MATHEMATICS

L. D. NIKOLENKO

SOME CRITERIA FOR NONOSCILLATION OF A FOURTH-ORDER DIFFERENTIAL EQUATION

(Presented by Academician N. N. Bogolyubov on 15 XII 1956)

In the present note, in a certain sense, the results obtained in article (1) for a second-order differential equation are generalized to the fourth-order differential equation

\[ \frac{d^{4}y}{dx^{4}}+\frac{d}{dx}\,[a(x)y']+b(x)y=0,\qquad x_{0}\leq x<\infty \tag{1} \]

(the functions \(a(x)\) and \(b(x)\) are assumed to be real and continuous for \(x\geq x_{0}\)).

Following Sternberg \((^{2})\), we shall call a fourth-order equation nonoscillatory if, beginning with some sufficiently large \(x_{0}\), every nontrivial solution of it has, for \(x\geq x_{0}\), no more than one double zero. Correspondingly, we shall call a fourth-order equation oscillatory if, for every finite \(x_{0}\), there exists a nontrivial solution of this equation having, for \(x\geq x_{0}\), more than one double zero. The point \(x=x_{1}\) is called a double zero of the solution \(y(x)\) if \(y(x_{1})=y'(x_{1})=0\).

In obtaining the nonoscillation criteria formulated below for equation (1), a lemma is used which is analogous to Sturm’s comparison theorem for a second-order differential equation.

Lemma 1. If \(a_{1}(x)\leq a_{2}(x)\), \(b_{1}(x)\geq b_{2}(x)\), beginning with some sufficiently large \(x\), then from the nonoscillation of the equation

\[ \frac{d^{4}y}{dy^{4}}+\frac{d}{dx}\,[a_{2}(x)y']+b_{2}(x)y=0 \]

there follows the nonoscillation of the equation

\[ \frac{d^{4}y}{dx^{4}}+\frac{d}{dx}\,[a_{1}(x)y']+b_{1}(x)y=0. \]

The proof of the lemma is based on Sternberg’s criterion \((^{2})\). From Lemma 1 there immediately follows the following simple criterion for nonoscillation of equation (1):

If, beginning with some sufficiently large \(x\), \(a(x)\leq 0\), \(b(x)\geq 0\), then equation (1) is nonoscillatory.

It is known that the differential equation \(y''+\frac{\alpha}{x^{2}}y=0\) is oscillatory for \(\alpha>1/4\) and nonoscillatory for \(\alpha\leq 1/4\); therefore, in article (1), when studying the oscillation of a second-order differential equation, the equation

\[ y''+\frac{1}{4x^{2}}y=0. \]

Study of the fourth-order equation

\[ \frac{d^{4}y}{dx^{4}}+\frac{d}{dx}\left(\frac{\alpha}{x^{2}}y'\right)+\frac{\beta}{x^{4}}y=0 \]

shows that this equation is nonoscillatory for \(\beta \geq \omega(\alpha)\) and oscillatory for \(\beta<\omega(\alpha)\), where

\[ \omega(\alpha)=\frac{9}{4}\alpha-\frac{9}{16}\quad \text{for } \alpha\leq \frac{5}{2}; \qquad \omega(\alpha)=\frac{1}{4}(\alpha+2)^2\quad \text{for } \alpha\geq \frac{5}{2}. \]

In connection with this, when studying the oscillation of equation (1), we shall take, as the comparison equation, the differential equation

\[ \frac{d^{4}y}{dx^{4}}+\frac{d}{dx}\left(\frac{\alpha}{x^{2}}\frac{dy}{dx}\right)+\frac{\omega(\alpha)}{x^{4}}\,y=0, \tag{2} \]

which is nonoscillatory for every finite \(\alpha\).

In accordance with Lemma 1, the following criterion for nonoscillation of equation (1) holds.

Theorem 1. If there exists an \(\alpha\) such that, beginning with some sufficiently large \(x\),
\[ a(x)\leq \frac{\alpha}{x^{2}}, \qquad b(x)\geq \frac{\omega(\alpha)}{x^{4}}, \]
then equation (1) is nonoscillatory.

To establish a more refined criterion for nonoscillation of equation (1), we have considered the equation

\[ \frac{d^{4}y}{dx^{4}} +\frac{d}{dx}\left[\left(\frac{\alpha}{x^{2}}+\varphi(x)\right)y'\right] +\left[\frac{\omega(\alpha)}{x^{4}}+\psi(x)\right]y=0. \tag{3} \]

If

\[ \int^{\infty} x\ln x\,|\varphi(x)|\,dx<\infty, \qquad \int^{\infty} x^{3}\ln x\,|\psi(x)|\,dx<\infty, \tag{4} \]

then, as the study of the asymptotic behavior of solutions of the differential equation (3) shows, this equation is nonoscillatory for \(\alpha\neq \frac{5}{2}\). Next put

\[ \varphi(x)=\max\left\{a(x)-\frac{\alpha}{x^{2}},\,0\right\}, \qquad \psi(x)=\min\left\{b(x)-\frac{\omega(\alpha)}{x^{4}},\,0\right\}. \tag{5} \]

If these functions satisfy conditions (4), then for \(\alpha\neq \frac{5}{2}\) the differential equation (3) is nonoscillatory and, consequently, in accordance with Lemma 1, equation (1) is also nonoscillatory, since for the functions \(\varphi(x)\) and \(\psi(x)\) defined by formulas (5) the inequalities

\[ a(x)\leq \frac{\alpha}{x^{2}}+\varphi(x),\qquad b(x)\geq \frac{\omega(\alpha)}{x^{4}}+\psi(x) \]

hold.

We thus arrive at the following criterion for nonoscillation of equation (1).

Theorem 2. If, for some finite \(\alpha\neq \frac{5}{2}\),

\[ \int^{\infty} x\ln x\max\left\{a(x)-\frac{\alpha}{x^{2}},\,0\right\}dx<\infty, \]

\[ \int^{\infty} x^{3}\ln x\left|\min\left\{b(x)-\frac{\omega(\alpha)}{x^{4}},\,0\right\}\right|dx<\infty, \]

then equation (1) is nonoscillatory.

Remark. The differential equation (1) is also nonoscillatory if

\[ \int^\infty x \ln^3 x \max \left\{ a(x)-\frac{5}{2x^2},\,0 \right\}\,dx<\infty, \]

\[ \int^\infty x^3 \ln^3 x \left|\min \left\{ b(x)-\frac{81}{16x^4},\,0 \right\}\right|\,dx<\infty \]

(this criterion corresponds to the case \(\alpha=5/2\)).

Institute of Mathematics
Academy of Sciences of the Ukrainian SSR

Received
4 XII 1956

References

\(^{1}\) L. D. Nikolenko, Ukr. Math. Zh., 7, 127 (1955).
\(^{2}\) R. Sternberg, Duke Math. J., 19, 311 (1952).