Sergei Yakovenko's blog: on Math and Teaching

Thursday, November 20, 2014

Lecture 6 (Nov. 21, 2014)

Filed under: Analytic ODE course — Sergei Yakovenko @ 8:20
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Zeros of solutions of linear equations

Nontrivial (i.e., not identically zero) solutions of linear ordinary differential equations obviously possess certain properties concerning their roots (points where these solutions vanish). The simplest, in a sense paradigmal property, is the following.

Prototheorem. Let u be a nontrivial solution of a sufficiently regular linear ordinary differential equation Lu=0 of order n>0. Then u cannot have a root of multiplicity greater or equal than n-1.

Here by regularity we mean the condition that the operator L=\partial^n+a_1(t)\partial^{n-1}+\cdots+a_{n-1}(t)\partial+a_n(t) has coefficients smooth enough to guarantee that any solution u(t) near any point a in the domain of its definition is uniquely determined by the initial conditions u(a),u'(s),\dots,u^{(n-1)}(a).

Indeed, if u has a root of multiplicity n, that is, all first n-1 derivatives of u at a vanish, then u^{(n)}(a)=0 by virtue of the equation and hence the by the uniqueness u(t) must be identically zero.

In particular, solutions of first order equation u'+a_1(t)u=0 are nonvanishing, solutions of any second order equation u''+a_1(t)u'+a_2(t)u=0 may have only simple roots etc.


What about quantitative versions?

Theorem (de la Vallee Poussin, 1929). Assume that the coefficients of the LODE

u^{(n)}+a_1(t)u^{(n-1)}+\cdots+a_{n-1}(t)u'+a_n(t)u=0,\qquad t\in[0,\ell],\qquad (\dag)

are explicitly bounded,  |a_k(t)|\leqslant A_k\in\mathbb R_+,\ \forall t\in[0,\ell],\ k=1,\dots,n.

Assume that the bounds are small relative to the length of the interval, i.e.,

\displaystyle \sum_{k=1}^n \frac{A_k}{k!}\ell^k<1.\qquad (*)

Then any nontrivial solution of the equation has no more than n-1 isolated roots on [0,\ell] .

Novikov’s counterexample

What about linear systems of the first order?

Consider the system \dot x=A(t)x with x=(x_1,\dots,x_n)\in  \mathbb R^n and the norm \|A(t)\| explicitly bounded on [0,\ell]. Consider all possible linear combinations u=\sum_k c_k x_k(t),\ c\in\mathbb R^n. Can one expect a uniform upper bound for the number of roots of all combinations?

Let a(t) be a polynomial having many zeros on [0,t]. Consider the 2\times 2-system of the form

\displaystyle \dot x_1=a(t)x_1,\qquad \dot x_2=(\dot a+ a^2)x_1.

The first equation defines a nonvanishing function x_1(t), the second equation – its derivative which vanishes at all roots of a(t).

By replacing a(t) by \varepsilon a(t) one can achieve an arbitrarily small sup-norm of the coefficients of this system on the segment [0,\ell] (or even any open complex neighborhood of this real segment). Thus no matter how small are the coefficients, the second component will have the specified number of isolated roots.


Complexification

What about complex valued versions? There is no Rolle theorem for them.

I will describe three possible replacements, Kim’s theorem (1963), nearest in the spirit, and two versions of the argument principle.

Theorem (W. Kim)
Assume that an analytic LODE

u^{(n)}+a_1(z)u^{(n-1)}+\cdots+a_{n-1}(z)u'+a_n(z)u=0,\qquad z\in D\subseteq\mathbb C

is defined in a convex compact subset D of diameter \ell and the condition (*) holds. Then this equation is disconjugate in D: any solution has at most n-1 isolated roots.

This result follows from the interpolation inequality of the following type: if u(z) is a function holomorphic in D and has n isolated roots there, then \|u\|_D\leqslant \frac{\ell^n}{n!}\|u^{(n)}\| (the maximum modulus norm is assumed).


Consider the equation (\dag) on the real interval but with complex-valued coefficients (and solutions). Solutions will be then real parameterized curves u:[0,\ell]\to\mathbb C which only exceptionally rarely have roots. Instead of counting roots, one can measure their rotation around the origin 0\in\mathbb C, which is defined as R(u)=|\mathrm{Arg}~u(\ell)-\mathrm{Arg}~u(0)| for any continuous choice of the argument.

Theorem. Assume that

\displaystyle \sum_{k=1}^n \frac{A_k}{k!}\ell^k<\frac12.

Then rotation of any nontrivial solution u is explicitly bounded: R(u)<\pi (n+1).

If an analytic LODE with explicitly bounded coefficients is defined, say, on a triangle D, then application of this result to the sides of the triangle yields an explicit upper bound for the number of isolated roots of analytic solutions inside the triangle.

Reference

S. Yakovenko, On functions and curves defined by differential equations, §2.

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1 Comment »

  1. […] previous results imply that one can produce an explicit upper bound for the variation of argument of any nontrivial […]

    Pingback by Lecture 8 (Nov. 26, 2014) | Sergei Yakovenko's blog: on Math and Teaching — Sunday, November 30, 2014 @ 11:40 | Reply


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