# American Institute of Mathematical Sciences

June  2020, 40(6): 3789-3812. doi: 10.3934/dcds.2020044

## 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

Received  March 2019 Revised  June 2019 Published  October 2019

Fund Project: The first author was partially supported by The Center for Nonlinear Analysis at Rutgers University. The second author was partially supported by NSF Grant DMS-1201328 and by The Lefschetz Center for Dynamical Systems at Brown University

We consider a class of nonautonomous delay-differential equations in which the time-varying coefficients have an oscillatory character, with zero mean value, and whose frequency approaches
 $+\infty$
as
 $t\to\pm\infty$
. Typical simple examples are
 $x'(t) = \sin (t^q)x(t-1) \qquad\text{and}\qquad x'(t) = e^{it^q}x(t-1), \;\;\;\;\;\;\;\;{(*)}$
where
 $q\ge 2$
is an integer. Under various conditions, we show the existence of a unique solution with any prescribed finite limit
 $\lim\limits_{t\to-\infty}x(t) = x_-$
at
 $-\infty$
. We also show, under appropriate conditions, that any solution of an initial value problem has a finite limit
 $\lim\limits_{t\to+\infty}x(t) = x_+$
at
 $+\infty$
, and thus we establish the existence of a class of heteroclinic solutions. We term this limiting phenomenon, and thus the existence of such solutions, "asymptotic homogenization." Note that in general, proving the existence of a bounded solution of a given delay-differential equation on a semi-infinite interval
 $(-\infty,-T]$
is often highly nontrivial.
Our original interest in such solutions stems from questions concerning their smoothness. In particular, any solution
 $x: \mathbb{R}\to \mathbb{C}$
of one of the equations in
 $(*)$
with limits
 $x_\pm$
at
 $\pm\infty$
is
 $C^\infty$
, but it is unknown if such solutions are analytic. Nevertheless, one does know that any such solution of the second equation in
 $(*)$
can be extended to the lower half plane
 $\{z\in \mathbb{C}\:|\: \mathop{{{\rm{Im}}}} z<0\}$
as an analytic function.
Citation: John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044
##### References:
 [1] N. G. de Bruijn, Asymptotic Methods in Analysis, Dover Publications, Inc., New York, 1981.  Google Scholar [2] J. Dieudonné, Foundations of Modern Analysis, Pure and Applied Mathematics, Vol. 10-I, Academic Press, New York-London, 1969.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] A. Zygmund, Trigonometric Series, Vols. I, II, 2nd edition, Cambridge Univ. Press, New York, 1959.  Google Scholar

show all references

##### References:
 [1] N. G. de Bruijn, Asymptotic Methods in Analysis, Dover Publications, Inc., New York, 1981.  Google Scholar [2] J. Dieudonné, Foundations of Modern Analysis, Pure and Applied Mathematics, Vol. 10-I, Academic Press, New York-London, 1969.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] A. Zygmund, Trigonometric Series, Vols. I, II, 2nd edition, Cambridge Univ. Press, New York, 1959.  Google Scholar
In every summation we assume that $j,k\in \mathbb{Z}\setminus\{0\}$.
 $\begin{array}{lcl} \Gamma_1(t) & = & {\frac{B(t^2)}{2t},}\\ \\ \Omega_1(t) & = & {\frac{B(t^2)}{2t^2},}\\ \\ \Gamma_2(t) & = & {\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{4 \omega^2j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg) +\sum\limits_j\frac{a_ja_{-j}}{4 \omega^2j^2}\bigg(\frac{e^{i \omega j(2t-1)}}{t}\bigg),}\\ \\ \Omega_2(t) & = & {\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{ia_ja_kk}{2 \omega j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg)}\\ \\ & & {+\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{2 \omega^2 j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^3}\bigg)}\\ \\ & & {+\sum\limits_j\frac{a_ja_{-j}}{4 \omega^2j^2}\bigg(\frac{e^{i \omega j(2t-1)}}{t^2}\bigg),}\\ \\ \Gamma_3(t) & = & {\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{8 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg),}\\ \\ \Omega_3(t) & = & {-\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{4 \omega^2j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg)A((t-2)^2)}\\ \\ & & {+\sum\limits_{j,k}\frac{a_ja_{-j}a_k(j-2k)}{4 \omega^2j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg)}\\ \\ & & {+\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{4 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^3}\bigg),}\\ \\ \Omega_4(t) & = & {-\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{8 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg)A((t-3)^2),} \end{array}$
 $\begin{array}{lcl} \Gamma_1(t) & = & {\frac{B(t^2)}{2t},}\\ \\ \Omega_1(t) & = & {\frac{B(t^2)}{2t^2},}\\ \\ \Gamma_2(t) & = & {\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{4 \omega^2j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg) +\sum\limits_j\frac{a_ja_{-j}}{4 \omega^2j^2}\bigg(\frac{e^{i \omega j(2t-1)}}{t}\bigg),}\\ \\ \Omega_2(t) & = & {\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{ia_ja_kk}{2 \omega j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg)}\\ \\ & & {+\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{2 \omega^2 j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^3}\bigg)}\\ \\ & & {+\sum\limits_j\frac{a_ja_{-j}}{4 \omega^2j^2}\bigg(\frac{e^{i \omega j(2t-1)}}{t^2}\bigg),}\\ \\ \Gamma_3(t) & = & {\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{8 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg),}\\ \\ \Omega_3(t) & = & {-\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{4 \omega^2j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg)A((t-2)^2)}\\ \\ & & {+\sum\limits_{j,k}\frac{a_ja_{-j}a_k(j-2k)}{4 \omega^2j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg)}\\ \\ & & {+\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{4 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^3}\bigg),}\\ \\ \Omega_4(t) & = & {-\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{8 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg)A((t-3)^2),} \end{array}$
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