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Transactions of the American Mathematical Society

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A Hopf global bifurcation theorem for retarded functional differential equations


Author: Roger D. Nussbaum
Journal: Trans. Amer. Math. Soc. 238 (1978), 139-164
MSC: Primary 34K15; Secondary 47H15, 58F14
DOI: https://doi.org/10.1090/S0002-9947-1978-0482913-0
MathSciNet review: 482913
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Abstract: We prove a result concerning the global nature of the set of periodic solutions of certain retarded functional differential equations. Our main theorem is an analogue, for retarded F.D.E.'s, of a result by J. Alexander and J. Yorke for ordinary differential equations.


References [Enhancements On Off] (What's this?)

  • [1] J. Alexander and J. Yorke, Global bifurcation of periodic orbits (to appear). MR 0474406 (57:14046)
  • [2] K. Borsuk, Theory of retracts, Polish Sci. Publ., Warsaw, 1967. MR 0216473 (35:7306)
  • [3] Shui-Nee Chow and J. Mallet-Paret, Fuller's index and global Hopf's bifurcation (to appear). MR 0492560 (58:11665)
  • [4] K. Cooke and J. Yorke, Equations modelling population growth and gonorrhea epidemiology, Math. Biosci. 16 (1973), 75-101. MR 0312923 (47:1478)
  • [5] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367. MR 0044116 (13:373c)
  • [6] F. B. Fuller, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967), 133-148. MR 0209600 (35:497)
  • [7] R. B. Grafton, A periodicity theorem for autonomous functional differential equations, J. Differential Equations 6 (1969), 87-109. MR 0243176 (39:4500)
  • [8] -, Periodic solutions of certain Lienard equations with delay, J. Differential Equations 11 (1972), 519-527. MR 0293207 (45:2286)
  • [9] A. Granas, Theory of compact vector fields, Rozprawy Mat. 30 (1962), 93 pp. MR 0149253 (26:6743)
  • [10] J. Hale, Functional differential equations, Springer-Verlag, New York, 1971. MR 0466837 (57:6711)
  • [11] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, rev. ed., Amer. Math. Soc., Providence, R.I., 1957. MR 0089373 (19:664d)
  • [12] J. Ize, Bifurcation theory for Fredholm operators, Mem. Amer. Math. Soc., no. 174, 1976. MR 0425696 (54:13649)
  • [13] -, Global bifurcation of periodic orbits, Communicaciones Tecnicas of C.I.M.A.S., Vol. 5, Series B., No. 8 (1974). (Spanish)
  • [14] G. S. Jones, The existence of periodic solutions of $ f'(x) = - \alpha f(x - 1)[1 + f(x)]$, J. Math. Anal. Appl. 4 (1962), 440-469. MR 0141837 (25:5234)
  • [15] -, Periodic motions in Banach space and applications to functional differential equations, Contrib. Differential Equations 3 (1964), 75-106. MR 0163039 (29:342)
  • [16] J. Kaplan and J. Yorke, Ordinary differential equations which yield periodic solutions of differential-delay equations, J. Math. Anal. Appl. 48 (1974), 317-324. MR 0364815 (51:1069)
  • [17] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
  • [18] J. Mallet-Paret, Generic periodic solutions of functional differential equations (to appear). MR 0442995 (56:1370)
  • [19] R. May, Stability and complexity in model ecosystems, Princeton Univ. Press, Princeton, N.J., 1973.
  • [20] L. Nirenberg, Topics in nonlinear functional analysis, New York Univ. Lecture Notes, 1973-1974. MR 0488102 (58:7672)
  • [21] R. D. Nussbaum, A global bifurcation theorem with applications to functional differential equations, J. Functional Analysis 19 (1975), 319-338. MR 0385656 (52:6516)
  • [22] -, Periodic solutions of some nonlinear autonomous functional differential equations. II, J. Differential Equations 14 (1973), 360-394. MR 0372370 (51:8586)
  • [23] -, Global bifurcation of periodic solutions of some autonomous functional differential equations, J. Math. Anal. Appl. 55 (1976), 699-725. MR 0430473 (55:3478)
  • [24] -, The range of periods of period solutions of $ x'(t) = - \alpha f(x(t - 1))$, J. Math. Anal. Appl. 58 (1977), 280-292. MR 0445086 (56:3431)
  • [25] -, Periodic solutions of differential-delay equations with two time lags (submitted).
  • [26] W. V. Petryshyn, On the approximation-solvability of equations involving A-proper and pseudo-A-proper mappings, Bull. Amer. Math. Soc. 81 (1975), 223-312. MR 0388173 (52:9010)
  • [27] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487-513. MR 0301587 (46:745)
  • [28] G. T. Whyburn, Topological analysis, Princeton Univ. Press, Princeton, N.J., 1958. MR 0099642 (20:6081)

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DOI: https://doi.org/10.1090/S0002-9947-1978-0482913-0
Article copyright: © Copyright 1978 American Mathematical Society

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