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

  • O. Arino (a1) and R. Benkhalti (a2)

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.

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2Arino, O. and Seguier, P.. Existence of oscillating solutions for certain differential equations with delay. Lecture Notes in Mathematics 730 (1979), 4664.
3Benkhalti, R.. Méthodes théoriques et numériques dans la détermination de phénomènes de ifurcation globale. (Doctorat de l'Université, Pau, 1986).
4Cooke, K. L.. A course on functional differential equations (Cortona: C.I.M.E., 1979).
5Chow, S. N. and Hale, J.. Methods of bifurcation theory (New York: Springer, 1982).
6Chow, S. N. and Mallet-Paret, J.. The Fuller index and global Hopf bifurcation. J. Differential Equations 29 (1978), 6685.
7Hale, J.. Theory of functional differential equations (New York: Springer, 1977).
8Hale, J.. Asymptotic behavior of the solutions of differential equations. Proceedings of the symposium on nonlinear oscillations (Kiev: IUTAM, 1961).
9Nussbaum, R. D.. A Hopf global bifurcation theorem for retarded functional differential equations. Trans. Amer. Math. Soc. 238 (1978), 139164.

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

  • O. Arino (a1) and R. Benkhalti (a2)

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