MATHEMATICS

I. T. Kiguradze

ON THE OSCILLATION OF SOLUTIONS OF SOME ORDINARY DIFFERENTIAL EQUATIONS

(Presented by Academician I. G. Petrovskii, 20 XII 1961)

In the present paper we consider the differential equation

\[ \frac{d^m u}{dt^m}+F(u^2,t)u=0, \tag{1} \]

where \(m \geqslant 2\), and the function \(F(y,t)\) is continuous for \(0 \leqslant t,\ y < \infty\) and satisfies the following conditions:

\[ F(y,t)\geqslant 0;\qquad F(y_1,t)\leqslant F(y_2,t),\quad \text{when } y_1<y_2. \tag{2} \]

We shall say that equation (1) is strictly nonlinear if, for large \(y\), \(F(y,t)\), in addition to (2), satisfies the condition

\[ \frac{F(y^2,t)}{\varphi(y)}\geqslant F(c^2,t),\qquad c<0, \tag{3} \]

where the function \(\varphi(y)\) is absolutely continuous in the interval \((0,\infty)\) and

\[ \varphi(y)>0,\quad \varphi'(y)\geqslant 0,\quad \int^\infty \frac{dy}{y\varphi(y)}<\infty. \tag{4} \]

A solution \(u(t)\) of equation (1) is called oscillatory if it has an infinite number of zeros, and otherwise nonoscillatory. In the present paper some criteria for the oscillation of solutions of equation (1) are established.

Lemma 1. Let \(u(t)\) be a continuous, nonnegative function in the interval \((0,\infty)\), and suppose it has continuous derivatives up to order \(m\) inclusive, which preserve their sign in this interval. If \(u^{(m)}(t)\leqslant 0\), then there exists a number \(p\), \(0\leqslant p\leqslant \left[\frac{m-1}{2}\right]\), such that

\[ u^{(k)}(t)\geqslant 0\quad (k=0,1,\ldots,l), \]
\[ (-1)^{m+k-1}u^{(k)}(t)\geqslant 0\quad (k=l+1,\ldots,m); \tag{5} \]

\[ 0\leqslant u^{(l)}(t)\leqslant \frac{l!}{t^l}u(t), \tag{6} \]

where

\[ l=2p+\frac{1+(-1)^m}{2}. \]

Let us indicate how the second part of the lemma is proved. According to (6), the function \(u^{(l)}(t)\) is nonnegative and nonincreasing. Therefore we have the inequality \(tu^{(l)}(t)\leqslant u^{(l-1)}(t)\), by virtue of which, from the identities

\[ tu^{(l-k)}(t)=(1+k)u^{(l-k-1)}(t)-(1+k)u^{(l-k-1)}(0)- \]

\[ -\int_0^t \left[ku^{(l-k)}(\tau)-\tau u^{(l-k+1)}(\tau)\right]\,d\tau \quad (k=1,2,\ldots,l-1) \]

we prove by induction that

\[ tu^{(l-k)}(t)\leqslant (1+k)u^{(l-k-1)}(t)\quad (k=1,2,\ldots,l-1), \]

whence inequality (6) follows immediately.

Let \(u(t)\) be some solution of equation (1). If \(u(t)\) or \(-u(t)\) satisfies inequalities (5), we shall say that \(u(t)\) belongs to the class \(A_p\).

From the lemma formulated above it is clear that every nonoscillatory solution of equation (1) belongs to one of the classes \(A_p\),
\[ 0 \leq p \leq \left[\frac{m-1}{2}\right]. \]

Theorem 1. If \(m\) is odd, then, in order that all solutions of equation (1) belonging to the class \(A_0\) tend to zero as \(t \to \infty\), it is necessary and sufficient that
\[ \int^\infty F(c^2,t)t^{m-1}\,dt=\infty \tag{7} \]
for every \(c>0\).

Proof. If \(u(t)\in A_0\), \(u(t)>c>0\) for \(t\geq 0\), then, taking (5) into account, from the equality
\[ \sum_{k=0}^{m-1}(-1)^k A_{m-1}^k t^{m-k-1}u^{(m-k-1)} +\int_{t_0}^{t} F(u^2,\tau)\tau^{m-1}u\,d\tau=c_0 \]
we find
\[ \int_{t_0}^{\infty} F(c^2,t)t^{m-1}\,dt<\frac{c_0}{c}, \]
which contradicts condition (7). The contradiction obtained shows that if the integral
\[ \int^\infty F(c^2,t)t^{m-1}\,dt \]
diverges for every \(c>0\), then all solutions of equation (1) belonging to the class \(A_0\) tend to zero as \(t\to\infty\).

It can be shown that if, for some \(c>0\), the integral
\[ \int^\infty F(c^2,t)t^{m-1}\,dt \]
converges, then the integral equation
\[ u(t)=\frac{c}{2}+\frac{1}{(m-1)!}\int_t^\infty (\tau-t)^{m-1}F(u^2,\tau)u(\tau)\,d\tau \]
and, consequently, the differential equation (1) has a solution \(u(t)\) tending to a finite, nonzero limit as \(t\to\infty\). Taking into account that \(u(t)\in A_0\), the necessity of condition (7) becomes obvious.

For \(m=3\) and \(F(y,t)\equiv F(t)\), Theorem 1 implies Villari’s theorem \((^1)\). The following theorem is proved in an analogous way.

Theorem 2. If \(m\) is even, then, in order that all solutions of equation (1) belonging to the class \(A_0\) be bounded, it is necessary and sufficient that condition (7) hold for every \(c>0\).

For the case \(m=2\) the analogous theorem belongs to Nehari \((^2)\).

Following Kneser \((^3)\), we shall say that condition (A) is satisfied for equation (1) if equation (1) has an oscillatory solution and every nonoscillatory solution tends monotonically to zero as \(t\to\infty\).

It is clear from the lemma that, in the case of even \(m\), the fulfillment of condition (A) means that all solutions of equation (1) are oscillatory.

Theorem 3. Let \(\varphi(t)\) be some function satisfying conditions (4). If, for every \(c>0\),
\[ \int^\infty F(c^2,t)\frac{t^{m-1}}{\varphi(t)}\,dt=\infty, \tag{8} \]
then condition (A) is satisfied for equation (1).

Proof. Let \(m\) be odd, \(u(t)\in A_0\). According to Theorem 1, \(\lim_{t\to\infty}u(t)=0\). Thus, to prove the theorem it is enough to show that for even \(m\) every solution is oscillatory, while for odd \(m\) it either belongs to the class \(A_0\), or is oscillatory. Suppose the contrary. Let \(u(t)\in A_p\), \(p\ge \dfrac{1+(-1)^{m-1}}{2}\). Without loss of generality, we may assume that \(u(t)>0\) and all its derivatives up to order \(m-1\) inclusive preserve their sign on the interval \((0,\infty)\).

Multiplying equation (1) by \(t^{m-1}/\varphi(t)u\) and integrating, we find

\[ \frac{f(t)}{\varphi(t)u} -\int_{t_0}^{t} f(\tau)\,d(\varphi(\tau)u)^{-1} +\int_{t_0}^{t} F(u^2,\tau)\frac{\tau^{m-1}}{\varphi(\tau)}\,d\tau = c_1+A_{m-1}^{m-l}\int_{t_0}^{t}\frac{\tau^{l-1}u^{(l)}}{\varphi(\tau)u}\,d\tau, \tag{9} \]

where

\[ f(t)=\sum_{k=0}^{m-l-1}(-1)^k A_{m-1}^k t^{m-k-1}u^{(m-k-1)}. \]

By virtue of (5) and (6), from (9) it follows that

\[ \int_{t_0}^{t} F(u^2,\tau)\frac{\tau^{m-1}}{\varphi(\tau)}\,d\tau \le c_1+(m-1)!\,l\int_{t_0}^{\infty}\frac{d\tau}{\tau\varphi(\tau)} = c_2. \tag{10} \]

Since \(u(t)>c>0\), hence we obtain

\[ \int_{t_0}^{\infty} F(c^2,\tau)\frac{\tau^{m-1}}{\varphi(\tau)}\,d\tau>\infty, \]

which contradicts condition (8).

The result obtained from Theorem 3 for the equation

\[ \frac{d^m u}{dt^m}+F(t)u=0, \tag{1_0} \]

is a generalization of the theorems of Kneser \((^3)\), Villari \((^1)\), and Anan’eva—Balaganskii \((^4)\).

Applying the results obtained by V. A. Kondrat’ev in \((^5)\) for equation \((1_0)\), one can prove the following theorem:

Theorem 4. 1) If

\[ \int^{\infty} F(t)\frac{t^{m-1}}{\varphi(t)}\,dt=\infty,\qquad F(t)\ge 0, \]

then condition (A) is satisfied for equation \((1_0)\).

2) If

\[ \int^{\infty}|F(t)|t^{m-1}\,dt>\infty, \]

then the solutions of equation \((1_0)\) are nonoscillatory.

3) If

\[ \int^{\infty} F(t)\frac{t^{m-1}}{\varphi(t)}\,dt=-\infty,\qquad m\ge 3,\qquad F(t)\le 0, \]

then there exists a fundamental system consisting of

\[ \frac{3+(-1)^m}{2} \]

nonoscillatory and

\[ m-\frac{3+(-1)^m}{2} \]

oscillatory solutions.

For \(\varphi(t)\equiv t\) this theorem is proved in V. A. Kondrat’ev’s work \((^5)\).

Theorem 5. If equation (1) is strictly nonlinear, then condition (7) is necessary and sufficient for condition (A) to hold for equation (1).

Proof. Necessity follows from Theorems 1 and 2. Let us prove sufficiency. Suppose the contrary. Let, under condition (7), equation (1) have a solution \(u(t)\in A_p\), \(p\ge \dfrac{1+(-1)^{m-1}}{2}\). It is clear that \(\lim_{t\to\infty}u(t)=\infty\); therefore, for large \(t\), (3) will be satisfied.

From equation (1) we easily find

\[ \frac{f(t)}{u\varphi(u)} -\int_{t_0}^{t} f(\tau)\,d(u\varphi(u))^{-1} +\int_{t_0}^{t}\frac{F(u^2,\tau)}{\varphi(u)}\,\tau^{m-1}\,d\tau = c_3+ A_{m-1}^{m-l}\int_{t_0}^{t}\frac{\tau^{l-1}u^{(l)}}{u\varphi(u)}\,d\tau . \tag{11} \]

Taking into account (3), (5), and (6), from (11) we obtain

\[ \int_{t_0}^{\infty} F(c^2,\tau)\,\tau^{m-1}\,d\tau \leq c_3+ (m-1)!\int_{u(t_0)}^{\infty}\frac{du}{u\varphi(u)}<\infty, \]

which contradicts condition (7).

When \(m=2\), Theorem 5 yields the theorems of Atkinson \({}^{6}\) and Jones \({}^{7}\).

We shall say that for equation (1) condition (B) is satisfied if every solution \(u(t)\) either oscillates, or

\[ \lim_{t\to\infty} u^{(m-1)}(t)=0. \]

The following theorem is easily proved:

Theorem 6. For condition (B) to hold, it is necessary and sufficient that

\[ \int^{\infty} F(c^2t^{2m-2},t)\,t^{m-1}\,dt=\infty \]

for every \(c>0\).

The author expresses deep gratitude to L. G. Magnaradze for valuable advice in the preparation of the present article.

CITED LITERATURE

\({}^{1}\) C. Villari, Ann. mat. pura ed appl., 51, 301 (1960).
\({}^{2}\) Z. Nehari, Trans. Am. Math. Soc., 95, 101 (1960).
\({}^{3}\) Kneser, Math. Ann., 42 (3), 409 (1893).
\({}^{4}\) G. V. Ananeva, V. I. Balaganskii, UMN, 14, No. 1, 135 (1959).
\({}^{5}\) V. A. Kondratev, Tr. Mosk. matem. obshch., 10, 418 (1961).
\({}^{6}\) F. V. Atkinson, Pacific J. Math., 5, 643 (1955).
\({}^{7}\) J. Jones, Quart. J. Math., 7, No. 28, 306 (1956).