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Asymptotic homogenization for delay-differential equations and a question of analyticity
1. | Division of Applied Mathematics, Brown University, Providence, RI 02912, USA |
2. | Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA |
$ +\infty $ |
$ t\to\pm\infty $ |
$ x'(t) = \sin (t^q)x(t-1) \qquad\text{and}\qquad x'(t) = e^{it^q}x(t-1), \;\;\;\;\;\;\;\;{(*)} $ |
$ q\ge 2 $ |
$ \lim\limits_{t\to-\infty}x(t) = x_- $ |
$ -\infty $ |
$ \lim\limits_{t\to+\infty}x(t) = x_+ $ |
$ +\infty $ |
$ (-\infty,-T] $ |
$ x: \mathbb{R}\to \mathbb{C} $ |
$ (*) $ |
$ x_\pm $ |
$ \pm\infty $ |
$ C^\infty $ |
$ (*) $ |
$ \{z\in \mathbb{C}\:|\: \mathop{{{\rm{Im}}}} z<0\} $ |
References:
[1] |
N. G. de Bruijn, Asymptotic Methods in Analysis, Dover Publications, Inc., New York, 1981. |
[2] |
J. Dieudonné, Foundations of Modern Analysis, Pure and Applied Mathematics, Vol. 10-I, Academic Press, New York-London, 1969.
![]() |
[3] |
J. Mallet-Paret and R. D. Nussbaum,
Analyticity and nonanalyticity of solutions of delay-differential equations, SIAM J. Math. Anal., 46 (2014), 2468-2500.
doi: 10.1137/13091943X. |
[4] |
J. Mallet-Paret and R. D. Nussbaum,
Intricate structure of the analyticity set for solutions of a class of integral equations, J. Dynam. Differential Equations, 31 (2019), 1045-1077.
doi: 10.1007/s10884-019-09746-1. |
[5] |
J. Mallet-Paret and R. D. Nussbaum, Analytic solutions of delay-differential equations, in preparation. Google Scholar |
[6] |
R. D. Nussbaum,
Periodic solutions of analytic functional differential equations are analytic, Michigan Math. J., 20 (1973), 249-255.
doi: 10.1307/mmj/1029001104. |
[7] |
A. Zygmund, Trigonometric Series, Vols. I, II, 2nd edition, Cambridge Univ. Press, New York, 1959. |
show all references
References:
[1] |
N. G. de Bruijn, Asymptotic Methods in Analysis, Dover Publications, Inc., New York, 1981. |
[2] |
J. Dieudonné, Foundations of Modern Analysis, Pure and Applied Mathematics, Vol. 10-I, Academic Press, New York-London, 1969.
![]() |
[3] |
J. Mallet-Paret and R. D. Nussbaum,
Analyticity and nonanalyticity of solutions of delay-differential equations, SIAM J. Math. Anal., 46 (2014), 2468-2500.
doi: 10.1137/13091943X. |
[4] |
J. Mallet-Paret and R. D. Nussbaum,
Intricate structure of the analyticity set for solutions of a class of integral equations, J. Dynam. Differential Equations, 31 (2019), 1045-1077.
doi: 10.1007/s10884-019-09746-1. |
[5] |
J. Mallet-Paret and R. D. Nussbaum, Analytic solutions of delay-differential equations, in preparation. Google Scholar |
[6] |
R. D. Nussbaum,
Periodic solutions of analytic functional differential equations are analytic, Michigan Math. J., 20 (1973), 249-255.
doi: 10.1307/mmj/1029001104. |
[7] |
A. Zygmund, Trigonometric Series, Vols. I, II, 2nd edition, Cambridge Univ. Press, New York, 1959. |
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