GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 5, 1999, 401-414

OSCILLATION AND NONOSCILLATION CRITERIA FOR A SECOND ORDER LINEAR EQUATION

T. CHANTLADZE, N. KANDELAKI, AND A. LOMTATIDZE

Abstract. New oscillation and nonoscillation criteria are established for the equation

u′′ + p(t)u = 0,

where p : ]1, +∞[ → R is the locally integrable function. These criteria generalize and complement the well known criteria of E. Hille, Z. Nehari, A. Wintner, and P. Hartman.

We shall consider the equation

u′′ + p(t)u = 0,  (0.1)

where the function p : ]1, +∞[ → R is Lebesgue integrable on each finite segment from [1, +∞[ . By a solution of equation (0.1) is understood a function u : [1, +∞[ → R which is absolutely continuous together with its first derivative on each finite segment from [1, +∞[ and which satisfies almost everywhere equation (0.1). Equation (0.1) is called oscillatory if there exists its solution with an infinite number of zeros and nonoscillatory otherwise.

Below we shall give some new oscillation and nonoscillation criteria for equation (0.1). The paper is organized as follows: the main results are formulated in Section 1; Section 2 contains remarks and comments; the auxiliary propositions are presented in Section 3, while the proofs of the main results can be found in Section 4.

Before we proceed to the formulation of the main results we want to introduce some notation.

Let

c(t) = \frac{1}{t} ∫_{1}^{t} ∫_{1}^{s} p(ξ) dξ ds for t ≥ 1.

401

1991 Mathematics Subject Classification. 34C10, 34K15, 34K25.

Key words and phrases. Second order linear equation, oscillatory equation, nonoscillatory equation.

1072-947X/99/0900-0401$16.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

402 T. CHANTLADZE, N. KANDELAKI, AND A. LOMTATIDZE

Below it will always be assumed that there exists a finite limit

c_{0} =^{def} lim_{t→+∞} c(t).  (0.2)

We set

Q(t) = t(c_{0} − ∫_{1}^{t} p(s) ds), H(t) = \frac{1}{t} ∫_{1}^{t} s^{2}p(s) ds for t ≥ 1,

Q_{∗} = lim inf_{t→+∞} Q(t), Q^{∗} = lim sup_{t→+∞} Q(t),

H_{∗} = lim inf_{t→+∞} H(t), H^{∗} = lim sup_{t→+∞} H(t).

1. Formulation of the Main Results

Theorem 1.1. Let

lim sup_{t→+∞} \frac{t}{ln t}(c_{0} − c(t)) > \frac{1}{4} .  (1.1)

Then equation (0.1) is oscillatory.

Corollary 1.1. Let Q_{∗} > −∞ and

lim sup_{t→+∞} \frac{1}{ln t} ∫_{1}^{t} sp(s) ds > \frac{1}{4} .

Then equation (0.1) is oscillatory.

Corollary 1.2. Let

lim inf_{t→+∞} [Q(t) + H(t)] > 1/2 .  (1.2)

Then equation (0.1) is oscillatory.

Theorem 1.2. Let

lim sup_{t→+∞} [Q(t) + H(t)] > 1.  (1.3)

Then equation (0.1) is oscillatory.

Corollary 1.2 readily implies (see equality (4.1) below) that if Q_{∗} > \frac{1}{4} then equation (0.1) is oscillatory, while from Theorem 1.1 it follows (see equalities (2.2) and (2.3) below) that the condition H_{∗} > \frac{1}{4} also guarantees the oscillation of equation (0.1) (see also [11]). Hence we shall limit our consideration to the case with Q_{∗} ≤ \frac{1}{4} and H_{∗} ≤ \frac{1}{4}.

OSCILLATION AND NONOSCILLATION CRITERIA 403

Theorem 1.3. Let either

0 ≤ Q_{∗} ≤ 1/4 and H^{∗} > 1/2 (1 + \sqrt{1 − 4Q_{∗}}),  (1.4)

or

0 ≤ H_{∗} ≤ 1/4 and Q^{∗} > 1/2 (1 + \sqrt{1 − 4H_{∗}}).  (1.5)

Then equation (0.1) is oscillatory.

Theorem 1.4. Let

0 ≤ Q_{∗} ≤ 1/4 and 0 ≤ H_{∗} ≤ 1/4 .  (1.6)

Then each of the conditions

Q^{∗} > Q_{∗} + 1/2 (\sqrt{1 − 4Q_{∗}} + \sqrt{1 − 4H_{∗}})  (1.7)

and

H^{∗} > H_{∗} + 1/2 (\sqrt{1 − 4Q_{∗}} + \sqrt{1 − 4H_{∗}})  (1.8)

guarantees the oscillation of equation (0.1).

When condition (1.6) is fulfilled, Theorem 1.2 can be formulated in a more precise way.

Theorem 1.5. Let condition (1.6) be fulfilled and

lim sup_{t→+∞} [Q(t) + H(t)] > H_{∗} + Q_{∗} + 1/2 (\sqrt{1 − 4Q_{∗}} + \sqrt{1 − 4H_{∗}}).

Then equation (0.1) is oscillatory.

To conclude the paragraph, we shall give two theorems on nonoscillation. In [3] and [12] it was respectively proved that if

−3/4 < Q_{∗} and Q^{∗} < 1/4 ,  (1.9)

or

−3/4 < H_{∗} and H^{∗} < 1/4 ,  (1.10)

then equation (0.1) is nonoscillatory. The theorem below complements these results.

Theorem 1.6. Let either

−∞ < Q_{∗} ≤ −3/4 and Q^{∗} < Q_{∗} − 1 + \sqrt{1 − 4Q_{∗}},  (1.11)

or

−∞ < H_{∗} ≤ −3/4 and H^{∗} < H_{∗} − 1 + \sqrt{1 − 4H_{∗}}.  (1.12)

Then equation (0.1) is nonoscillatory.

Our next theorem is an attempt to reverse Corollary 1.1.

Theorem 1.7. Let there exist a finite limit

p_{0} =^{def} lim_{t→+∞} \frac{1}{ln t} ∫_{1}^{t} sp(s) ds < \frac{1}{4}  (1.13)

and

G^{∗} < G_{∗} + \sqrt{1 − 4p_{0}},

where

G_{∗} = lim inf_{t→+∞} ln t[\frac{1}{ln t} ∫_{1}^{t} sp(s) ds − p_{0}],

G^{∗} = lim sup_{t→+∞} ln t[\frac{1}{ln t} ∫_{1}^{t} sp(s) ds − p_{0}].

Then equation (0.1) is nonoscillatory.

Corollary 1.3. Let

−∞ < lim sup_{t→+∞} ∫_{1}^{t} sp(s) ds < lim inf_{t→+∞} ∫_{1}^{t} sp(s) ds + 1 < +∞.

Then equation (0.1) is nonoscillatory.

2. Remarks

Among a great number of papers dealing with the oscillation of equation (0.1) we shall mention only those having a direct connection with the aboveformulated theorems.

A. Wintner [2] and P. Hartman [4] proved respectively that if lim_{t→+∞} c(t) = +∞ or

−∞ < lim inf_{t→+∞} c(t) < lim sup_{t→+∞} c(t) ≤ +∞,

then equation (0.1) is oscillatory. Hence the case with the existence of the finite limit (0.2) seems to us the most interesting one to investigate.

Frequently, equation (0.1) is considered under the assumption that there exists a finite limit

∫_{1}^{+∞}p(s) ds =^{def} lim_{t→+∞} ∫_{1}^{t} p(s) ds,  (2.1)

since in that case there also exist limit (0.2) (and c_{0} = ∫_{1}^{+∞}p(s) ds). However

it is clear that (2.1) is not the necessary condition for (0.2). An example will be given below to show that Theorem 1.1 also covers the case with

p_{∗} = lim inf_{t→+∞} ∫_{1}^{t} p(s) ds < c_{0} < lim sup_{t→+∞} ∫_{1}^{t} p(s) ds = p^{∗}.

In the particular case, where p(t) ≥ 0 for t > 1, Corollary 1.1 was proved in [10], while Theorems 1.3 and 1.4 in [12]. In order that the function Q be bounded from below, it is necessary that c_{0} = p^{∗}. Since

c(t) = ∫_{1}^{t} p(s) ds − \frac{H(t)}{t} − \frac{1}{t} ∫_{1}^{t} \frac{1}{s} H(s) ds for t ≥ 1,

for the function H to be bounded from below it is necessary that c_{0} = p_{∗}. Therefore condition (2.1) is necessary for Theorem 1.4, while the conditions c_{0} = p^{∗} and c_{0} = p_{∗} are necessary for the fulfilment of conditions (1.4) and (1.5) of Theorem 1.3, respectively. Note that the oscillation criterion Q_{∗} > \frac{1}{4} (originating from E. Hille [1]) is somewhat more general than in [5], since it does not demand that (2.1) be fulfilled.

Theorems 1.3 and 1.4 generalize and improve the well-known oscillation criterion from E. Hille [1]. In this paper, the case is considered when p(t) ≥ 0 for t > 1 and an example is given, showing that the constant 1 in the oscillation criterion Q^{∗} > 1 cannot be decreased. However, in this example Q_{∗} = 0 and H_{∗} = 0. If Q_{∗} > 0 or H_{∗} > 0, then, as follows from Theorems 1.3 and 1.4, the constant 1 can be decreased.

One can easily verify that for any pair of numbers (x_{0}, y_{0}), where x_{0} ≤ y_{0}, there is a function p : ]1, +∞[ → R such that (0.2) holds and Q_{∗} = x_{0}, Q^{∗} = y_{0} (H_{∗} = x_{0} and H^{∗} = y_{0}). Therefore Theorems 1.3–1.6 are meaningful.

For (1.9) and (1.11) condition (2.1) is necessary.

By using the equality

c(t) = c(τ) + ∫_{τ}^{t} \frac{ln s}{s^{2}} [\frac{1}{ln s} ∫_{1}^{s} ξp(ξ) dξ] ds for t, τ > 1  (2.2)

one can readily find that if there exists a finite limit lim_{t→+∞} \frac{1}{ln t} ∫_{1}^{t} sp(s) ds,

then (0.2) holds. Thus for Theorem 1.7 condition (0.2) is necessary.

As is known (see [8] and [9]), for equation (0.1) to be oscillatory when p(t) = \frac{μ}{t} sin t for t ≥ 1, it is necessary and sufficient that |μ| < \frac{1}{\sqrt{2}}. This example shows that only condition (1.13) is not enough for the nonoscillation of equation (0.1).

It might seem at first glance that for conditions (1.10) and (1.12) it is not required that (0.2) be fulfilled. However, using equalities (2.2) and

\frac{1}{ln t} ∫_{1}^{t} sp(s) ds = \frac{1}{ln t} H(t) + \frac{1}{ln t} ∫_{1}^{t} \frac{1}{s} H(s) ds for t > 1 (2.3)

it is easy to show that if the function H is bounded (from both sides), then (0.2) holds. A similar situation arises for the oscillation criterion H_{∗} > \frac{1}{4} (originating from Z. Nehari [6]). By (2.2) and (2.3) one can easily verify that in that case either (0.2) holds or this limit is equal to +∞. However in the latter case, equation (0.1) will be oscillatory by virtue of A. Wintner’s above mentioned theorem [2]. Therefore for this criterion condition (0.2) is also necessary in a certain sense.

Finally, if we rewrite equation (0.1) as

v′′(x) = −\frac{1}{λ^{2}} t^{2(1−λ)}[p(t) − \frac{1 − λ^{2}}{4t^{2}} ] v(x),

then, after transforming u(t) = \frac{t^{\frac{1−λ}{2}}}{\sqrt{λ}} v(x), x = t^{λ}, λ > 0, and applying Theorems 1.3 and 1.4, we can generalize and improve the oscillation criteria of Z. Nehari [6].

Example. Let λ ≠ 0 and γ be real numbers,

g(t) = −γ \frac{ln t}{t} + \frac{λ}{1 + ln t} (sin ln^{2} t − 1) for t ≥ 1

and

p(t) = 2g′(t) + tg′′(t) for t ≥ 1.

It is easy to verify that

∫_{1}^{t} p(s) ds = g(t) + tg′(t) + γ, c(t) = g(t) + \frac{t − 1}{t} (γ − λ) + λ for t ≥ 1,

lim inf_{t→+∞} ∫_{1}^{t} p(s) ds = γ − 2|λ|, lim sup_{t→+∞} ∫_{1}^{t} p(s) ds = γ + 2|λ|, c_{0} = γ,

\frac{t}{ln t} (c_{0} − c(t)) = −\frac{t}{ln t} g(t) + \frac{1}{ln t} (γ − λ) for t ≥ 1,

−∞ = lim inf_{t→+∞} \frac{t}{ln t} (c_{0} − c(t)), lim sup_{t→+∞} \frac{t}{ln t} (c_{0} − c(t)) = γ for λ < 0,

γ = lim inf_{t→+∞} \frac{t}{ln t} (c_{0} − c(t)), lim sup_{t→+∞} \frac{t}{ln t} (c_{0} − c(t)) = +∞ for λ > 0.

Thus, by Theorem 1.1, if λ > 0 or λ < 0 and γ > \frac{1}{4} equation (0.1) is oscillatory.

3. Some Auxiliary Propositions

In this paragraph we establish some properties of solutions of equation (0.1). Throughout this paper it is assumed that the function p : ]1, +∞[ → R is Lebesgue integrable on each finite segment from [1, +∞[ . For the convenience of reference we shall give one proposition without proving it (see, for example, [7], Lemma 7.1, p. 365).

Lemma 3.1. Let equation (0.1) be nonoscillatory and u(t) ≠ 0 for t ≥ a be its some solution. Then

∫^{+∞}ρ^{2}(s) ds < +∞

and

ρ(t) = c_{0} − ∫_{1}^{t} p(s) ds + ∫_{t}^{+∞}ρ^{2}(s) ds for t ≥ a,  (3.1)

where

ρ(t) = \frac{u′(t)}{u(t)} for t ≥ a.  (3.2)

Lemma 3.2. Let equation (0.1) be nonoscillatory and 0 ≤ Q_{∗} ≤ \frac{1}{4}. Then for each solution u of equation (0.1) the estimate

lim inf_{t→+∞} \frac{tu′(t)}{u(t)} ≥ \frac{1}{2} (1 − \sqrt{1 − 4Q_{∗}})  (3.3)

is valid.

Proof. Let u(t) ≠ 0 for t ≥ a be some solution of equation (0.1). By Lemma 3.1 equality (3.1) is fulfilled, where the function ρ is defined by (3.2). We set

r = lim inf_{t→+∞} tρ(t).

If r = +∞, then there is nothing to prove. Therefore it will be assumed that r < +∞. If Q_{∗} = 0, then estimate (3.3) is trivial by virtue of (3.1). So it will assumed that Q_{∗} > 0. For arbitrary ε ∈]0, Q_{∗}[ we choose t_{ε} > a such that

Q(t) > Q_{∗} − ε for t ≥ t_{ε}.  (3.4)

Now from (3.1) we have tρ(t) > Q_{∗} −ε for t ≥ t_{ε}. Hence we readily conclude that r ≥ Q_{∗}. Choose t_{1ε} > t_{ε} such that tρ(t) > r − ε for t ≥ t_{1ε}. Taking this and inequality (3.4) into account, from (3.1) we find

tρ(t) ≥ Q_{∗} − ε + (r − ε)^{2} for t ≥ t_{1ε}.

Therefore r ≥ Q_{∗} − ε + (r − ε)^{2} and, since ε was arbitrary, we have r^{2} − r + Q_{∗} ≤ 0. Now by simple calculations we conclude that (3.3) is valid.

Lemma 3.3. Let equation (0.1) be nonoscillatory and 0 ≤ H_{∗} ≤ \frac{1}{4}. Then for each solution u of equation (0.1) the estimate

lim sup_{t→+∞} \frac{tu′(t)}{u(t)} ≤ \frac{1}{2} (1 + \sqrt{1 − 4H_{∗}} )  (3.5)

holds.

Proof. Let u(t) ≠ 0 for t ≥ a be some solution of equation (0.1). We introduce the function ρ by equality (3.2) and set

M = lim sup_{t→+∞} tρ(t).

If M ≤ 0, then there is nothing to prove. Hence it will be assumed that M > 0.

Clearly, ρ′(t) = −p(t) − ρ^{2}(t) for t ≥ a. By multiplying both sides of this equality by t^{2} and integrating from τ > a to t we obtain

tρ(t) = − H(t) + \frac{1}{t} ∫_{τ}^{t} sρ(s)(2 − sρ(s)) ds +

+ \frac{1}{t} ∫_{1}^{τ} s^{2}p(s) ds + \frac{τ^{2}}{t} ρ(τ) for t > τ > a.  (3.6)

Since sρ(s)(2 − sρ(s)) ≤ 1 for s ≥ a, (3.6) implies

tρ(t) ≤ 1 − H(t) + \frac{1}{t} ∫_{1}^{τ} s^{2}p(s) ds + \frac{τ^{2}}{t} ρ(τ) for t > τ > a,

and therefore M ≤ 1 − H_{∗}. Thus estimate (3.5) is valid for H_{∗} = 0.

Now let us assume that H_{∗} > 0. For arbitrary 0 < ε < min{H_{∗}, 1 − M} we choose t_{ε} > a such that tρ(t) < M + ε, H(t) > H_{∗} − ε for t ≥ t_{ε}. Now (3.6) (for τ = t_{ε}) implies

tρ(t) ≤ −H_{∗} +ε+(M +ε)(2−M −ε)+ \frac{1}{t} ∫_{1}^{t_{ε}} s^{2}p(s) ds+ \frac{1}{t} t_{ε}^{2}ρ(t_{ε}) for t > t_{ε}.

Hence we easily find that M ≤ −H_{∗} + ε + (M + ε)(2 − M − ε) and therefore M^{2} − M + H_{∗} ≤ 0. Now by simple calculations we conclude that (3.5) holds.

Lemma 3.4. For equation (0.1) to be nonoscillatory it is necessary and sufficient that the equation

\begin{aligned}v′′ & = −\frac{1}{t^{2}} (Q^{2}(t) + 2αQ(t) + α(α − 1)) v − \frac{2}{t} (α + Q(t)) v′ \\ ( v′′ & = −\frac{1}{t^{2}} (H^{2}(t) + 2(1 − α)H(t) + α(α − 1)) v + \frac{2}{t} (H(t) − α) v′ ), \end{aligned} (3.7)

where α is some real number, be nonoscillatory.

Proof. It is easy to verify that if v is a solution of equation (3.7), then the function u defined by the equality

u(t) = t^{α}v(t) exp [ ∫_{1}^{t} \frac{Q(s)}{s} ds] for t ≥ 1

( u(t) = t^{α}v(t) exp [ − ∫_{1}^{t} \frac{H(s)}{s} ds] for t ≥ 1 )

is a solution of equation (0.1). Therefore these equations are simultaneously either oscillatory or nonoscillatory.

Lemma 3.5. For equation (0.1) to be nonoscillatory it is necessary and sufficient that the equation

v′′ = −\frac{1}{t^{2}} [G^{2}(t) − (2α − 1)G(t) + p_{0} + α(α − 1)] v +

+\frac{2}{t} (α − G(t)) v′,  (3.8)

where

G(t) = ∫_{1}^{t} sp(s) ds − p_{0} ln t for t ≥ 1  (3.9)

be nonoscillatory for some numbers α and p_{0}.

Proof. It is easy to verify that if v is a solution of equation (3.8), then the function u defined by the equality

u(t) = t^{α}v(t) exp [ − ∫_{1}^{t} \frac{G(s)}{s} ds] for t ≥ 1

is a solution of equation (0.1).

4. Proof of the Main Results

Proof of Theorem 1.1. Let us assume the opposite, i.e., that u(t) ≠ 0 for t ≥ a is a solution of equation (0.1). Equality (3.1), where the function ρ is defined by (3.2), is fulfilled by virtue of Lemma 3.1. The integration of (3.1) from a to t gives

t(c_{0} − c(t)) = ∫_{a}^{t} \frac{sρ(s)(1 − sρ(s))}{s} ds − t ∫_{t}^{+∞}ρ^{2}(s) ds + aρ(a) +

+ ∫_{1}^{a} sp(s) ds for t ≥ a.

Since sρ(s)(1 − sρ(s)) ≤ \frac{1}{4} for s ≥ a, the latter equality implies

\frac{t}{ln t} (c_{0} − c(t)) ≤ \frac{1}{4} + \frac{1}{ln t} [aρ(a) + ∫_{1}^{a} sp(s) ds] for t ≥ a,

which contradicts condition (1.1).

By virtue of the equality

\frac{t}{ln t} (c_{0} − c(t)) = \frac{1}{ln t} Q(t) + \frac{1}{ln t} ∫_{1}^{t} sp(s) ds for t > 1,

one can easily show that Corollary 1.1 is valid.

Proof of Corollary 1.2. It is easy to find that

Q(t) + H(t) = \frac{2}{t} ∫_{1}^{t} Q(s) ds + \frac{c_{0}}{t} for t ≥ 1  (4.1)

and

\frac{t}{ln t} (c_{0} − c(t)) = \frac{1}{ln t} ∫_{1}^{t} \frac{Q(s)}{s} ds + \frac{c_{0}}{ln t} =

= \frac{1}{ln t} [\frac{1}{t} ∫_{1}^{t} Q(s) ds + ∫_{1}^{t} \frac{1}{s^{2}} ( ∫_{1}^{s} Q(ξ) dξ) ds] + \frac{c_{0}}{ln t} for t > 1. (4.2)

Using (1.2) and (4.1) we obtain

lim inf_{t→+∞} \frac{1}{t} ∫_{1}^{t} Q(s) ds > \frac{1}{4} .

Hence by (4.2) we conclude that (1.1) is fulfilled. Therefore equation (0.1) will be oscillatory by virtue of Theorem 1.1.

Proof of Theorem 1.2. Let us assume the opposite, i.e., that u(t) ≠ 0, t ≥ a, is a solution of equation (0.1). Equality (3.1), where the function ρ is defined by (3.2), holds by virtue of Lemma 3.1. Clearly,

ρ′(t) = −p(t) − ρ^{2}(t) for t ≥ a.

By multiplying both sides of this equality by t^{2} and integrating from τ ≥ a to t we obtain equality (3.6).

Now, using (3.1), we find that

Q(t) + H(t) = \frac{1}{t} ∫_{τ}^{t} sρ(s)(2 − sρ(s)) ds − t ∫_{t}^{+∞}ρ^{2}(s) ds +

+ \frac{1}{t} ∫_{1}^{τ} s^{2}p(s) ds + \frac{1}{t} τ^{2}ρ(τ) for t ≥ a.  (4.3)

Hence we have

Q(t) + H(t) ≤ 1 + \frac{1}{t} ∫_{1}^{τ} s^{2}p(s) ds + \frac{1}{t} τ^{2}ρ(τ) for t ≥ a,

which contradicts condition (1.3).

Proof of Theorem 1.3. Let us assume the opposite, i.e., that u(t) ≠ 0, t ≥ a, is a solution of equation (0.1). Then (3.1) and (3.6), where the function ρ is defined by (3.2), are valid. By Lemma 3.2 when (1.4) is fulfilled and by Lemma 3.3 when (1.5) is fulfilled, for any sufficiently small ε > 0 there exists t_{ε} > a such that,

tρ(t) > r − ε and tρ(t) < M + ε for t ≥ t_{ε},  (4.4)

respectively, where

r = \frac{1}{2} (1 − \sqrt{1 − 4Q_{∗}}), M = \frac{1}{2} (1 + \sqrt{1 − 4H_{∗}} ).  (4.5)

Hence, if (1.4) is fulfilled, (3.1) and (3.6) imply

H(t) ≤ −r + ε + 1 + \frac{1}{t} t_{ε}^{2}ρ(t_{ε}) + \frac{1}{t} ∫_{1}^{t_{ε}} s^{2}p(s) ds for t ≥ t_{ε},

and if (1.5) is fulfilled, (3.1) and (3.6) imply Q(t) ≤ M + ε for t ≥ t_{ε}, which contradicts the conditions of the theorem.

Proof of Theorem 1.4. Let us assume that (1.8) is fulfilled (the case, where (1.7) is fulfilled, is proved similarly). Let u(t) ≠ 0, t ≥ a, be a solution of equation (0.1). Then (3.6), where the function ρ is defined by equality (3.2), is valid. We shall assume that H_{∗} > 0, since for H_{∗} = 0 condition (1.8) is equivalent to condition (1.4) of Theorem 1.3 which has been proved above. By Lemmas 3.2 and 3.3, for arbitrary ε ∈]0, 1 − M[ , there exists t_{ε} > a such that (4.4) holds, where r and M are the numbers defined by equalities (4.5). Since M + ε < 1, we have

sρ(s)(2 − sρ(s)) < (M + ε)(2 − M − ε) for s ≥ t_{ε}.  (4.6)

Taking into account (4.4) and (4.6), from (3.6) we obtain

H(t) ≤ −r + ε + (M + ε)(2 − M − ε) + \frac{1}{t} ∫_{1}^{t_{ε}} s^{2}p(s) ds + \frac{1}{t} t_{ε}^{2}ρ(t_{ε}) for t ≥ t_{ε}.

Hence we easily conclude that H^{∗} ≤ −r + M(2 − M), which contradicts condition (1.8).

The proof of Theorem 1.5 repeats that of Theorem 1.2 with the only difference that one should use (4.4) and (4.6) in equality (4.3).

Proof of Theorem 1.6. Assume that (1.11) ((1.12)) is fulfilled. Choose ε > 0 such that

Q^{∗} + ε < Q_{∗} − ε − 1 + \sqrt{1 − 4(Q_{∗} − ε)}
( H^{∗} + ε < H_{∗} − ε − 1 + \sqrt{1 − 4(H_{∗} − ε)} ).

We set

α = [\frac{1}{2} (1 + \sqrt{1 − 4(Q^{∗} + ε)})]^{2} ( α = 1 − \frac{1}{4} (1 + \sqrt{1 − 4(H^{∗} + ε)})^{2} ).

It is easy to verify that

Q^{∗} + ε = \sqrt{α} − α ( H^{∗} + ε = \sqrt{1 − α} − (1 − α) )  (4.7)

and

Q_{∗} − ε > −\sqrt{α} − α ( H_{∗} − ε > −\sqrt{1 − α} − (1 − α) ).  (4.8)

Choose t_{ε} > 1 such that

Q_{∗} − \frac{ε}{2} < Q(t) < Q^{∗} + \frac{ε}{2} for t ≥ t_{ε}

( H_{∗} − \frac{ε}{2} < H(t) < H^{∗} + \frac{ε}{2} for t ≥ t_{ε} ).

Now by (4.7) and (4.8) we have

−\sqrt{α} − α < Q(t) < \sqrt{α} − α for t ≥ t_{ε}

( − \sqrt{1 − α} − (1 − α) < H(t) < \sqrt{1 − α} − (1 − α) for t ≥ t_{ε} ),

i.e.,

Q^{2}(t) + 2αQ(t) + α(α − 1) < 0 for t ≥ t_{ε}
( H^{2}(t) + 2(1 − α)H(t) + α(α − 1) < 0 for t ≥ t_{ε} ).

Thus equation (3.7) is nonoscillatory. Therefore, by Lemma 3.4, equation (0.1) is nonoscillatory.

Proof of Theorem 1.7. Let the function G be defined by equality (3.9). Choose ε > 0 and t_{ε} > 1 such that G^{∗} + 2ε < G_{∗} + \sqrt{1 − 4p_{0}} and G_{∗} − ε < G(t) < G^{∗} + ε for t ≥ t_{ε}.

We set

α = G_{∗} − ε + \frac{1}{2} + \frac{1}{2} \sqrt{1 − 4p_{0}}.

Clearly,

α − \frac{1}{2} − \frac{1}{2} \sqrt{1 − 4p_{0}} < G(t) < α − \frac{1}{2} + \frac{1}{2} \sqrt{1 − 4p_{0}} for t ≥ t_{ε},

i.e.,

G^{2}(t) − (2α − 1)G(t) + p_{0} + α(α − 1) < 0 for t ≥ t_{ε}.

Thus equation (3.8) is nonoscillatory. Therefore by Lemma 3.5 equation (0.1) is nonoscillatory.

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(Received 01.06.1998)

Authors’ addresses:

T. Chantladze, N. Kandelaki N. Muskhelishvili Institute of Computational Mathematics Georgian Academy of Sciences 8, Akuri St., Tbilisi 380093 Georgia

A. Lomtatidze Department of Mathematical Analysis Faculty of Natural Sciences Masaryk University Janačkovo nám. 2a, 662 95 Brno Czech Republic