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Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm

  • Jiřina Šišoláková EMAIL logo
Published/Copyright: June 12, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

We study linear differential equations whose coefficients consist of products of powers of natural logarithm and general continuous functions. We derive conditions that guarantee the non-oscillation of all non-trivial solutions of the treated type of equations. The conditions are formulated as a non-oscillation criterion, which is the counterpart of a previously obtained oscillation theorem. Therefore, from the presented main result, it follows that the analysed equations are conditionally oscillatory. The used method is based on averaging techniques for the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This article is finished by new corollaries and examples.

Keywords: linear equation; oscillation theory; non-oscillation; Riccati equation; Prüfer angle
MSC 2010: 34C10

1 Motivations and main results

We study the second-order linear differential equations, i.e. equations in the form

(1.1) ( r ( t ) x ′ ( t ) ) ′ + s ( t ) x ( t ) = 0 ,

where r > 0 , s are the continuous functions on a neighbourhood of ∞. Linear equations form a very intensively studied type of differential equations. Linear differential equations are classified as oscillatory (any solution has infinitely many zeros in all neighbourhood of ∞) or non-oscillatory (any non-trivial solution has the biggest zero). This classification is based on the famous Sturm theory. Concerning the oscillation theory of equation (1.1), we refer to [1,2] and references cited therein.

The basic motivations of our research come from [3–5]. The most relevant results are explicitly formulated in the following, where e is the base of the natural logarithm, R e ≔ [ e , ∞ ) , log denotes the natural logarithm, and p > 0 is arbitrary. In [3–5], linear differential equations of the form

(1.2) log p t r ( t ) x ′ ( t ) ′ + log p t t 2 s ( t ) x ( t ) = 0

are analysed.

Theorem 1.1

Let us consider equation (1.2), where r : R e → ( 0 , ∞ ) and s : R e → R are continuous and periodic functions with period α > 0 .

  1. If

    1 α ∫ e e + α r ( τ ) d τ 1 α ∫ e e + α s ( τ ) d τ > 1 4 ,

    then equation (1.2) is oscillatory.

  2. If

    1 α ∫ e e + α r ( τ ) d τ 1 α ∫ e e + α s ( τ ) d τ < 1 4 ,

    then equation (1.2) is non-oscillatory.

Proof

See [5].□

Theorem 1.2

Let a continuously differentiable function f : R e → ( 0 , ∞ ) and a continuous function g : R e → [ 1 , ∞ ) satisfy

(1.3) lim t → ∞ f ′ ( t ) g ( t ) = 0 , lim t → ∞ f ( t ) g 2 ( t ) t = 0 .

Let us consider equation (1.2), where continuous functions r : R e → ( 0 , ∞ ) and s : R e → R satisfy

(1.4) limsup t → ∞ ∫ t t + f ( t ) r ( τ ) d τ f ( t ) g ( t ) < ∞ and limsup t → ∞ ∫ t t + f ( t ) ∣ s ( τ ) ∣ d τ f ( t ) g ( t ) < ∞ .

Let

(1.5) r f ≔ liminf t → ∞ 1 f ( t ) ∫ t t + f ( t ) r ( τ ) d τ ∈ R and s f ≔ liminf t → ∞ 1 f ( t ) ∫ t t + f ( t ) s ( τ ) d τ ∈ R .

If

(1.6) r f s f > 1 4 ,

then equation (1.2) is oscillatory.

Proof

See [3].□

The aim of this article is to prove a non-oscillation criterion, which generalizes Theorem 1.1, (II), and which is a non-oscillation counterpart of Theorem 1.2. To prove such a result, we use lemmas from [4] about a modification of the adapted Prüfer angle. Theorem 1.1 shows that the studied equations are conditionally oscillatory. Thus, the non-oscillation counterpart of Theorem 1.2 means that the considered equations remain conditionally oscillatory also for more general coefficients. This fact is documented by Theorem 1.4. Concerning the notion of the conditional oscillation together with other basic types of conditionally oscillatory differential equations, we point out at least relevant articles [6–10] (see also [11–13] for generalizations and [14–16] for perturbed equations).

The announced non-oscillation counterpart reads as follows.

Theorem 1.3

Let a continuously differentiable function F : R e → ( 0 , ∞ ) and a continuous function G : R e → [ 1 , ∞ ) satisfy

(1.7) lim t → ∞ G ( t ) = ∞ , limsup t → ∞ ∣ F ′ ( t ) G ( t ) ∣ < ∞ , and limsup t → ∞ F ( t ) G 2 ( t ) t < ∞ .

Let us consider equation (1.2), where continuous functions r : R e → ( 0 , ∞ ) and s : R e → R satisfy

(1.8) lim t → ∞ ∫ t t + F ( t ) r ( τ ) d τ F ( t ) G ( t ) = 0 and lim t → ∞ ∫ t t + F ( t ) ∣ s ( τ ) ∣ d τ F ( t ) G ( t ) = 0 .

Let

(1.9) r F ≔ limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) r ( τ ) d τ and s F ≔ limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) s ( τ ) d τ .

If

(1.10) r F s F < 1 4 ,

then equation (1.2) is non-oscillatory.

The combination of Theorems 1.2 and 1.3 gives the next new result, which is a direct generalization of Theorem 1.1.

Theorem 1.4

Let a continuously differentiable function h : R e → ( 0 , ∞ ) and a continuous function H : R e → [ 1 , ∞ ) satisfy

(1.11) lim t → ∞ H ( t ) = ∞ , lim t → ∞ h ′ ( t ) H ( t ) = 0 , and lim t → ∞ h ( t ) H 2 ( t ) t = 0 .

Let us consider equation (1.2), where continuous functions r : R e → ( 0 , ∞ ) and s : R e → R satisfy

(1.12) lim t → ∞ ∫ t t + h ( t ) r ( τ ) d τ h ( t ) H ( t ) = 0 and lim t → ∞ ∫ t t + h ( t ) ∣ s ( τ ) ∣ d τ h ( t ) H ( t ) = 0 .

Let

(1.13) r h ≔ lim t → ∞ 1 h ( t ) ∫ t t + h ( t ) r ( τ ) d τ ∈ R and s h ≔ lim t → ∞ 1 h ( t ) ∫ t t + h ( t ) s ( τ ) d τ ∈ R .

  1. If

    (1.14) r h s h > 1 4 ,

    then equation (1.2) is oscillatory.

  2. If

    (1.15) r h s h < 1 4 ,

    then equation (1.2) is non-oscillatory.

In this paragraph, we complete a short literature overview (many results are proved for more general half-linear equations). Concerning the (conditional) oscillation of perturbed differential equations, we refer to [17–20] (and also [21–24]). Relevant oscillation and non-oscillation results about difference equations are presented in [25–32] (see also [33,34]). Similar results for dynamic equations on time scales are obtained in [35–38] (see also [39,40]). We remark that the considered oscillation theory of dynamic equations on time scales is less developed than its special case given by difference equations. Concerning relevant oscillation and non-oscillation criteria for differential equations that are not linear or half-linear, we refer at least to [41–44].

This article has the next three parts. In the forthcoming section, we state the used methods that are the modified Riccati transformation and the generalized adapted Prüfer angle. All auxiliary results are collected in Section 3. The last part consists of the proofs of Theorems 1.3 and 1.4 and corollaries with examples, which illustrate our results.

2 Methods

Now, we briefly describe the applied combination of the modified Riccati transformation and the generalized adapted Prüfer angle. Let us consider equation (1.2). For a non-trivial solution x of equation (1.2) satisfying x ( t ) ≠ 0 , the Riccati transformation

w ( t ) = log p t r ( t ) ⋅ x ′ ( t ) x ( t )

gives the so-called Riccati equation

(2.1) w ′ ( t ) + log p t t 2 s ( t ) + r ( t ) log p t w 2 ( t ) = 0 .

Then, considering equation (2.1), the modified transformation

v ( t ) = t log p t w ( t )

leads to

v ′ ( t ) = log t − p t log t v ( t ) − 1 t s ( t ) − r ( t ) t v 2 ( t ) .

Using the adapted Prüfer transformation

x ( t ) = ρ ( t ) sin φ ( t ) , x ′ ( t ) = ρ ( t ) r ( t ) t cos φ ( t ) ,

we obtain the equation for the adapted Prüfer angle in the form:

(2.2) φ ′ ( t ) = 1 t r ( t ) cos 2 φ ( t ) − log t − p log t cos φ ( t ) sin φ ( t ) + s ( t ) sin 2 φ ( t ) .

For details about the derivation of equation (2.2), see [4] (and also [3]).

Our results are based on an averaging technique given by F . Thus, let us consider a solution φ : R e → R of equation (2.2) and let the averaging function ψ : R e → R be given by the following formula:

(2.3) ψ ( t ) ≔ 1 F ( t ) ∫ t t + F ( t ) φ ( τ ) d τ , t ∈ R e .

3 Auxiliary results

In this section, we collect all used lemmas. For the averaging function ψ , we prove the following auxiliary results. We remark that the considered functions F and G are taken from the statement of Theorem 1.3. These functions satisfy (1.7) and (1.8), which are used only in the proofs of the lemmas below.

Lemma 3.1

If φ : R e → R is a solution of equation (2.2), then

(3.1) lim t → ∞ t F ( t ) G ( t ) sup s ∈ [ t , t + F ( t ) ] ∣ φ ( s ) − ψ ( t ) ∣ = 0 .

In particular,

(3.2) lim t → ∞ ∣ φ ( t ) − ψ ( t ) ∣ = 0 .

Proof

For any t ∈ R e , the definition of ψ (see (2.3)) and the continuity of φ give ψ ( t ) = φ ( ρ ( t ) ) for some ρ ( t ) ∈ [ t , t + F ( t ) ] . Hence, we obtain

t F ( t ) G ( t ) sup s ∈ [ t , t + F ( t ) ] ∣ φ ( s ) − ψ ( t ) ∣ = t F ( t ) G ( t ) sup s ∈ [ t , t + F ( t ) ] ∣ φ ( s ) − φ ( ρ ( t ) ) ∣ = t F ( t ) G ( t ) sup s ∈ [ t , t + F ( t ) ] ∫ ρ ( t ) s φ ′ ( τ ) d τ ≤ t F ( t ) G ( t ) ∫ t t + F ( t ) ∣ φ ′ ( τ ) ∣ d τ = t F ( t ) G ( t ) ∫ t t + F ( t ) r ( τ ) τ cos 2 φ ( τ ) − log τ − p τ log τ cos φ ( τ ) sin φ ( τ ) + s ( τ ) τ sin 2 φ ( τ ) d τ ≤ 1 F ( t ) G ( t ) ∫ t t + F ( t ) r ( τ ) cos 2 φ ( τ ) + 1 + p log τ ∣ cos φ ( τ ) sin φ ( τ ) ∣ + ∣ s ( τ ) ∣ sin 2 φ ( τ ) d τ ≤ 1 F ( t ) G ( t ) ∫ t t + F ( t ) ( r ( τ ) + 1 + p + ∣ s ( τ ) ∣ ) d τ ,

for any t ∈ R e . From lim t → ∞ G ( t ) = ∞ (see (1.7)) and (1.8), we have (3.1). Considering (1.7), we also have

(3.3) liminf t → ∞ t F ( t ) G ( t ) > 0 ,

which implies (3.2).□

Remark 1

From (3.3) in the proof of Lemma 3.1, one can see the inequality

(3.4) limsup t → ∞ F ( t ) G ( t ) t < ∞ ,

which is used later.

Lemma 3.2

If φ : R e → R is a solution of equation (2.2), then

(3.5) lim t → ∞ t ψ ′ ( t ) − 1 F ( t ) ∫ t t + F ( t ) r ( τ ) d τ cos 2 ψ ( t ) + log t − p log t cos ψ ( t ) sin ψ ( t ) − 1 F ( t ) ∫ t t + F ( t ) s ( τ ) d τ sin 2 ψ ( t ) = 0 .

Proof

For any t > e , we have

ψ ′ ( t ) = 1 F ( t ) ∫ t t + F ( t ) φ ( τ ) d τ ′ = − F ′ ( t ) F 2 ( t ) ∫ t t + F ( t ) φ ( τ ) d τ + ( 1 + F ′ ( t ) ) φ ( t + F ( t ) ) − φ ( t ) F ( t ) = 1 F ( t ) ∫ t t + F ( t ) φ ′ ( τ ) d τ + F ′ ( t ) F ( t ) φ ( t + F ( t ) ) − 1 F ( t ) ∫ t t + F ( t ) φ ( τ ) d τ .

Therefore, we obtain (see (1.7) and (3.1) in Lemma 3.1)

limsup t → ∞ t ψ ′ ( t ) − 1 F ( t ) ∫ t t + F ( t ) φ ′ ( τ ) d τ = limsup t → ∞ t F ′ ( t ) F ( t ) φ ( t + F ( t ) ) − 1 F ( t ) ∫ t t + F ( t ) φ ( τ ) d τ = limsup t → ∞ ∣ F ′ ( t ) G ( t ) ∣ t F ( t ) G ( t ) ∣ φ ( t + F ( t ) ) − ψ ( t ) ∣ = limsup t → ∞ ∣ F ′ ( t ) G ( t ) ∣ ⋅ lim t → ∞ t F ( t ) G ( t ) ∣ φ ( t + F ( t ) ) − ψ ( t ) ∣ = 0 ,

i.e.

(3.6) lim t → ∞ t ψ ′ ( t ) − 1 F ( t ) ∫ t t + F ( t ) φ ′ ( τ ) d τ = 0 .

Since

1 F ( t ) ∫ t t + F ( t ) φ ′ ( τ ) d τ = 1 F ( t ) ∫ t t + F ( t ) 1 τ r ( τ ) cos 2 φ ( τ ) − log τ − p log τ cos φ ( τ ) sin φ ( τ ) + s ( τ ) sin 2 φ ( τ ) d τ ,

it suffices to show that the limits (cf. (3.5) and (3.6))

(3.7) lim t → ∞ t F ( t ) ∫ t t + F ( t ) r ( τ ) τ cos 2 φ ( τ ) d τ − 1 F ( t ) ∫ t t + F ( t ) r ( τ ) d τ cos 2 ψ ( t ) = 0 ,

(3.8) lim t → ∞ t F ( t ) ∫ t t + F ( t ) log τ − p τ log τ cos φ ( τ ) sin φ ( τ ) d τ − log t − p log t cos ψ ( t ) sin ψ ( t ) = 0 ,

and

(3.9) lim t → ∞ t F ( t ) ∫ t t + F ( t ) s ( τ ) τ sin 2 φ ( τ ) d τ − 1 F ( t ) ∫ t t + F ( t ) s ( τ ) d τ sin 2 ψ ( t ) = 0

are valid. To calculate each one of the aforementioned limits (in fact, to estimate the corresponding limit superior in the three cases), we use the Lipschitz continuity of trigonometric functions. More precisely, we use the well-known inequalities

(3.10) ∣ cos 2 x 1 − cos 2 x 2 ∣ ≤ ∣ x 1 − x 2 ∣ , x 1 , x 2 ∈ R ,

(3.11) ∣ cos x 1 sin x 1 − cos x 2 sin x 2 ∣ ≤ ∣ x 1 − x 2 ∣ , x 1 , x 2 ∈ R ,

and

(3.12) ∣ sin 2 x 1 − sin 2 x 2 ∣ ≤ ∣ x 1 − x 2 ∣ , x 1 , x 2 ∈ R .

At first, we prove (3.7). It holds (see (1.7), (1.8), (3.1) in Lemma 3.1, (3.4), and (3.10))

limsup t → ∞ t F ( t ) ∫ t t + F ( t ) r ( τ ) τ cos 2 φ ( τ ) d τ − 1 F ( t ) ∫ t t + F ( t ) r ( τ ) d τ cos 2 ψ ( t ) ≤ limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) r ( τ ) d τ cos 2 ψ ( t ) − 1 F ( t ) ∫ t t + F ( t ) r ( τ ) cos 2 φ ( τ ) d τ + limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) r ( τ ) cos 2 φ ( τ ) d τ − t F ( t ) ∫ t t + F ( t ) r ( τ ) τ cos 2 φ ( τ ) d τ ≤ limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) r ( τ ) ∣ cos 2 ψ ( t ) − cos 2 φ ( τ ) ∣ d τ + limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) r ( τ ) 1 − t τ cos 2 φ ( τ ) d τ ≤ limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) r ( τ ) ∣ ψ ( t ) − φ ( τ ) ∣ d τ + 1 F ( t ) ∫ t t + F ( t ) r ( τ ) 1 − t t + F ( t ) d τ ≤ limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) r ( τ ) F ( t ) G ( t ) t d τ + 1 F ( t ) ∫ t t + F ( t ) r ( τ ) F ( t ) t + F ( t ) d τ ≤ limsup t → ∞ 1 F ( t ) G ( t ) ∫ t t + F ( t ) r ( τ ) F ( t ) G 2 ( t ) t d τ + 1 F ( t ) G ( t ) ∫ t t + F ( t ) r ( τ ) F ( t ) G ( t ) t d τ = 0 ,

i.e. (3.7) is valid. Next, we obtain (see (1.7), (3.1) in Lemma 3.1, (3.4), and (3.11))

limsup t → ∞ t F ( t ) ∫ t t + F ( t ) log τ − p τ log τ cos φ ( τ ) sin φ ( τ ) d τ − log t − p log t cos ψ ( t ) sin ψ ( t ) ≤ limsup t → ∞ log t − p log t cos ψ ( t ) sin ψ ( t ) − 1 F ( t ) ∫ t t + F ( t ) log t − p log t cos φ ( τ ) sin φ ( τ ) d τ + limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) log t − p log t cos φ ( τ ) sin φ ( τ ) d τ − t F ( t ) ∫ t t + F ( t ) log τ − p τ log τ cos φ ( τ ) sin φ ( τ ) d τ ≤ ( 1 + p ) limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) ∣ cos ψ ( t ) sin ψ ( t ) − cos φ ( τ ) sin φ ( τ ) ∣ d τ + limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) 1 − t τ d τ + p limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) 1 log t − t τ log τ d τ ≤ ( 1 + p ) limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) ∣ ψ ( t ) − φ ( τ ) ∣ d τ + limsup t → ∞ 1 − t t + F ( t ) + p limsup t → ∞ ( t + F ( t ) ) log ( t + F ( t ) ) − t log t t log 2 t ≤ ( 1 + p ) limsup t → ∞ F ( t ) G ( t ) t 1 F ( t ) ∫ t t + F ( t ) t ∣ ψ ( t ) − φ ( τ ) ∣ F ( t ) G ( t ) d τ + limsup t → ∞ F ( t ) t + p limsup t → ∞ log t + F ( t ) t = 0 ,

which guarantees (3.8). Similarly, as in the derivation of (3.7), we have (see (1.7), (1.8), (3.1) in Lemma 3.1, (3.4), and (3.12))

limsup t → ∞ t F ( t ) ∫ t t + F ( t ) s ( τ ) τ sin 2 φ ( τ ) d τ − 1 F ( t ) ∫ t t + F ( t ) s ( τ ) d τ sin 2 ψ ( t ) ≤ limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) s ( τ ) d τ sin 2 ψ ( t ) − 1 F ( t ) ∫ t t + F ( t ) s ( τ ) sin 2 φ ( τ ) d τ + limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) s ( τ ) sin 2 φ ( τ ) d τ − t F ( t ) ∫ t t + F ( t ) s ( τ ) τ sin 2 φ ( τ ) d τ ≤ limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) ∣ s ( τ ) ∣ ∣ sin 2 ψ ( t ) − sin 2 φ ( τ ) ∣ d τ + limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) ∣ s ( τ ) ∣ 1 − t τ sin 2 φ ( τ ) d τ ≤ limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) ∣ s ( τ ) ∣ ∣ ψ ( t ) − φ ( τ ) ∣ d τ + 1 F ( t ) ∫ t t + F ( t ) ∣ s ( τ ) ∣ 1 − t t + F ( t ) d τ ≤ limsup t → ∞ 1 F ( t ) ∫ t t + F ( t ) ∣ s ( τ ) ∣ F ( t ) G ( t ) t d τ + 1 F ( t ) ∫ t t + F ( t ) ∣ s ( τ ) ∣ F ( t ) t + F ( t ) d τ ≤ limsup t → ∞ 1 F ( t ) G ( t ) ∫ t t + F ( t ) ∣ s ( τ ) ∣ F ( t ) G 2 ( t ) t d τ + 1 F ( t ) G ( t ) ∫ t t + F ( t ) ∣ s ( τ ) ∣ F ( t ) G ( t ) t d τ = 0 ,

which yields the validity of (3.9). The proof is complete.□

4 Proof of main results and their corollaries

To prove Theorem 1.3, we need the following known lemmas.

Lemma 4.1

If a solution φ : R e → R of equation (2.2) satisfies

(4.1) limsup t → ∞ φ ( t ) < ∞ ,

then equation (1.2) is non-oscillatory.

Proof

See [4, Lemma 3.1, (B)].□

Lemma 4.2

Let X > 0 and Y ∈ R be such that X Y < 1 / 4 . If ϑ : R e → R is a solution of the equation

(4.2) ϑ ′ ( t ) = 1 t X cos 2 ϑ ( t ) − log t − p log t cos ϑ ( t ) sin ϑ ( t ) + Y sin 2 ϑ ( t ) ,

then

(4.3) limsup t → ∞ ϑ ( t ) < ∞ .

Proof

See [4, Lemma 3.4, (B)].□

Now, we can prove the main results.

The proof of Theorem 1.3

Let φ : R e → R be a solution of equation (2.2) and let the corresponding averaging function ψ : R e → R be defined in (2.3). Taking into account (3.5) in Lemma 3.2, one can consider ω : R e → R satisfying

(4.4) lim t → ∞ ∣ ω ( t ) ∣ = 0

and

(4.5) ψ ′ ( t ) = 1 t ω ( t ) + 1 F ( t ) ∫ t t + F ( t ) r ( τ ) d τ cos 2 ψ ( t ) − log t − p log t cos ψ ( t ) sin ψ ( t ) + 1 F ( t ) ∫ t t + F ( t ) s ( τ ) d τ sin 2 ψ ( t ) .

In fact, the function ω is given by the whole term in the absolute value in (3.5).

From (1.10), it follows the existence of ε > 0 for which

(4.6) ( r F + 2 ε ) ( s F + 2 ε ) < 1 4 .

Let us consider t so large that (see (4.4))

(4.7) ∣ ω ( t ) ∣ < ε

and (see (1.9))

(4.8) 1 F ( t ) ∫ t t + F ( t ) r ( τ ) d τ < r F + ε , 1 F ( t ) ∫ t t + F ( t ) s ( τ ) d τ < s F + ε .

Hence, from (4.5), (4.7), and (4.8), we obtain

(4.9) ψ ′ ( t ) < 1 t ε + ( r F + ε ) cos 2 ψ ( t ) − log t − p log t cos ψ ( t ) sin ψ ( t ) + ( s F + ε ) sin 2 ψ ( t ) = 1 t ( r F + 2 ε ) cos 2 ψ ( t ) − log t − p log t cos ψ ( t ) sin ψ ( t ) + ( s F + 2 ε ) sin 2 ψ ( t ) .

If we put X ≔ r F + 2 ε and Y ≔ s F + 2 ε , then (4.9) gives

(4.10) ψ ′ ( t ) < 1 t X cos 2 ψ ( t ) − log t − p log t cos ψ ( t ) sin ψ ( t ) + Y sin 2 ψ ( t ) ,

where (see (4.6))

(4.11) X Y < 1 4 .

Considering (4.11) and comparing equation (4.2) with (4.10) for ψ ≡ ϑ , Lemma 4.2 gives the inequality (see (4.3))

limsup t → ∞ ψ ( t ) < ∞ .

In addition, from Lemma 3.1, we have (see (3.2))

(4.12) limsup t → ∞ φ ( t ) = limsup t → ∞ ψ ( t ) < ∞ .

Now, we apply Lemma 4.1 (cf. (4.1) and (4.12)), which proves the non-oscillation of equation (1.2).

The proof of Theorem 1.4

It suffices to consider the intersection of the assumptions of Theorems 1.2 and 1.3 for h ≡ f ≡ F and H ≡ g ≡ G , where (1.3) and (1.7) ⇒ (1.11), (1.4) and (1.8) ⇒ (1.12), (1.5) and (1.9) ⇒ (1.13), (1.6) ⇒ (1.14), and (1.10) ⇒ (1.15).

To substantiate the novelty of Theorems 1.3 and 1.4 in several cases (e.g. for all p > 0 ), we formulate new corollaries below.

Corollary 4.1

Let a continuously differentiable function F : R e → ( 0 , ∞ ) and a continuous function G : R e → [ 1 , ∞ ) satisfy (1.7). Let us consider the linear equation

(4.13) log t r ( t ) x ′ ( t ) ′ + log t t 2 s ( t ) x ( t ) = 0 ,

where r : R e → ( 0 , ∞ ) and s : R e → R are the bounded and continuous functions. Let r F , s F be defined in (1.9). If r F s F < 1 / 4 , then equation (4.13) is non-oscillatory.

Proof

It suffices to put p = 1 in equation (1.2) and use Theorem 1.3, where (1.8) follows from the boundedness of r and s and from the first limit in (1.7).□

Corollary 4.2

Let a continuously differentiable function h : R e → ( 0 , ∞ ) and a continuous function H : R e → [ 1 , ∞ ) satisfy (1.11). Let us consider equation (4.13), where r : R e → ( 0 , ∞ ) and s : R e → R are bounded and continuous functions. Let r h and s h be defined in (1.13).

  1. If r h s h > 1 / 4 , then equation (4.13) is oscillatory.

  2. If r h s h < 1 / 4 , then equation (4.13) is non-oscillatory.

Proof

It suffices to put p = 1 in equation (1.2). Then, the corollary follows from Theorem 1.4, where (1.12) is guaranteed by the boundedness of r and s and (1.11).□

Corollaries 4.1 and 4.2 are new also in the case when the first coefficient is constant and the second one does not change its sign. Thus, we mention the following two corollaries for a different choice of p and for concrete auxiliary functions together with examples of equations whose oscillation behaviour does not follow from any previously known result.

Corollary 4.3

Let a > 1 . Let us consider the linear equation

(4.14) ( log 2 t x ′ ( t ) ) ′ + log t t 2 s ( t ) x ( t ) = 0 ,

where s : R e → ( 0 , ∞ ) is a continuous function. If

(4.15) limsup t → ∞ 1 t a ∫ t t + t a s ( τ ) d τ < 1 4 ,

then equation (4.14) is non-oscillatory.

Proof

It suffices to consider Theorem 1.3 for r ( t ) = 1 , F ( t ) = t a , and, e.g. G ( t ) = log t , t ∈ R e . One can easily verify that (1.7) is valid and that (1.8) follows from the choice of r , the positivity of s , the first limit in (1.7), and (4.15).□

Example 1

Let A ∈ ( 0 , 1 / 4 ) and let us consider equation (4.14) for

s ( t ) ≔ A + t − 2 n , t ∈ [ 2 n , 2 n + n ] , n ∈ N ; A + n − ( t − 2 n − n ) , t ∈ ( 2 n + n , 2 n + 2 n ] , n ∈ N ; A , t ∉ [ 2 n , 2 n + 2 n ] , n ∈ N ,

where t ∈ R e . For any a > 1 , it holds

limsup t → ∞ 1 t a ∫ t t + t a s ( τ ) d τ = limsup n → ∞ 1 2 n / a ∫ 2 n 2 n + 2 n / a s ( τ ) d τ ≤ A + limsup n → ∞ 1 2 n / a ∫ 2 n 2 n + 2 n n d τ = A < 1 4 ;

i.e. (4.15) is fulfilled for any a > 1 . Corollary 4.3 says that equation (4.14) is non-oscillatory. Note that, for all A ≤ 0 , the non-oscillation of equation (4.14) follows from the famous Sturm comparison theorem if one proves its non-oscillation for some A > 0 .

Corollary 4.4

Let us consider equation (4.14), where s : R e → ( 0 , ∞ ) is a bounded and continuous function. Let a > 1 and let

s a ≔ lim t → ∞ 1 t a ∫ t t + t a s ( τ ) d τ ∈ R .

  1. If s a > 1 / 4 , then equation (4.14) is oscillatory.

  2. If s a < 1 / 4 , then equation (4.14) is non-oscillatory.

Proof

Now, it suffices to consider Theorem 1.4 for r ( t ) = 1 , h ( t ) = t a , and, e.g. H ( t ) = log t , t ∈ R e . It is seen that (1.11) and (1.12) are true (consider the boundedness of the coefficients).□

Example 2

Let A > B > 0 and let us consider equation (4.14) for

s ( t ) ≔ A + B t − 2 n n , t ∈ [ 2 n , 2 n + n ] , n ∈ N , n ≥ 4 ; A + B − B t − 2 n − n n , t ∈ ( 2 n + n , 2 n + 3 n ] , n ∈ N , n ≥ 4 ; A − B + B t − 2 n − 3 n n , t ∈ ( 2 n + 3 n , 2 n + 4 n ] , n ∈ N , n ≥ 4 ; A , t ∉ [ 2 n , 2 n + 4 n ] , n ∈ N , n ≥ 4 ,

where t ∈ R e . For all a > 1 , we obtain

liminf t → ∞ 1 t a ∫ t t + t a s ( τ ) d τ = liminf n → ∞ 1 2 n / a ∫ 2 n 2 n + 2 n / a s ( τ ) d τ ≥ A − limsup n → ∞ 1 2 n / a ∫ 2 n + 2 n 2 n + 4 n B d τ = A ,

limsup t → ∞ 1 t a ∫ t t + t a s ( τ ) d τ = limsup n → ∞ 1 2 n / a ∫ 2 n 2 n + 2 n / a s ( τ ) d τ ≤ A + limsup n → ∞ 1 2 n / a ∫ 2 n 2 n + 2 n B d τ = A .

Thus,

s a = lim t → ∞ 1 t a ∫ t t + t a s ( τ ) d τ = A ∈ R .

We can apply Corollary 4.4. If A > 1 / 4 , then equation (4.14) is oscillatory; and if A < 1 / 4 , then equation (4.14) is non-oscillatory. We remark that the case A = 1 / 4 remains an open problem.

Remark 2

Corollaries 4.1–4.4 improve some results from [4,5,8,9,45,46] in a certain sense.

Remark 3

In Example 2, there is mentioned that the border case A = 1 / 4 is not solved. In general, the limiting case

(4.16) lim t → ∞ 1 h ( t ) ∫ t t + h ( t ) r ( τ ) d τ lim t → ∞ 1 h ( t ) ∫ t t + h ( t ) s ( τ ) d τ = 1 4

remains unsolved (see Theorem 1.4). In some articles (see, e.g. [47]), there is described a conjecture that the oscillation of the considered (modified Euler type) conditionally oscillatory equations is not solvable for non-periodic coefficients in the limiting case. More precisely, the conjecture says that equation (1.2) is oscillatory for some almost periodic or even limit periodic functions r and s satisfying (4.16) and that equation (1.2) is non-oscillatory for other almost (or limit) periodic functions r and s satisfying (4.16).

  1. Funding information: This research was supported by Czech Science Foundation under Grant GA20-11846S.

  2. Author contributions: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.

  3. Conflict of interest: The author declares no conflicts of interest.

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Received: 2023-11-14
Revised: 2024-03-31
Accepted: 2024-04-11
Published Online: 2024-06-12

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2024-0012
Keywords for this article
linear equation; oscillation theory; non-oscillation; Riccati equation; Prüfer angle
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  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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