edu/special/02/h2/abstr.html