Periodic solutions of special differential equations: an example in non-linear functional analysis

Abstract

SynopsisWe consider differential-delay equations which can be written in the formThe functions fi and gk are all assumed odd. The equationis a special case of such equations with q = N + 1 (assuming f is an odd function). We obtain an essentially best possible theorem which ensures the existence of a non-constant periodic solution x(t) with the properties (1) x(t)≧0 for 0≦t≦q, (2) x(–t) = –x(t) for all t and (3) x(t + q) = –x(t) for all t. We also derive uniqueness and constructibility results for such special periodic solutions. Our theorems answer a conjecture raised in [8].