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Abstract differential equations by Samuel Zaidman

By Samuel Zaidman

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Originally published in 1891. Mallet-Paret, J. (1976) Negatively invariant sets of compact maps and an extension of a theorem of Cartwright. J. Differential Eqns 22, 331-348. Man´e, R. (1981) On the dimension of the compact invariant sets of certain nonlinear maps. In Lecture Notes in Math. 898, 230-242. Massatt, P. (1983) Attractivity properties of α-contractions. J. Differential Eqns 48, 326-333. L. (1950) The existence of periodic solutions of systems of differential equations. Duke Math. J. 17 457-475.

Also, the 2 function x(t) = −t3 , t < 0, and x(t) =≡ 0, t ≥ 0, dx dt (t) = −3t . In fact, since x ≤ 1 for t ≥ −1, it is clear that x satisfies the equation for t ≥ 0. Since x is monotone decreasing for t ≤ 0, |xt | = x(t − 1) = −(t − 1)3 and dx (t) = −3t2 It is easy to verify that dt −3t2 = f ((t − 1)3 ) for t < 0. 4) dt where σ : R → [0, 1] is smooth, σ (0) = 0 and σ(0) = 1. 4) can be written as G(ϕ) = ϕ(−σ(ϕ(0)) for ϕ ∈ C([−1, 0] , R) and G is not locally lipschitz in a neighborhood of zero . In fact assume that there exist positive constants k and ρ such that |G(ϕ1 ) − G(ϕ2 )| ≤ k |ϕ1 − ϕ2 | for |ϕ1 | , |ϕ2 | < ρ √ Let ϕ(θ) = (−1+ 1 + θ), for θ ∈ [−1, 0] , where is a positive constant such that |ϕ| < ρ .

1962) Nonlinear Oscillations. D. , Princeton. Myshkis, A. D. (1951) Lineare Differentialgleichungen mit nacheilenden Argumentom. Deutscher Verlag. Wiss. Berlin, 1955. Translation of the 1951 Russian edition. Nussbaum, R. (1972) Some asymptotic fixed point theorems. Trans. Am. Math. Soc. 171,349-375. Nussbaum, R. (1974) Periodic solutions of some nonlinear autonomous functional differential equations. Ann. Math. Pura Appl. 10, 263-306. ¨ (1908) La mathematique dans ses rapports avec la physique. Picard, E Actes du IVe congr`es international des Math´ematiciens, Rome.

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