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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
\[ x'(t)=\sum_{i=1}^n(-1)^{[il/n]}f(x(t-r_i)), \quad 1\leq l\leq n-1, \] 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 J. L. Kaplan and J. A. Yorke [J. Math. Anal. Appl. 48, 317-324 (1974; Zbl 0293.34102)].

MSC:

34K13 Periodic solutions to functional-differential equations
34C25 Periodic solutions to ordinary differential equations
34K05 General theory of functional-differential equations

Citations:

Zbl 0293.34102
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References:

[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
[5] Ekeland, I., An index theory for periodic solutions of convex Hamiltonian systems, (Proceedings of Symposia in Pure Mathematics, 45 (1986)), 395-423, Part 1 · Zbl 0596.34023
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[7] Meyer, K. R.; Hall, G. R., Introduction to Hamiltonian Dynamical Systems and the n-body Problem (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0743.70006
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