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# Asymptotic homogenization for delay-differential equations and a question of analyticity

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.

Mathematics Subject Classification: Primary: 34K12, 34K25, 37C29; Secondary: 26E05, 26E10, 37C60.

 Citation: • • Table 1.  In every summation we assume that $j,k\in \mathbb{Z}\setminus\{0\}$.

 $\begin{array}{lcl} \Gamma_1(t) & = & {\frac{B(t^2)}{2t},}\\ \\ \Omega_1(t) & = & {\frac{B(t^2)}{2t^2},}\\ \\ \Gamma_2(t) & = & {\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{4 \omega^2j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg) +\sum\limits_j\frac{a_ja_{-j}}{4 \omega^2j^2}\bigg(\frac{e^{i \omega j(2t-1)}}{t}\bigg),}\\ \\ \Omega_2(t) & = & {\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{ia_ja_kk}{2 \omega j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg)}\\ \\ & & {+\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{2 \omega^2 j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^3}\bigg)}\\ \\ & & {+\sum\limits_j\frac{a_ja_{-j}}{4 \omega^2j^2}\bigg(\frac{e^{i \omega j(2t-1)}}{t^2}\bigg),}\\ \\ \Gamma_3(t) & = & {\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{8 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg),}\\ \\ \Omega_3(t) & = & {-\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{4 \omega^2j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg)A((t-2)^2)}\\ \\ & & {+\sum\limits_{j,k}\frac{a_ja_{-j}a_k(j-2k)}{4 \omega^2j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg)}\\ \\ & & {+\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{4 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^3}\bigg),}\\ \\ \Omega_4(t) & = & {-\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{8 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg)A((t-3)^2),} \end{array}$
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Tables(1)

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