## 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

### Keywords:

periodic solutions; delay equations; Hamiltonian systems

Zbl 0293.34102
Full Text:

### 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 [6] Mawhin, J.; Willen, M., Critical Point Theory and Hamiltonian Systems (1992), Springer-Verlag: Springer-Verlag New York [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 [8] Rockafellar, R. T., Convex Analysis (1970), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0229.90020
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