Summary
We develop here some new fixed point theorems and apply them to the question of existence of nontrivial periodic solutions of nonlinear, autonomous functional differential equations. We prove that the standard results of G. S. Jones and R. B. Grafton can be obtained by our methods, and we prove periodicity results for some equations, for instance a neutral functional differential equation, which appear inaccessible by previous techniques.
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F. E. Browder,On a generalization of the Schauder fixed point theorem, Duke Math. Jour.,26 (1959), pp. 291–304.
F. E. Browder,Another generalization of the Schauder fixed point theorem, Duke Math. Jour.,32 (1965), pp. 399–406.
F. E. Browder,A further generalization of the Schauder fixed point theorem, Duke Math. Jour.,32 (1965), pp. 575–578.
F. E. Browder,Asymptotic fixed point theorems, Math. Ann.,185 (1970), pp. 38–61.
W. J. Cunningham,A nonlinear differential-difference equation of growth, Proc. Nat. Acad. Sci. U.S.A.,40 (1954), pp. 708–713.
G. Darbo,Punti uniti in trasformazioni a condiminio non compatto, Rend. Sem. Mat. Univ. Padova,24 (1955), pp. 353–367.
R. B. Grafton,A periodicity theorem for autonomous functional differential equations, Jour. Diff. Eqns.,6 (1969), pp. 87–109.
R. B. Grafton,Periodic solutions of certain Liénard equations with delay, Jour. Diff. Eqns.,11 (1972), pp. 519–527.
R. B. Grafton,Liénard equations with delay: existence and stability of periodic solutions, Abstract of talk at the Park City, Utah, Symposium on Functional Differential Equations, March 1972.
A. Halanay -J. Yorke,Some new results and problems in the theory of differential-delay equations, SIAM Review,13 (1971), pp. 55–80.
J. Hale,Functional Differential Equations, Springer-Verlag, New York, 1971.
J. Hale -C. Perello,The neighborhood of a singular point of functional differential equations, Contrib. Diff. Eqns.,3 (1964), pp. 351–375.
S. Kakutani -L. Markus,On the nonlinear difference-differential equation y′(t)= =[A − By(t − τ)]y(t), Contrib. Theory Nonlinear Oscillations,4 (1958), pp. 1–18.
J. Kaplan - J. Yorke,On the stability of a periodic solution of a differential-delay equation, to appear.
V. Klee,Some topological properties of convex sets, Trans. Amer. Math. Soc.,78 (1955), pp. 30–45.
C. Kuratowski,Sur les espaces complets, Fund. Math.,15 (1930), pp. 301–309.
G. S. Jones,The existence of periodic solutions of f′(x)=− αf(x − 1)[1 + f(x)], Jour. Math. Anal. Appl.,5 (1962), pp. 435–450.
G. S. Jones,On the nonlinear differential difference equation f′(x) =− αf(x − 1)[1 + f(x)], Jour. Math. Anal. Appl.,4 (1962), pp. 440–469.
G. S. Jones,Periodic motions in Banach space and applications to functional differential equations, Contrib. Diff. Eqns.,3 (1964), pp. 75–106.
R. D. Nussbaum,The fixed point index and asymptotic fixed point theorems for k-set-contractions, Bull. Amer. Math. Soc.,75 (1969), pp. 490–495.
R. D. Nussbaum,Asymptotic fixed point theorems for local condensing maps, Math. Ann.,191 (1971), pp. 181–195.
R. D. Nussbaum,The fixed point index for local condensing maps, Ann. Mat. Pura Appl.,89 (1971), pp. 217–258.
R. D. Nussbaum,Some asymptotic fixed point theorems, Trans. Amer. Math. Soc.,171 (1972), pp. 349–375.
R. D. Nussbaum,A generalization of the Ascoli theorem and an application to functional differential equations, Jour. Math. Anal. Appl.,35 (1971), pp. 600–610.
R. D. Nussbaum,Existence and uniqueness theorems for some functional differential equations of neutral type, Jour. Diff. Eqns.,11 (1972), pp. 607–623.
E. M. Wright,A nonlinear difference-differential equation, Jour. Reine Angewandte Math.,494 (1955), pp. 66–87.
R. B. Brown,The Lefschetz Fixed Point Theorem, Scott, Foresman and Company, Glenview, Illinois, 1971.
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Partially supported by NSFGP 20228 and a Rutgers Research Council Faculty Fellowship.
Entrata in Redazione il 10 gennaio 1973.
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Nussbaum, R.D. Periodic solutions of some nonlinear autonomous functional differential equations. Annali di Matematica 101, 263–306 (1974). https://doi.org/10.1007/BF02417109
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DOI: https://doi.org/10.1007/BF02417109