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Wright's equation has no solutions of period four

Published online by Cambridge University Press:  14 November 2011

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

Let N:ℝ→ℝ be a locally Lipschitzian map such that (y + l)N(y)>0 for all y ≠ –1 and such that N(y)=1 + y for – 1 ≦ y ≦ 3. For any positive number α the equation y'(t) αy(t–1)N(y(t)) has, aside from the constantsolutions y(t) ≡ –1, and y(t) ≡–1 solution y(t) such that y(t + 4) = y(t) for all real t If N(y) = 1 + y for all y, one obtains Wright's equation, which isknown to have periodic solutions of minimal period p (depending on α) arbitrarily close to 4. Some results concerning nonexistence of periodic solutions of period 4 of other differential-delay equations are also proved. In all cases the method of proof consists in analysing an associated fourth-order system of ordinary differential equationsand showing that this system has no nonconstant periodic solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 3-4 , 1989 , pp. 281 - 288
Copyright
Copyright © Royal Society of Edinburgh 1989

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