Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

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

Roger D. Nussbauma1

a1 Department of Mathematics, Rutgers, The State University of New Jersey, U.S.A.

Synopsis

We consider differential-delay equations which can be written in the form

S0308210500010490_eqnU1

The functions fi and gk are all assumed odd. The equation

S0308210500010490_eqnU2

is 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≦tq, (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].

(Received September 09 1977)

Footnotes

† Partially supported by a grant from the National Science Foundation.

‡ Work carried out while a Visiting Member at the Courant Institute of Mathematical Sciences.