Mathematics

T. M. Karaseva

On a Certain “Exact Estimate” of the Multipliers of Second-Order Differential Equations with Periodic Coefficients

(Presented by Academician I. G. Petrovskii on 26 II 1958)

We shall consider a differential equation of the form

\[ y'' + p(x)y = 0 \qquad (-\infty < x < \infty;\ p(x+T)=p(x)). \tag{1} \]

Let \(\varphi(x)\) and \(\psi(x)\) denote two solutions of this equation, determined by the conditions \(\varphi(0)=1,\ \varphi'(0)=0,\ \psi(0)=0,\ \psi'(0)=1\). As is known, the quantity

\[ A = {}^{1}/_{2}[\varphi(T)+\psi'(T)] \tag{2} \]

is called the Lyapunov constant, and the roots of the equation

\[ \rho^2 - 2A\rho + 1 = 0, \tag{3} \]

which we shall denote by \(\rho_1\) and \(\rho_2\) \((\rho_1\rho_2=1)\), are called the multipliers. According to Floquet’s theorem, equation (1) has two solutions \(y_1(x)\) and \(y_2(x)\) such that

\[ y_1(x+T)=\rho_1 y_1(x), \qquad y_2(x+T)=\rho_2 y_2(x) \qquad (-\infty < x < \infty), \tag{4} \]

if \(A \ne \pm 1\). If \(|A|<1\), then both multipliers are equal to one in modulus and both solutions (4) are bounded on the whole axis. If, however, \(|A|>1\), then one of the multipliers, say \(\rho_1\), is greater than one in absolute value, and the solution \(y_1(x)\) increases exponentially as \(x \to \infty\). In this case the rate of growth is characterized by the magnitude of the multiplier \(\rho_1\).

A. M. Lyapunov \((^{1,2})\) considered the family of equations

\[ y''+\lambda p(x)y=0 \]

and showed that the real \(\lambda\)-axis contains an infinite number of intervals within which \(A=A(\lambda)\) is less than one in absolute value. These intervals are called zones of stability and are numbered from left to right in such a way that the leftmost interval containing positive \(\lambda\)’s is assigned number 1. Between each pair of adjacent stability zones there lies a zone of instability, which is either an isolated point or a closed interval; moreover, to the right of the first stability zone lies the first instability zone, etc. The function \(p(x)\), by definition, belongs to the \(k\)-th zone of stability (instability) if the number \(\lambda=1\) belongs to the \(k\)-th zone of stability (instability).

1. In the works of V. A. Yakubovich \((^{3,4})\) and V. I. Burdina \((^5)\), some estimates of the absolute values of the multipliers are given. In the present work an estimate is obtained for the absolute value of a multiplier which, in a number of cases, is more convenient.

Denote by \(Q_\alpha\) the totality of all real functions \(q(x)\) \((q(x+T)\equiv q(x))\), quadratically integrable on the interval \((0,T)\) and satisfying the conditions:

\[ 1)\quad T\int_0^T q^2(x)\,dx=\alpha;\qquad 2)\quad \int_0^T q(x)\,dx=0. \tag{5} \]

The totality of all generalized functions \(p(x)\) of the form \(p(x)=q'(x)\) \((q\in Q_\alpha)\) will be denoted by \(P_\alpha\). We note that if \(p(x)\in P_\alpha\), then \(\int_0^T p(x)\,dx=0\).

Theorem 1. If the function \(p(x)\in P_\alpha\) belongs to an instability zone, then the number \(n\) of this zone does not exceed the integer part of the number \([2\sqrt{\alpha}/\pi]\). Whatever \(n<[2\sqrt{\alpha}/\pi]\) may be \((n=1,2,\ldots)\), there exists a function \(p\in P_\alpha\) belonging to the \(n\)-th instability zone.

This assertion is “sharp” in the sense that the number \([2\sqrt{\alpha}/\pi]\) cannot be replaced by a smaller number.

Corollary. If the function \(p(x)\in P_\alpha\) belongs to an instability zone, then the number \(n\) of this zone does not exceed the number \([2\sqrt{\alpha}/\pi+1]\).

Theorem 2. If the function \(p(x)\in P_\alpha\) belongs to the \(n\)-th instability zone \((n=1,2,\ldots;\ n\leq 2\sqrt{\alpha}/\pi)\), then for the multipliers corresponding to the function \(p(x)\) the estimate

\[ |\rho_{1,2}|\leq \exp\left[\pi\sqrt{\frac{2n}{\pi}\sqrt{\alpha}-n^2}\right] \tag{6} \]

holds.

This estimate is sharp. It is attained for the function \(q=q_n(x)\) \((q_n\in Q_\alpha)\)

\[ q_n=\frac{\sqrt{\alpha}}{T}\operatorname{ctg}\beta,\qquad x=\frac{T}{n\pi}\left(\beta-\frac{\sin 2\beta}{2}\right) \quad (-\infty<\beta<\infty), \tag{7} \]

since the multipliers corresponding to the function \(p_n=q'_n\) are equal to

\[ \rho_{1,2}=(-1)^n \exp\left[\pm \pi\sqrt{\frac{2n}{\pi}\sqrt{\alpha}-n^2}\right] \qquad \left(n\leq \left[\frac{2\sqrt{\alpha}}{\pi}\right]\right). \tag{8} \]

It is easy to see that the function \(p_n(x)\) indeed belongs to the \(n\)-th instability zone. In fact, the multiplier \(\rho_{1,2}(\lambda)\) corresponding to the function \(\lambda p_n=\lambda q'_n\) is equal to

\[ \rho_{1,2}(\lambda)=(-1)^n \exp\left[\pm \pi\sqrt{\frac{2n\lambda}{\pi}\sqrt{\alpha}-n^2}\right]. \]

When \(\lambda\) runs through the values from 0 to 1, the multiplier \(\rho_1(\lambda)\) first describes an arc of the unit circle of length \(n\pi\), and then becomes and remains real.

For the proof of Theorems 1 and 2 the following variational problem was solved.

To each function \(p(x)\in P_\alpha\) there corresponds the Lyapunov number \(A=A(p)\). It is required to find \(\sup A(p)\) over all functions \(p(x)\in P_\alpha\) belonging to the \(n\)-th instability zone.

1. The totality of functions entering the \(n\)-th stability zone and the \(n\)-th instability zone will be called the \(n\)-th zone. There exists a whole series of criteria (for example, the number of zeros on the interval \((0\leq x\leq T)\) of solutions of the equation \(y''+p(x)y=0\)) that make it possible to establish whether a function \(p(x)\) belongs to the \(n\)-th zone. Therefore, a generalization of the stability criterion obtained by M. G. Krein (⁶) is of known practical interest.

Theorem 3. If \(p(x)\subset P_{\alpha}\) belongs to the \(n\)-th zone and \(\alpha<n^{2}\pi^{2}/4\), then all solutions of equation (1) are bounded.

This criterion is “sharp” in the sense that the number \(n^{2}\pi^{2}/4\) cannot be replaced by any smaller number.

If \(\alpha<\pi^{2}/4\), then the requirement that \(p(x)\) belong to the first zone is automatically satisfied.

The author takes this opportunity to express his gratitude to M. G. Krein and G. Ya. Lyubarskii for valuable advice and discussion of the work.

Kharkov Automobile and Highway Institute

Received
21 II 1958

References Cited

\(^{1}\) A. M. Lyapunov, C. R., 128, 910 (1899).
\(^{2}\) A. M. Lyapunov, C. R., 128, 1085 (1899).
\(^{3}\) V. A. Yakubovich, DAN, 87, 345 (1952).
\(^{4}\) V. A. Yakubovich, Prikl. matem. i mekh., 18, 533 (1954).
\(^{5}\) V. I. Burdina, DAN, 93, 603 (1953).
\(^{6}\) M. G. Krein, Prikl. matem. i mekh., 19, 641 (1955).