This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
J. Alexander and J. Yorke, "Global bifurcation of periodic orbits", Amer. J. Math. 100 (1978), 263–292
W. Alt, "Some periodicity criteria for functional differential equations", Manuscripta Math. 23 (1978), 295–318.
K. Borsuk, Theory of Retracts, Polish Sci. Publ., Warsaw, 1967.
F.E. Browder, "Local and global properties of nonlinear mappings in Banach spaces", Institute Naz. di Alta Math. Symposia Math. 2(1968), 13–35.
F.E. Browder, "On a generalization of the Schauder fixed point theorem", Duke Math. J. 26 (1959), 291–303.
F.E. Browder, "A further generalization of the Schauder fixed point theorem", Duke Math. J. 32 (1965), 575–578.
F.E. Browder, "Asymptotic fixed point theorems", Math. Ann. 185 (1970), 38–60.
Shui-Nee Chow, "Existence of periodic solutions of autonomous functional differential equations", J. Differential Equations 15 (1974), 350–378.
Shui-Nee Chow and J. Hale, "Periodic solutions of autonomous equations", J. Math. Anal. Appl. 66 (1978), 495–506
Shui-Nee Chow and J. Mallet-Paret, "Fuller's index and global Hopf's bifurcation", to appear.
Shui-Nee Chow and J. Mallet-Paret, "Integral averaging and bifurcation", J. Differential Equations 26(1977), 112–159.
D. Darbo, "Punti uniti in trasformazioni a condiminio non compatto", Rend. Sem. Math. Univ. Padova 24 (1955), 84–92.
R.F. Brown, The Lefschetz Fixed Point Theorem, Scott Foresman Co., Henview, Illinois, (1971).
A. Dold, "Fixed point index and fixed point theorems for Euclidean neighborhood retracts", Topology 4 (1965), 1–8.
J. Eells and G. Fournier, "La théorie des points fixes des applications a itérée condensante", Bull. Soc. Math. France, Memoire 45 (1976), 91–120.
C. Fenske, "Analytische Theorie des Abbildungsgrades für Abbildungen in Banachräumen", Math. Nachr. 48 (1971), 279–290.
C. Fenske and H.-O. Peitgen, "On fixed points of zero index in asymptotic fixed point theory", Pacific J. Math. 66 (1976) 391–410
G. Fournier and H.-O. Peitgen, "On some fixed point principles for cones in linear normed spaces", Math. Ann. 225 (1977), 205–218
L. Glass and M. Mackey, "Oscillation and chaos in physiological control systems", Science 197 (1977), 287–289.
R.B. Grafton, "A periodicity theorem for autonomous functional differential equations", J. Differential Equations 6 (1969), 87–109.
R.B. Grafton, "Periodic solutions of certain Liénard equations with delay", J. Differential Equations 11 (1972) 519–527.
K.P. Hadeler and J. Tomiuk, "Periodic solutions of difference-differential equations", Arch. Rat. Mech. Anal. 65 (1977), 87–95.
U. an der Heiden, "Periodic solutions of a nonlinear second order differential equation with delay", to appear.
J. Hale, Theory of Functional Differential Equations, Springer Verlag, New York, Heidelberg, Berlin, (1977).
J. Hale, "Nonlinear oscillations in equations with delays", to appear.
J. Ize, "Bifurcation theory for Fredholm operators", Memoirs Amer. Math. Soc. 174 (1976).
G.S. Jones, "The existence of periodic solutions of f'(x) = −αf(x−1)[1 + f(x)]", J. Math. Anal. Appl. 5 (1962), 435–450.
G.S. Jones, "On the nonlinear differential difference equation f'(x) =-αf(x-1)[1 + f(x)]", J. Math. Anal. Appl. 4 (1962), 440–469.
G.S. Jones, "Periodic motions in Banach space and applications to functional differential equations", Contributions to Differential Equations 3 (1964), 75–106.
J. Kaplan and J. Yorke, "Ordinary differential equations which yield periodic solutions of differential-delay equations", J. Math. Anal. Appl. 48 (1974), 317–324.
J. Kaplan and J. Yorke, "On the stability of a periodic solution of a differential delay equation", SIAM J. Math. Anal. 6 (1975), 268–282.
J. Kaplan and J. Yorke, "On the nonlinear differential delay equation x'(t) = −f(x(t), x(t−1))", J. Differential Equations 23 (1977), 293–314.
M. Krasnosel'skii and P.P. Zabreiko, "Iterations of operators and the fixed point index", Soviet Math. Doklady 12 (1971), 294–298.
M. Krasnosel'skii, Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen, The Netherlands, (1964).
J. Leray, "Théorie des points fixes: indice total et nombre de Lefschetz", Bull. Soc. Math. France 87 (1959), 221–233.
A.D. Myschkis, "Lineare Differentialgleichungen mit nacheilendem Argument", German translation of Russian original, VEB Deutscher Verlag der Wissenschaften, volume 17, Berlin, (1955).
R.D. Nussbaum, "The fixed point index for local condensing maps", Ann. Mat. Pura Appl. 89 (1971), 217–258.
R.D. Nussbaum, "Some asymptotic fixed point theorems", Transactions AMS 171 (1972), 349–375.
R.D. Nussbaum, "Generalizing the fixed point index", Math. Ann. 228 (1977), 259–278.
R.D. Nussbaum, "Periodic solutions of some nonlinear autonomous functional differential equations", Annali di Math. pura e appl. 101 (1974), 263–306.
R.D. Nussbaum, "Periodic solutions of some nonlinear autonomous functional differential equations, II", J. Differential Equations 14 (1973), 360–394.
R.D. Nussbaum, "A correction of ‘Periodic solutions of some nonlinear autonomous functional differential equations, II',", J. Differential Equations 16 (1974), 548–549.
R.D. Nussbaum, "A global bifurcation theorem with applications to functional differential equations", J. Functional Analysis 19 (1975), 319–338.
R.D. Nussbaum, "The range of periods of periodic solutions of x'(t) = −αf(x(t−1))", J. Math. Anal. Appl. 58 (1977), 280–292.
R.D. Nussbaum, "Periodic solutions of analytic functional differential equations are analytic", Michigan Math. J. 20 (1973), 249–255.
R.D. Nussbaum, "Global bifurcation of periodic solutions of some autonomous functional differential equations", J. Math. Anal. Appl. 55 (1976), 699–725.
R.D. Nussbaum, "A Hopf global bifurcation theorem for retarded functional differential equations", Transactions of Amer. Math. Soc. 238 (1978), 139–163.
R.D. Nussbaum, "Integral equations from the theory of epidemics", in Nonlinear Systems and Applications, edited by V. Lakshmikantham, Academic Press, New York, (1977), pp. 235–255.
R.D. Nussbaum, "Periodic solutions of special differential-delay equations: an example in nonlinear functional analysis", to appear in Proc. Royal Soc. Edinburgh.
R.D. Nussbaum, "Differential-delay equations with two time lags", to appear in Memoirs of the Amer. Math Soc..
R.D. Nussbaum, "Uniqueness and nonuniqueness for periodic solutions of x′(t)=−g(x(t−1))", to appear.
H.-O. Peitgen, "On continua of periodic solutions for functional differential equations", Rocky Mountain J. 7 (1977), 609–617.
B.N. Sadovskii, "Limit compact and condensing operators", Russian Math. Surveys 27 (1972), 85–155.
A. Somolinos, "Periodic solutions of the sunflower equation", Quart. Applied Math. 35 (January, 1978), 465–478.
H. Steinlein, "Ein Satz über den Leray-Schauderschen Abbildungsgrad", Math. Z. 120 (1972), 176–208.
H. Steinlein, "Über die verallgemeinerten Fixpunktindizes von Iterierten verdichtender Abbildungen", Manuscripta Math. 8 (1972), 251–266.
H.-O. Walther, "A theorem on the amplitudes of periodic solutions of differential delay equations with applications to bifurcation", J. Diff. Equat. 29 (1978), 396–404
H.-O. Walther, "Über Ejektivität und periodische Lösungen bei autonomen Funktionaldifferentialgleichungen mit verteilter Verzögerung", Habilitationsschrift, Univ. München, (1977).
E.M. Wright, "A nonlinear difference-differential equation", J. Reine Angew. Math. 194 (1955), 66–87.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1979 Springer-Verlag
About this paper
Cite this paper
Nussbaum, R.D. (1979). Periodic solutions of nonlinear autonomous functional differential equations. In: Peitgen, HO., Walther, HO. (eds) Functional Differential Equations and Approximation of Fixed Points. Lecture Notes in Mathematics, vol 730. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064325
Download citation
DOI: https://doi.org/10.1007/BFb0064325
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-09518-7
Online ISBN: 978-3-540-35129-0
eBook Packages: Springer Book Archive
