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Multiple periodic solutions of differential delay equations via Hamiltonian systems. II. (English) Zbl 1136.34330

This paper is the second part of the author’s study of the delay differential equation

${x}^{\text{'}}\left(t\right)=-\sum _{j=1}^{n-1}f\left(x\left(t-j\right)\right)·\phantom{\rule{2.em}{0ex}}\left(1\right)$

The problem considered here is the same as that in Part I [G. H. Fei Nonlinear Anal., Theory Methods Appl. 65, No. 1 (A), 25–39 (2006; Zbl 1136.34329)] that is to establish a lower bound for the number of geometrically different nonconstant periodic solutions. The major difference between this paper and [op. cit.] is that $n\ge 2$ is an odd integer. The author’s work shows that Kaplan and Yorke’s original idea can be used to search for periodic solutions of the delay differential equation (1). The study extends and refines the related known results on this topic.

##### MSC:
 34K13 Periodic solutions of functional differential equations 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 58E05 Abstract critical point theory