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Multiple periodic solutions to differential delay equations created by asymptotically linear Hamiltonian systems. (English) Zbl 0918.34066
The author considers the following differential delay equation $$x'(t)=\sum_{i=1}^n(-1)^{[il/n]}f(x(t-r_i)), \quad 1\le l\le n-1,$$ where $l$ and $n$ are relatively prime, $r_i$ are positive constants and $[]$ denotes the integer part. Assuming that $f\in C^1$ is an odd function with positive derivative and $f(x)/x$ converges as $x$ tends to $+\infty ,$ some existence and multiplicity results for periodic solutions are proved. As a corollary these results yield to a proof of a conjecture due to {\it J. L. Kaplan} and {\it J. A. Yorke} [J. Math. Anal. Appl. 48, 317-324 (1974; Zbl 0293.34102)].

MSC:
34K13Periodic solutions of functional differential equations
34C25Periodic solutions of ODE
34K05General theory of functional-differential equations
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References:
[1] Jibin Li and He, X. Z. Proof and generalization of Kaplan-Yorke’s conjecture on periodic solution of differential delay equations. J. math Analysis Applic. (to appear). · Zbl 0983.34061
[2] Jibin Li and He, X. Z. Periodic solutions of some differential delay equations created by high-dimensional Hamiltonian systems. J. math. Analysis Applic. (to appear). · Zbl 0946.34063
[3] Kaplan, J. L.; Yorke, J. A.: Ordinary differential equations which yield periodic solutions of differential-delay equations. J. math. Analysis applic. 48, 317-324 (1994) · Zbl 0293.34102
[4] Amann, H.; Zehnder, E.: Periodic solutions of asymptotically linear Hamiltonian systems. Manuscripta math. 32, 149-189 (1980) · Zbl 0443.70019
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[6] Mawhin, J.; Willen, M.: Critical point theory and Hamiltonian systems. (1992)
[7] Meyer, K. R.; Hall, G. R.: Introduction to Hamiltonian dynamical systems and the n-body problem. (1992) · Zbl 0743.70006
[8] Rockafellar, R. T.: Convex analysis. (1970) · Zbl 0193.18401