Reports of the Academy of Sciences of the USSR

1. Volume 113, No. 4

MATHEMATICS

V. A. KONDRAT'EV

SUFFICIENT CONDITIONS FOR NONOSCILLATION AND OSCILLATION OF SOLUTIONS OF THE EQUATION \(y''+p(x)y=0\)

(Presented by Academician I. G. Petrovsky, 8 X 1956)

It is known \((^1,^6,^7)\)* that the solutions of the equation

\[ y''+p(x)y=0 \tag{1} \]

are nonoscillatory for \(x \geq x_0\), if there exists a continuously differentiable function \(\theta(x)\) such that

\[ \theta' + \theta^2 + p \leq 0,\qquad x \geq x_0 . \tag{2} \]

Using this, it is easy to obtain a series of sufficient conditions under which the solutions of equation (1) will be nonoscillatory.

Theorem 1. If for \(x \geq x_0\) there exist a positive differentiable function \(r(x)\) and a constant \(\nu \geq 0\) such that, for some constant \(C\), the inequality

\[ \frac{r'}{2}-\frac{r'}{2}\sqrt{\nu} \leq -\int_{x_0}^{x}\left[p-\frac{(1-\nu)r'^2}{4r^2}\right]r\,dx+C \leq \frac{r'}{2}+\frac{r'}{2}\sqrt{\nu},\qquad x\geq x_0, \]

is satisfied, then the solutions of equation (1) are nonoscillatory.

Under the conditions of the theorem,

\[ \theta_1(x)=\frac{1}{r}\left(\int_{a}^{x}\left[p-\frac{(1-\nu)r'^2}{4r^2}\right]r\,dx+C\right) \]

satisfies condition (2). Indeed, if we denote

\[ -\int_{x_0}^{x}\left[p-\frac{(1-\nu)r'^2}{4r^2}\right]r\,dx+C=u, \]

we obtain:

\[ \theta_1' + \theta_1^2 + p = (ru' + r'u + u^2 + pr^2)/r^2 = [-pr^2+\tfrac14(1-\nu)r'^2-r'u+u^2+pr^2]/r^2 = [u-\tfrac12(r'-r'\sqrt{\nu})][u-\tfrac12(r'+r'\sqrt{\nu})]/r^2 \leq 0. \]

Corollary 1. If \(\nu=1\), Petropavlovsky’s theorem \((^2)\) is obtained.

Corollary 2. If \(r=x\), it follows that, for nonoscillation, it is sufficient that, for some \(\nu \geq 0\), the inequality

\[ 0\leq -\int_{c}^{x} xp\,dx-\frac{1-\nu}{4}\ln x\leq \sqrt{\nu} \]

be satisfied.

Corollary 3. If \(r=x^\alpha\), \(\alpha\geq 0\), then we obtain the sufficient condition for nonoscillation

\[ 0\leq -\int_{a}^{x}px^\alpha\,dx+C\leq \alpha x^{\alpha-1} \]

for some \(C\) and \(\alpha\geq 0\).

* In note \((^7)\) an elementary proof of this fact is given.

Theorem 2. If there exists a differentiable function \(r(x)\) such that

\[ 0 \leqslant -\int_a^x \left(px-\frac{1}{4x}\right) r\,dx+C \leqslant r'x,\qquad a\leqslant x\leqslant b, \]

for at least one constant \(C\), then the solutions of equation (1) are nonoscillatory.

When the conditions of the theorem are fulfilled, it is also easy to find a function \(\theta(x)\) satisfying the inequality \(\theta'+\theta^2+p\leqslant 0\). Such a function, for example, is

\[ \theta(x)=\frac{1}{x^2}\left[-\int_a^x \left(px-\frac{1}{4x}\right) r\,dx+C\right]+\frac{1}{2x}. \]

Corollary. If we take \(r=(\ln x)^\alpha,\ \alpha\geqslant 0\), then we obtain that, for nonoscillation, it is sufficient that

\[ 0 \leqslant -\int_a^x \left(px-\frac{1}{4x}\right)(\ln x)^\alpha\,dx+C \leqslant \alpha(\ln x)^{\alpha-1}. \]

Theorem 2 gives an admissible measure, for preserving nonoscillation, of the deviation of the function \(p(x)\) from \(1/4x^2\), while Theorem 1 gives a measure of such a deviation from zero.

Theorem 3. If there exists a differentiable function \(r(x)>0\) such that

\[ \lim_{x\to+\infty}\int_a^x \left(pr-\frac{r'^2}{4r}\right)dx=+\infty, \]

then every solution of equation (1) has an infinite number of zeros on \([a,\infty)\).

Suppose the contrary. Then there will exist, continuous for \(x\geqslant x_0\), a solution of the equation \(\theta'+\theta^2+p=0\) (3). The function \(u=\theta r\) satisfies the equation \(ru'-r'u+u^2+pr^2=0\). Hence \(u'-r'^2/4r+pr\leqslant 0\). Integrating, we obtain \(\lim_{x\to+\infty}u(x)=-\infty\), whence it follows that all functions \(\theta(x)\) are negative for \(x\geqslant x_1\), which is impossible (4).

From this theorem, taking \(r=x^\alpha\) \((\alpha<1)\), one obtains Hille’s result (5): if

\[ \int^\infty px^\alpha dx=\infty\quad(\alpha<1), \]

then all solutions of equation (1) have an infinite number of zeros. Let us note that it can be proved that, if \(\int^\infty (px^\alpha)\,dx<\infty\) \((0\leqslant\alpha<1)\), then the solutions have few zeros; the series

\[ \sum_{n=1}^{\infty}\frac{1}{(a_{n+1}-a_n)^{1-\alpha}}, \]

where \(a_n\) is the sequence of zeros of an arbitrary solution, converges.

Theorem 4. If there exists a differentiable function \(r(x)>0\) such that

\[ \lim_{x\to+\infty}\int_a^x \left[\left(px-\frac{1}{4x}\right)r-\frac{xr'^2}{4r}\right]dx=+\infty, \]

then all solutions of equation (1) have an infinite number of zeros for \(a\leqslant x<+\infty\).

The proof is analogous to the proof of Theorem 3; instead of the function \(\theta r\), one must consider \(u=(\theta-1/2x)rx\).

Corollary. If for the function \(r(x)\) we take \((\ln x)^\alpha,\ \alpha<1\), then we obtain the sufficient condition for oscillation:

\[ \int_a^\infty \left(px-\frac{1}{4x}\right)(\ln x)^\alpha dx=+\infty. \]

Theorem 5. If the solutions of the equation \(y''+\varphi(x)y=0\) have on \([a,\infty)\) an infinite number of zeros, and moreover
\[ \left|\int^\infty \varphi\,dx\right|<\infty,\qquad \left|\int^\infty p\,dx\right|<\infty \]
and
\[ \int_x^\infty p\,dx \geq \int_x^\infty \varphi\,dx \geq 0, \]
then the solutions of equation (1) also have an infinite number of zeros.

Suppose the contrary. Then the equation \(\theta' + \theta^2 + p = 0\) has an indefinitely continuable solution. Making the substitution
\[ \psi(x)=\theta-\int_x^\infty p\,dx, \tag{*} \]
we obtain
\[ \psi' + \left[\psi+\int_x^\infty p\,dx\right]^2=0. \tag{**} \]

Among the indefinitely continuable solutions of this equation there is a positive one; otherwise, since by virtue of (*) \(\psi\) is a monotonically nonincreasing function, all \(\theta(x)\), by virtue of (), would be negative beginning with some \(x\), which is impossible \((^4)\). Denote the positive solution by \(\psi_1\). The function
\[ \sigma(x)=\psi_1+\int_x^\infty \varphi\,dx \]
must satisfy the condition \(\sigma' + \sigma^2+\varphi \leq 0\), but this is impossible, since it was assumed that the equation \(y''+\varphi(x)y=0\) has oscillating solutions. From this theorem, for example, it follows that if
\[ \int_x^\infty p\,dx \geq \frac{1+\varepsilon}{4x},\qquad \varepsilon>0, \]
then all solutions of equation (1) are oscillating.

Theorem 6. If for some \(x_0\) and \(\alpha<1\) the inequality
\[ \int_{x_0}^{x} p x^\alpha dx \geq C>-\infty \]
holds, and
\[ \int^\infty \frac{g_+^2(x)}{x^\alpha}\,dx=\infty,\qquad \int^\infty \frac{g_-^2(x)}{x^\alpha}\,dx=\infty, \]
where
\[ g(x)=-\int_{x_0}^{x} px^\alpha dx+ \left[\frac{\alpha^2}{4(\alpha-1)}-\frac{\alpha}{2}\right]x^{\alpha-1} \]
(\(g_+\) and \(g_-\) are, respectively, the positive and negative parts of \(g(x)\)), then the solutions of equation (1) have an infinite number of zeros.

Suppose that the conditions of the theorem are satisfied, but the solutions of (1) have a finite number of zeros. Then the equation \(\theta' + \theta^2 + p = 0\) has an indefinitely continuable solution. The function \(u=\theta x^\alpha\) satisfies the equation
\[ u' + \frac{(u-\alpha x^{\alpha-1}/2)^2}{x^\alpha} +px^\alpha-\frac{\alpha^2 x^{\alpha-2}}{4}=0 \]
or
\[ u(x)-u(x_0)+\int_{x_0}^{x}\frac{(u-\alpha x^{\alpha-1}/2)^2}{x^\alpha}\,dx +\int_{x_0}^{x}px^\alpha dx -\frac{\alpha^2x^{\alpha-1}}{4(\alpha-1)} +\frac{\alpha^2x_0^{\alpha-1}}{4(\alpha-1)}=0. \]

Among its solutions extendable without bound there will be one such \(u_1(x)\) that

\[ \int^\infty \frac{(u_1-\alpha x^{\alpha-1}/2)^2}{x^\alpha}\,dx<\infty, \]

otherwise all \(u(x)\to -\infty\) and \(\theta(x)\) would be \(<0\) for large \(x\), and this is impossible by (4).

We have

\[ u_1(x)=\lambda+\int_x^\infty \frac{(u_1-\alpha x^{\alpha-1}/2)^2}{x^\alpha}\,dx+ \frac{\alpha^2 x^{\alpha-1}}{4(\alpha-1)}\int_{x_0}^x -p x^\alpha\,dx, \]

where

\[ \lambda=u(x_0)-\int_{x_0}^\infty \frac{(u_1-\alpha x^{\alpha-1}/2)^2}{x^\alpha}\,dx- \frac{\alpha^2 x_0^{\alpha-1}}{4(\alpha-1)}. \]

Suppose \(\lambda\geq 0\). Then

\[ \int_{x_0}^x \frac{(u_1-\alpha x^{\alpha-1}/2)^2}{x^\alpha}\,dx = \int_{x_0}^x \frac{1}{x^\alpha} \left[ \lambda+g(x)+ \int_x^\infty \frac{(u_1-\alpha x^{\alpha-1}/2)^2}{x^\alpha}\,dx \right]^2 dx \geq \int_{x_0}^x \frac{g_+^2}{x^\alpha}\,dx \to +\infty . \]

If, however, \(\lambda<0\), then

\[ \int_{x_0}^x \frac{(u_1-\alpha x^{\alpha-1}/2)^2}{x^\alpha}\,dx \geq \int_{x_0}^x \frac{[|\lambda|/2+g_-(x)]^2}{x^\alpha}\,dx \geq \int_{x_0}^x \frac{g_-^2}{x^\alpha}\,dx \to +\infty, \]

but

\[ \int_{x_0}^\infty \frac{(u_1-\alpha x^{\alpha-1}/2)^2}{x^\alpha}\,dx<\infty . \]

The contradiction obtained proves the theorem.

The conditions of the theorem, for example, will be satisfied if

\[ \int_{x_0}^x px\,dx \]

is a periodic function. We note that the requirement of divergence of

\[ \int^\infty \frac{g_+^2}{x^\alpha}\,dx \]

and

\[ \int^\infty \frac{g_-^2}{x^\alpha}\,dx \]

cannot be omitted.

Theorem 7. If the solutions of the equations

\[ y''+f_i(x)y=0 \qquad (i=1,2,\ldots,n) \]

are nonoscillatory, then the solutions of the equation

\[ y''+\left(\sum_{i=1}^n c_i f_i\right)y=0 \]

are also nonoscillatory for \(c_i\geq 0\), \(\sum_{i=1}^n c_i\leq 1\).

Indeed, from the nonoscillation of the solutions of these equations it follows that for each \(i\) there exist \(\theta_i\) such that

\[ \theta_i' + \theta_i^2 + f_i \leq 0 . \]

Then

\[ \theta=\sum_{i=1}^n c_i\theta_i \]

satisfies the condition

\[ \theta' + \theta^2 + \sum_{i=1}^n c_i f_i \leq 0, \]

whence nonoscillation follows. For the case \(n=2\), see \((^2)\).

Moscow State University
named after M. V. Lomonosov

Received
5 X 1956

References

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