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Periodic solutions for: ẋ(t) = λf(x(t),x(t – 1))

Published online by Cambridge University Press:  14 November 2011

O. Arino  and
R. Benkhalti
O. Arino
Affiliation:
Department of Mathematics, University of Pau, 64000 Pau, France; and Department of Mathematics, College of Liberal Arts, University, MS 38677, U.S.A
R. Benkhalti
Affiliation:
Department of Mathematics & Computer Science, Pacific Lutheran University, Tacoma, WA 98447, U.S.A.
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Synopsis

We present a new result on the existence of periodic solutions for the equation:

for all positive parameters λ sufficiently large. Our fundamental assumption is the following monotonicity property: if ø ≧ ψ (ø and ψ are data) then x(ø) ≧ x(ψ). The proof consists in applying the global Hopf bifurcation theorem. The main steps are: (i) a classical estimation of the periods; (ii) an a priori estimate for the solutions along a connected branch; (iii) a transformation acting on periodic solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 3-4 , 1988 , pp. 245 - 260
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

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