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)].


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


Zbl 0293.34102
Full Text: DOI


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