# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $$x'(t)=-\sum_{j=1}^{n-1}f(x(t-j)).\tag1$$ The problem considered here is the same as that in Part I [{\it 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 $n\geq 2$ 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
Full Text:
##### References:
 [1] Abbondandolo, A.: Morse theory for asymptotically Hamiltonian systems. Nonlinear analysis 39, No. 8, 997-1049 (2000) · Zbl 0942.37004 [2] Amann, H.; Zehnder, E.: Periodic solutions of an asymptotically linear Hamiltonian systems. Manuscripta math. 32, 149-189 (1980) · Zbl 0443.70019 [3] Benci, V.: On critical point theory for indefinite functionals in the presence of symmetries. Trans. amer. Math. soc. 274, 533-572 (1982) · Zbl 0504.58014 [4] K.C. Chang, Infinite dimensional Morse theory and multiple solution problems, Progress in Nonlinear Differential Equations and their Applications, vol. 6, 1993. · Zbl 0779.58005 [5] Conley, C.; Zehnder, E.: Morse type index theory for flows and periodic solutions for Hamiltonian equations. Comm. pure appl. Math. 37, 207-253 (1984) · Zbl 0559.58019 [6] O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, H.O. Walther, Delay equations, functional-, complex-, and nonlinear analysis, Applied Mathematical Science, vol. 110, Springer, Berlin, 1995. · Zbl 0826.34002 [7] Fannio, L. O.: Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete contin. Dynam. syst. 3, 251-264 (1997) · Zbl 0989.37060 [8] Fei, G.: Maslov-type index and periodic solution of asymptotically linear Hamiltonian systems which are resonant at infinity. J. diff. Eq. 121, 121-133 (1995) · Zbl 0831.34046 [9] G. Fei, Multiple periodic solutions of differential delay equations via Hamiltonian systems (I), preprint (2004). [10] J.K. Hale, S.M. Verduyn Lunel, Introduction to functional differential equations, Applied Mathematical Science, vol. 99, Springer, Berlin, 1993. · Zbl 0787.34002 [11] Kaplan, J.; Yorke, J.: Ordinary differential equations which yield periodic solutions of differential delay equations. J. math. Anal. appl. 48, 317-324 (1974) · Zbl 0293.34102 [12] Li, J.; He, X.: Multiple periodic solutions of differential delay equations created by asymptotically linear Hamiltonian systems. Nonlinear anal. T.M.A. 31, 45-54 (1998) · Zbl 0918.34066 [13] Li, J.; He, X.: Proof and generalization of kaplan-Yorke’s conjecture under the condition $f^{\prime}(0)$>0 on periodic solution of differential delay equations. Sci. China, ser. A 42, 957-964 (1999) · Zbl 0983.34061 [14] Li, J.; He, X.; Liu, Z.: Hamiltonian symmetric groups and multiple periodic solutions of differential delay equations. Nonlinear anal. T.M.A. 35, 457-474 (1999) · Zbl 0920.34061 [15] Li, S.; Liu, J. Q.: Morse theory and asymptotically linear Hamiltonian systems. J. diff. Eq. 78, 53-73 (1989) · Zbl 0672.34037 [16] Li, J.; Liu, Z.; He, X.: Periodic solutions of some differential delay created by Hamiltonian systems. Bull. austral. Math. soc. 60, 377-390 (1999) · Zbl 0946.34063 [17] Y. Long, E. Zehnder, Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, in: S. Albeverio, et al. (Eds.), Stochatic Processes, Physics and Geometry, Proceedings of Conference in Asconal/Locarno, Switzerland, World Scientific, Singapore, 1990, pp. 528 -- 563. [18] J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Science, vol. 74, Springer, Berlin, 1989. · Zbl 0676.58017 [19] Nussbaum, R.: Periodic solutions of special differential delay eqations: an example in non-linear functional analysis. Proc. R. Soc. Edinburgh 81A, 131-151 (1978) · Zbl 0402.34061 [20] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, 1986. · Zbl 0609.58002 [21] Szulkin, A.: Cohomology and Morse theory for strongly indefinite functionals. Math. Z. 209, 375-418 (1992) · Zbl 0735.58012 [22] Szulkin, A.; Zou, W.: Infinite dimensional cohomologygroups and periodic solutions of asymptotically linear Hamiltonian systems. J. diff. Eq. 174, 369-391 (2001) · Zbl 0997.37040