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

Published online by Cambridge University Press:  14 November 2011

Roger D. Nussbaum
Roger D. Nussbaum
Affiliation:
Department of Mathematics, Rutgers, The State University of New Jersey, U.S.A.
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Synopsis

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

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

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≦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].

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 81 , Issue 1-2 , 1978 , pp. 131 - 151
Copyright
Copyright © Royal Society of Edinburgh 1978

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