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A Hopf global bifurcation theorem for retarded functional differential equations. (English) Zbl 0389.34050

MSC:
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C25 Periodic solutions to ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
47J05 Equations involving nonlinear operators (general)
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[1] J. C. Alexander and James A. Yorke, Global bifurcations of periodic orbits, Amer. J. Math. 100 (1978), no. 2, 263 – 292. · Zbl 0386.34040
[2] Karol Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. · Zbl 0153.52905
[3] Shui Nee Chow and John Mallet-Paret, The Fuller index and global Hopf bifurcation, J. Differential Equations 29 (1978), no. 1, 66 – 85. · Zbl 0369.34020
[4] Kenneth L. Cooke and James A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Math. Biosci. 16 (1973), 75 – 101. · Zbl 0251.92011
[5] J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353 – 367. · Zbl 0043.38105
[6] F. Brock Fuller, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967), 133 – 148. · Zbl 0152.40204
[7] R. B. Grafton, A periodicity theorem for autonomous functional differential equations., J. Differential Equations 6 (1969), 87 – 109. · Zbl 0175.38503
[8] R. B. Grafton, Periodic solutions of certain Liénard equations with delay, J. Differential Equations 11 (1972), 519 – 527. · Zbl 0231.34063
[9] A. Granas, The theory of compact vector fields and some of its applications to topology of functional spaces. I, Rozprawy Mat. 30 (1962), 93. · Zbl 0111.11001
[10] Jack K. Hale, Functional differential equations, Springer-Verlag New York, New York-Heidelberg, 1971. Applied Mathematical Sciences, Vol. 3. · Zbl 0222.34063
[11] Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. · Zbl 0078.10004
[12] Jorge Ize, Bifurcation theory for Fredholm operators, Mem. Amer. Math. Soc. 7 (1976), no. 174, viii+128. · Zbl 0338.47032
[13] -, Global bifurcation of periodic orbits, Communicaciones Tecnicas of C.I.M.A.S., Vol. 5, Series B., No. 8 (1974). (Spanish)
[14] G. Stephen Jones, The existence of periodic solutions of \?\(^{\prime}\)(\?)=-\?\?(\?-1){1+\?(\?)}, J. Math. Anal. Appl. 5 (1962), 435 – 450. · Zbl 0106.29504
[15] G. Stephen Jones, Periodic motions in Banach space and applications to functional-differential equations, Contributions to Differential Equations 3 (1964), 75 – 106.
[16] James L. Kaplan and James A. Yorke, Ordinary differential equations which yield periodic solutions of differential delay equations, J. Math. Anal. Appl. 48 (1974), 317 – 324. · Zbl 0293.34102
[17] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. · Zbl 0148.12601
[18] John Mallet-Paret, Generic periodic solutions of functional differential equations, J. Diff. Equations 25 (1977), no. 2, 163 – 183. · Zbl 0358.34078
[19] R. May, Stability and complexity in model ecosystems, Princeton Univ. Press, Princeton, N.J., 1973.
[20] L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. With a chapter by E. Zehnder; Notes by R. A. Artino; Lecture Notes, 1973 – 1974.
[21] Roger D. Nussbaum, A global bifurcation theorem with applications to functional differential equations, J. Functional Analysis 19 (1975), no. 4, 319 – 338. · Zbl 0314.47041
[22] Roger D. Nussbaum, Periodic solutions of some nonlinear, autonomous functional differential equations. II, J. Differential Equations 14 (1973), 360 – 394. · Zbl 0311.34087
[23] Roger D. Nussbaum, Global bifurcation of periodic solutions of some autonomous functional differential equations, J. Math. Anal. Appl. 55 (1976), no. 3, 699 – 725. · Zbl 0363.34050
[24] Roger D. Nussbaum, The range of periods of periodic solutions of \?’(\?)=-\?\?(\?(\?-1)), J. Math. Anal. Appl. 58 (1977), no. 2, 280 – 292. · Zbl 0359.34066
[25] -, Periodic solutions of differential-delay equations with two time lags (submitted).
[26] W. V. Petryshyn, On the approximation-solvability of equations involving \?-proper and psuedo-\?-proper mappings, Bull. Amer. Math. Soc. 81 (1975), 223 – 312. · Zbl 0303.47038
[27] Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487 – 513. · Zbl 0212.16504
[28] Gordon Thomas Whyburn, Topological analysis, Princeton Mathematical Series. No. 23, Princeton University Press, Princeton, N. J., 1958. · Zbl 0080.15903
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