Periodic solutions of some nonlinear, autonomous functional differential equations. II

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.

[1]  S. Kakutani,et al.  I. On the Non-Linear Difference-Differential Equation y1(t) = [A - By(t-r) ly (t) , 1958 .

[2]  J. Hamilton Noncompact mappings and cones in Banach spaces , 1972 .

[3]  H. Steinlein Ein Satz über den Leray-Schauderschen Abbildungsgrad , 1972 .

[4]  Leo F. Boron,et al.  Positive solutions of operator equations , 1964 .

[5]  James A. Yorke,et al.  Some New Results and Problems in the Theory of Differential-Delay Equations , 1971 .

[6]  H. Steinlein Über die Verallgemeinerten Fixpunktindizes von Iterierten Verdichtender Abbildungen , 1973 .

[7]  E. M. Wright A non-linear difference-differential equation. , 1946 .

[8]  H. Amann On the number of solutions of nonlinear equations in ordered Banach spaces , 1972 .

[9]  E. M. Wright Linear difference-differential equations , 1948, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  Roger D. Nussbaum,et al.  A generalization of the Ascoli theorem and an application to functional differential equations , 1971 .

[11]  Roger D. Nussbaum,et al.  Periodic solutions of some nonlinear autonomous functional differential equations , 1974 .

[12]  C. Kuratowski,et al.  Sur les espaces complets , 1930 .

[13]  G.Stephen Jones,et al.  The existence of periodic solutions of f′(x) = − αf(x − 1){1 + f(x)} , 1962 .

[14]  G. S. Jones Periodic motions in banach space and applications to functional-differential equations. , 1962 .

[15]  R. B. Grafton,et al.  A periodicity theorem for autonomous functional differential equations , 1969 .

[16]  Existence and uniqueness theorems for some functional differential equations of neutral type , 1972 .

[17]  E. M. Wright II.—The Stability of Solutions of Non-linear Difference-differential Equations , 1950, Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences.

[18]  R. Nussbaum Asymptotic fixed point theorems for local condensing maps , 1971 .

[19]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[20]  J. Dugundji An extension of Tietze's theorem. , 1951 .

[21]  V. Klee Some topological properties of convex sets , 1955 .

[22]  R. Grafton Periodic solutions of certain Lie´nard equations with delay , 1972 .

[23]  G.Stephen Jones On the nonlinear differential-difference equation f′(x) = −αf(x − 1) {1 + f(x)} , 1962 .

[24]  R. Nussbaum Some asymptotic fixed point theorems , 1972 .

[25]  F. Browder A new generalization of the Schauder fixed point theorem , 1967 .

[26]  H. H. Schaefer,et al.  Topological Vector Spaces , 1967 .

[27]  R. Nussbaum The fixed point index for local condensing maps , 1971 .

[28]  Roger D. Nussbaum,et al.  The fixed point index and asymptotic fixed point theorems for $k$-set-contractions , 1969 .

[29]  F. Browder Asymptotic fixed point theorems , 1970 .

[30]  Gabriele Darbo,et al.  Punti uniti in trasformazioni a codominio non compatto , 1955 .

[31]  J. Hale Functional Differential Equations , 1971 .