GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 2, 1997, 129-138

OSCILLATION AND NONOSCILLATION CRITERIA FOR SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS

A. LOMTATIDZE

Abstract. Sufficient conditions for oscillation and nonoscillation of second-order linear equations are established.

1. Statement of the Problem and Formulation of Basic Results

Consider the differential equation

u′′ + p(t)u = 0,  (1)

where p : [0, +∞[→ [0, +∞[ is an integrable function. By a solution of equation (1) is understood a function u : [0, +∞[→] − ∞, +∞[ which is locally absolutely continuous together with its first derivative and satisfies this equation almost everywhere.

Equation (1) is said to be oscillatory if it has a nontrivial solution with an infinite number of zeros, and nonoscillatory otherwise.

It is known (see [1]) that if for some λ < 1 the integral ∫^{+∞} s^{λ}p(s)ds diverges, then equation (1) is oscillatory. Therefore, we shall always assume below that

∫^{+∞}s^{λ}p(s)ds < +∞ for λ < 1.

129 Introduce the notation

1991 Mathematics Subject Classification. 34C10.

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

1072-947X/97/0300-0129$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

h_{λ}(t) = t^{1−λ} ∫_{t}^{∞}s^{λ}p(s)ds for t > 0 and λ < 1,

h_{λ}(t) = t^{1−λ} ∫_{1}^{t} s^{λ}p(s)ds for t > 0 and λ > 1,

(2)

p_{∗}(λ) = lim inf_{t→+∞} h_{λ}(t), p^{∗}(λ) = lim sup_{t→+∞} h_{λ}(t).

In [1] it is proved that equation (1) is oscillatory if p^{∗}(0) > 1 or p_{∗}(0) > \frac{1}{4}, and nonoscillatory if p^{∗}(0) < \frac{1}{4}. The oscillation criteria for equation (1) written in terms of the numbers p_{∗}(λ) and p^{∗}(λ) have been established in [2]. Below we shall give the sufficient conditions for oscillation and nonoscillation of equation (1) which make the above-mentioned results of papers [1, 2] more precise and even extend them in some cases.

First of all, for the completenes of the picture we give a proposition, which slightly generalizes one of E. Hille’s theorems [1].

Proposition. Let either p_{∗}(0) > \frac{1}{4} or p_{∗}(2) > \frac{1}{4}. Then equation (1) is oscillatory.

Theorem 1. Let p_{∗}(0) ≤ \frac{1}{4} and p_{∗}(2) ≤ \frac{1}{4}. Moreover, let either

p^{∗}(λ) > \frac{λ^{2}}{4(1 − λ)} + \frac{1}{2}(1 + \sqrt{1 − 4p_{∗}(2)})  (3)

for some λ < 1 or

p^{∗}(λ) > \frac{λ^{2}}{4(λ − 1)} − \frac{1}{2}(1 − \sqrt{1 − 4p_{∗}(0)})  (4)

for some λ > 1. Then equation (1) is oscillatory.

Corollary 1. Let either

lim_{λ→1−}(1 − λ)p^{∗}(λ) > \frac{1}{4}  (5)

or

lim_{λ→1+}(λ − 1)p^{∗}(λ) > \frac{1}{4}  (6)

Then equation (1) is oscillatory.

OSCILLATION AND NON-OSCILLATION CRITERIA 131

Corollary 2 ([2]). For some λ ≠ 1 let

|1 − λ|p_{∗}(λ) > \frac{1}{4}  (7)

Then equation (1) is oscillatory.

Remark 1. Inequalities (5)–(7) are exact and cannot be weakened. Indeed, let p(t) = \frac{1}{4t^{2}} for t ≥ 1. Then |1 − λ|p_{∗}(λ) = \frac{1}{4}, and equation (1) has oscillatory solution u(t) = \sqrt{t} for t > 1.

Theorem 2. Let p_{∗}(0) ≤ \frac{1}{4} and p_{∗}(2) ≤ \frac{1}{4}. Moreover, let either

p_{∗}(0) > \frac{λ(2 − λ)}{4} and
p^{∗}(λ) > \frac{p_{∗}(0)}{1 − λ} + \frac{1}{2}(\sqrt{1 − 4p_{∗}(0)} + \sqrt{1 − 4p_{∗}(2)})  (8)

for some λ < 1 or

p_{∗}(2) > \frac{λ(2 − λ)}{4} and
p^{∗}(λ) > \frac{p_{∗}(2)}{λ − 1} + \frac{1}{2}(\sqrt{1 − 4p_{∗}(0)} + \sqrt{1 − 4p_{∗}(2)})  (9)

for some λ > 1. Then equation (1) is oscillatory.

Theorem 3. Let p_{∗}(0) ≠ 0 and p_{∗}(2) ≤ \frac{1}{4}. Moreover, for some 0 < λ < 1 let p_{∗}(λ) < \frac{1−λ^{2}}{4} and either

p_{∗}(λ) > \frac{p_{∗}(0)}{1 − λ} + \frac{λ}{2(1 − λ)}(\sqrt{1 − 4p_{∗}(0)} + \sqrt{1 − 4p_{∗}(2)})

and

p^{∗}(λ) > p_{∗}(λ) + \frac{1}{2}(λ + \sqrt{1 − 4p_{∗}(2)}) +
+ \sqrt{λ^{2} + 1 − 4(1 − λ)p_{∗}(λ) + 2λ\sqrt{1 − 4p_{∗}(2)}}

or

p_{∗}(λ) < \frac{p_{∗}(0)}{1 − λ} + \frac{λ}{2(1 − λ)}(\sqrt{1 − 4p_{∗}(0)} + \sqrt{1 − 4p_{∗}(2)})

and

p^{∗}(λ) > \frac{p_{∗}(0)}{1 − λ} + \frac{1}{2(1 − λ)}(\sqrt{1 − 4p_{∗}(0)} + \sqrt{1 − 4p_{∗}(2)}).  (10)

Then equation (1) is oscillatory.

132 A. LOMTATIDZE

Theorem 3′. Let p_{∗}(0) ≤ \frac{1}{4} and p_{∗}(2) ≤ \frac{1}{4}. Moreover, for some 0 < λ < 1 let condition (10) be fulfilled, and let p_{∗}(0) > \frac{1−λ^{2}}{4} . Then equation (1) is oscillatory.

Corollary 3. Let p_{∗}(0) ≤ \frac{1}{4}, p_{∗}(2) ≤ \frac{1}{4} and

p^{∗}(0) > p_{∗}(0) + \frac{1}{2}(\sqrt{1 − 4p_{∗}(0)} + \sqrt{1 − 4p_{∗}(2)}).

Then equation (1) is oscillatory.

Corollary 4. For some λ ∈ [0, \frac{1}{4}[ let

\frac{λ}{1 − λ} < p_{∗}(λ) < \frac{1}{4(1 − λ)}

and

p^{∗}(λ) > 1 + p_{∗}(λ) − \frac{1}{2}(1 − λ − \sqrt{(1 + λ)^{2} − 4(1 − λ)p_{∗}(λ)}).

Then equation (1) is oscillatory.

Theorem 4. For some λ ≠ 1 let

p_{∗}(λ) > \frac{(2λ − 1)(3 − 2λ)}{4|1 − λ|} and p^{∗}(λ) < \frac{1}{4|1 − λ|}.  (11)

Then equation (1) is nonoscillatory.

Remark 2. As will be seen from the proof, Theorem 4 remains also valid when the function p, generally speaking, does not have a constant sign. For such a case this result for λ = 0 is described in [3].

Corollary 5. Let p_{∗}(0) < \frac{1}{4} and p_{∗}(2) < \frac{1}{4}, and let the inequality

p^{∗}(λ) < \frac{1}{4|1 − λ|}

hold for some λ ∈] − ∞, 1 − \sqrt{\frac{1}{4} − p_{∗}(0)}[∪]1 + \sqrt{\frac{1}{4} − p_{∗}(2)}, +∞[ . Then equation (1) is nonoscillatory.

2. Some Auxiliary Propositions

Lemma 1. For equation (1) to be nonoscillatory, it is necessary and sufficient that for some λ ≠ 1 the equation

v′′ = \frac{1}{t^{2}} ( − h_{λ}^{2}(t) + λ sgn(1 − λ)h_{λ}(t))v − \frac{2 sgn(1 − λ)}{t}h_{λ}(t)v′, (12)

where h_{λ} is the function defined by (2), be nonoscillatory.

Proof. The equality ρ(t) = t^{λ} \frac{u′(t)}{u(t)} − t^{λ−1}h_{λ}(t) sgn(1 − λ) determines the relation between the nonoscillatory solution u of equation (1) and the solution ρ, defined in some neighborhood of +∞, of the equation

ρ′ = −t^{−λ}ρ^{2} + λt^{−1}ρ − 2 sgn(1 − λ)t^{−1}h_{λ}^{2}(t)ρ −
− t^{λ−2}h_{λ}^{2}(t) + λ sgn(1 − λ)t^{λ−1}h_{λ}(t).  (13)

On the other hand, the equality ρ(t) = t^{λ} \frac{v′(t)}{v(t)} determines the relation between the nonoscillatory solution v of equation (11) and the solution ρ defined in some neighborhood +∞ of equation (13). Thus nonoscillation of either of equation (1) or (12) results in nonoscillation of the other.

Lemma 2. Let equation (1) be nonoscillatory. Then there exists t_{0} > 0 such that the equation

ρ′ + p(t)ρ + ρ^{2} = 0  (14)

has a solution ρ : ]t_{0}, +∞[→ [0, +∞[; moreover,

ρ(t_{0}+) = +∞, (t − t_{0})ρ(t) < 1 for t_{0} < t < +∞,  (15)

lim_{t→+∞} t^{λ}ρ(t) = 0 for λ < 1  (16)

and

lim inf_{t→+∞} tρ(t) ≥ A, lim sup_{t→+∞} tρ(t) ≤ B,  (17)

where

A = \frac{1}{2}(1 − \sqrt{1 − 4p_{∗}(0)}), B = \frac{1}{2}(1 + \sqrt{1 − 4p_{∗}(2)})^{1}.  (18)

1Since equation (1) is nonoscillatory, we have p_{∗}(0) ≤ \frac{1}{4} and p_{∗}(2) ≤ \frac{1}{4} .

Proof. Since equation (1) is nonoscillatory, there exists t_{0} > 0 such that the solution u of equation (1) under the initial conditions u(t_{0}) = 0, u′(t_{0}) = 1 satisfies the inequalities

u(t) > 0, u′(t) ≥ 0 for t_{0} < t < +∞.

Clearly, the function ρ(t) = \frac{u′(t)}{u(t)} for t_{0} < t < +∞ is the solution of equation (14), and lim_{t→t_{o}+} = +∞. From (14) we have

\frac{−ρ′(t)}{ρ^{2}(t)} > 1 for t_{0} < t < +∞.

Integrating the above inequality from t_{0} to t, we obtain (t − t_{0})ρ(t) < 1 for t_{0} < t < +∞. In particular, equality (16) holds for any λ < 1.

Let us now show that inequalities (17) are valid. Assume p_{∗}(0) ≠ 0 and p_{∗}(2) ≠ 0 (inequalities (17) are trivial, otherwise). Let us introduce the notation

r = lim inf_{t→+∞} tρ(t), R = lim sup_{t→+∞} tρ(t).

From (14) we easily find that for any t_{1} > t_{0}

\begin{aligned}tρ(t) & = t ∫_{t}^{+∞}p(s)ds + t ∫_{t}^{+∞}ρ^{2}(s)ds, \\ tρ(t) & = \frac{t_{1}^{2}ρ(t_{1})}{t} − t^{−1} ∫_{t_{1}}^{t} s^{2}p(s)ds + t^{−1} ∫_{t_{1}}^{t} sρ(s)(2 − sρ(s))ds \\ & for t_{1} < t < +∞. \end{aligned} (19)

This implies that r ≥ p_{∗}(0) and R ≤ 1 − p_{∗}(2).

It is easily seen that for any 0 < ε < min{r, 1 − R} there exists t_{ε} > t_{1} such that

r − ε < tρ(t) < R + ε, t ∫_{t}^{+∞}p(s)ds > P_{∗}(0) − ε,

\frac{1}{t} ∫_{t_{1}}^{t} s^{2}p(s)ds > p_{∗}(2) − ε for t_{ε} < t < +∞.

Taking into account the above argument, from (19) we have

tρ(t) > p_{∗}(0) − ε + (r − ε)^{2} for t_{ε} < t < +∞,

tρ(t) < \frac{t_{ε}^{2}ρ(t_{ε})}{t} − p_{∗}(2) + ε + (R + ε)(2 − R − ε) for t_{ε} < t < +∞,

whence

r ≥ p_{∗}(0) + r^{2}, R ≤ −p_{∗}(2) + R(2 − R),

that is, r ≥ A and R ≤ B, where A and B are the numbers defined by equalities (18). Hence (17) holds.

Lemma 3. Let the functions g, q : [a, +∞[→ R be integrable in every finite interval, and let v : [a, +∞[→]0, +∞[ be absolutely continuous together with its first derivative on every compactum contained in [a, +∞[. Moreover, let the inequality

v′′(t) ≤ g(t)v(t) + q(t)v′(t)  (20)

hold almost everywhere in [a, +∞[. Then the equation u′′ = g(t)u + q(t)u′ is nonoscillatory.

3. Proof of the Basic Results

Proof of Theorem 1. Assume the contrary. Let equation (1) be nonoscillatory. Then, according to Lemma 2, equation (14) has the solution ρ : ]t_{0}, +∞[→ [0, +∞[ satisfying conditions (15)–(17). Suppose λ < 1 (λ > 1). Because of (17) we have that for any ε > 0 there exists t_{ε} > t_{0} such that

A − ε < tρ(t) < B + ε for t_{ε} < t < +∞.  (21)

Multiplying equality (14) by t^{λ}, integrating it from t to +∞ (from t_{ε} to t), and taking into account (15)–(17), we get

∫_{t}^{+∞}s^{λ}p(s)ds = t^{λ}ρ(t) + \frac{λ^{2}t^{λ−1}}{4(1 − λ)} − ∫_{t}^{+∞}(s^{\frac{λ}{2}} ρ(s) − \frac{1}{2}s^{\frac{λ}{2} −1})^{2}ds <

< t^{λ−1}(B + ε + \frac{λ^{2}}{4(1 − λ)}) for t_{ε} < t < +∞

( ∫_{t_{ε}}^{t} s^{λ}p(s)ds < t^{λ−1}(\frac{λ^{2}}{4(1 − λ)} − A + ε + t_{ε}^{λ}ρ(t_{ε}))

for t_{ε} < t < +∞ ),

whence we have p^{∗}(λ) ≤ \frac{λ^{2}}{4(1−λ)} + \frac{1}{2}(1 + \sqrt{1 − 4p_{∗}(2)}) (p^{∗}(λ) ≤ \frac{λ^{2}}{4(λ−1)} − \frac{1}{2}(1 − \sqrt{1 − 4p_{∗}(0)})), which contradicts equality (3) ((4)).

To convince ourselves that Corollary 1 is valid, let us note that (5) ((6)) imply

lim_{λ→1+} [(1 − λ)p^{∗}(λ) − \frac{λ^{2}}{4} − \frac{1 − λ}{2}(1 + \sqrt{1 − 4p_{∗}(2)})] > 0

( lim_{λ→1−} [(λ − 1)p^{∗}(λ) − \frac{λ^{2}}{4} − \frac{λ − 1}{2}(1 + \sqrt{1 − 4p_{∗}(0)})] > 0).

Consequently, (3) ((4)) is fulfilled for some λ < 1 (λ > 1). Thus, according to Theorem 1, equation (1) is oscillatory. As for Corollary 2, taking into account that the mapping λ ⟼ (1 − λ)p_{∗}(λ) for λ < 1 (λ ⟼ (λ − 1)p_{∗}(λ) for λ > 1 ) is non-decreasing (non-increasing), we easily find from (7) that (5) ((6)) is fulfilled for some λ.

Proof of Theorem 2. Assume the contrary. Let equation (1) be nonoscillatory. Then according to Lemma 2, equation (14) has the solution ρ : ]t_{0}, +∞[→ [0, +∞[ satisfying conditions (15)–(17). Suppose λ < 1 (λ > 1). By the conditions of the theorem, p_{∗}(0) > \frac{λ(2−λ)}{4} (p_{∗}(2) > \frac{λ(2−λ)}{4}), which implies that A > \frac{λ}{2} (B < \frac{λ}{2}). On account of (17), for any 0 < ε < A − \frac{λ}{2} (0 < ε < \frac{λ}{2} − B) there exists t_{ε} > t_{0} such that (21) holds.

Multiplying equality (14) by t^{λ}, integrating it from t to +∞ (from t_{ε} to t), and taking into account (15)–(17), we easily find that

t^{1−λ} ∫_{t}^{+∞}s^{λ}p(s)ds = tρ(t) + t^{1−λ} ∫_{t}^{+∞}s^{λ−2}sρ(s)(λ − sρ(s))ds <

< B + ε + \frac{1}{1 − λ}(A − ε)(λ − A + ε) for t_{ε} < t < +∞

(t^{1−λ} ∫_{t_{ε}}^{t} s^{λ}p(s)ds < ε − A + \frac{1}{λ − 1}(B + ε)(λ − B − ε) +

+ t^{1−λ}t_{ε}ρ(t_{ε}) for t_{ε} < t < +∞).

This implies

p^{∗}(λ) ≤ \frac{p_{∗}(0)}{1 − λ} + \frac{1}{2}(\sqrt{1 − 4p_{∗}(0)} + \sqrt{1 − 4p_{∗}(2)})

(p^{∗}(λ) ≤ \frac{p_{∗}(2)}{λ − 1} + \frac{1}{2}(\sqrt{1 − 4p_{∗}(0)} + \sqrt{1 − 4p_{∗}(2)})),

which contradicts condition (8) ((9)).

Proof of Theorems 3 and 3′. Assume the contrary. Let equation (1) be nonoscillatory. Then according to Lemma 2, equation (14) has the solution ρ : ]t_{0}, +∞[→ [0, +∞[ satifying conditions (15)–(17). Multiplying equality (14) by t^{λ}, integrating it from t to +∞, and taking into account (16), we easily obtain

tρ(t) = h_{λ}(t) − λt^{1−λ} ∫_{t}^{+∞}s^{λ−1}ρ(s)ds +

+ t^{1−λ} ∫_{t}^{+∞}s^{λ}ρ^{2}(s)ds for t_{0} < t < +∞,  (22)

where h_{λ} is the function defined by equality (2).

Introduce the notation

r = lim inf_{t→+∞} tρ(t).

On account of (17) we have r > 0. Therefore for any 0<ε<max{r, p_{∗}(λ)} there exists t_{ε} > t_{0} such that

r − ε < tρ(t) < B + ε, h_{λ}(t) > p_{∗}(λ) − ε for t_{ε} < t < +∞.

Owing to the above arguments, we find from (22) that

(1 − λ)h_{λ} < B + ε − (r − ε)^{2} for t_{ε} < t < +∞,

tρ(t) > p_{∗}(λ) − ε − \frac{λ}{1 − λ}(B + ε) + \frac{1}{1 − λ}(r − ε)^{2} for t_{ε} < t < +∞,

which implies

p^{∗}(λ) ≤ \frac{B − r^{2}}{1 − λ} ,  (23)

r ≥ p_{∗}(λ) − \frac{λ}{1 − λ}B + \frac{r^{2}}{1 − λ}.

The latter inequality results in r ≥ x_{1}, where x_{1} is the least root of the equation

\frac{1}{1 − λ}x^{2} − x + p_{∗}(λ) − \frac{λ}{1 − λ}B = 0.

Thus r ≥ max{A, x_{1}}. From (23) we have that if A < x_{1}, then

p^{∗}(λ) ≤ B + p_{∗}(λ) − x_{1},

but if A ≥ x_{1}, then

p^{∗}(λ) ≤ \frac{1}{1 − λ}B − \frac{1}{1 − λ}A^{2},

which contradicts the conditions of the theorem.

Proof of Theorem 4. From (11) we have that for some t_{0} > 0

\frac{(2λ − 1)(3 − 2λ)}{4|1 − λ|} < h_{λ}(t) < \frac{1}{4|1 − λ|} for t_{0} < t < +∞,

whence

h_{λ}^{2}(t) + \frac{2λ^{2} − 4λ + 1}{2|1 − λ|}h_{λ}(t) + \frac{(2λ − 1)(3 − 2λ)}{16(1 − λ)^{2}} < 0 for t_{0} < t < +∞.

Taking into consideration the latter inequality, we can easily see that (20) holds, where

v(t) = t^{\frac{1−2λ}{4(1−λ)}} for t_{0} < t < +∞,
g(t) = −\frac{1}{t^{2}} (h_{λ}^{2}(t) − λ sgn(1 − λ)h_{λ}(t)) for t_{0} < t < +∞,

and

q(t) = −\frac{2}{t} sgn(1 − λ)h_{λ}(t) for t_{0} < t < +∞.

Consequently, according to Lemmas 1 and 3, equation (1) is nonoscillatory.

References

1. E. Hille, Non-oscillation theorems. Trans. Amer. Math. Soc., 64(1948), 234–252.

2. Z. Nehari, Oscillation criteria for second order-linear differential equations. Trans. Amer. Math. Soc., 85(1957), 428–445.

3. A. Wintner, On the non-existence of conjugate points. Amer. J. Math., 73(1951), 368–380.

(Received 03.05.1995)

Author’s address:

N. Muskhelishvili Institute of

Computational Mathematics

Georgian Academy of Sciences

8, Akuri St., Tbilisi 380093

Georgia