Oscillation and Nonoscillation of Asymptotically Almost Periodic Half-Linear Difference Equations

Abstract

We analyse half-linear difference equations with asymptotically almost periodic coefficients. Using the adapted Riccati transformation, we prove that these equations are conditionally oscillatory. We explicitly find a constant, determined by the coefficients of a given equation, which is the borderline between the oscillation and the nonoscillation of the equation. We also mention corollaries of our result with several examples.

1. Introduction

We intend to study the oscillation behaviour of the second-order half-linear difference equation Δ𝑟_𝑘ΦΔ𝑦_𝑘+𝑝_𝑘Φ𝑦_𝑘+1=0, (1) where Φ(𝑦)=|𝑦|^𝑝−1sgn𝑦, 𝑝>1, and 𝑟_𝑘≠0 for all considered 𝑘. Throughout this paper, we consider integers 𝑘≥𝑚 for a sufficiently large 𝑚∈ℕ. Concerning the oscillation and nonoscillation of these equations, many results have been obtained in the last years. The basic facts of the oscillation theory for (1) are established in [1]. Several criteria for (non)oscillation of (1) are proved, for example, in [2–6] (for (1) with 𝑟_𝑘≡1, see [7–9]; for the continuous case, see [10–12]). Oscillatory properties of slightly more general difference equations are investigated, for example, in [13, 14]. A detailed literature overview can be found in [15].

The oscillation theory of half-linear difference equations is motivated by the continuous case and by the linear case.

Equation (1) is the discretization of the half-linear differential equation 𝑟(𝑡)Φ𝑦^′(𝑡)^′+𝑝(𝑡)Φ𝑦(𝑡)=0,𝑟(𝑡)>0. (2) The first occurrence of an equation of this form is dated back to 1961 (see [16]). But [17, 18] are regarded as the basic papers in this field. For a comprehensive overview of half-linear differential equations, we refer to [19] with references cited therein. Note that techniques needed in the discrete theory are often more complicated than their continuous counterparts. In this paper, we apply the Riccati method which is used in the continuous case as well. Comparing this paper with its continuous counterpart [20], one can see that the discrete method requires several nontrivial extra steps and is technically more exacting. Particularly, we cannot use the fact that the ranges of considered solutions are intervals.

Equation (1) is a generalization of the linear equation Δ𝑟_𝑘Δ𝑦_𝑘+𝑝_𝑘𝑦_𝑘+1=0,𝑟_𝑘≠0, (3) whose oscillation theory is well described in the literature (see [21, 22] and references therein). There are limitations of the tools known from the theory of linear equations (e.g., the transformation theory and the Casoratian identity). These limitations are more or less consequences of the fact that the solution space of half-linear equations is not additive. Nevertheless, it remains homogeneous (likewise in the linear case). This observation motivates the term “half-linear.”

Actually, we consider (1) in the form Δ𝑟_𝑘ΦΔ𝑦_𝑘+𝛾𝑠_𝑘(𝑘+1)^(𝑝)Φ𝑦_𝑘+1=0, (4) where {𝑟_𝑘}, {𝑠_𝑘} are positive sequences, 𝛾>0, and 𝑘^(𝑝) stands for the so-called generalized power function. Our objective is to prove that (4) is conditionally oscillatory and to find the corresponding oscillation constant. We recall that (4) is conditionally oscillatory if there exists Γ>0 with the property that (4) is oscillatory for 𝛾>Γ and nonoscillatory for 𝛾<Γ. In this case, Γ is called the oscillation constant of (4). Note that we consider the form with the generalized power function 1/(𝑘+1)^(𝑝) because we want to be consistent with previous results and our method is much more transparent in this case.

According to the best of our knowledge, the first result about the conditional oscillation of a second-order equation was proved by Kneser back in 1893 (see [23]). In [23], it is shown that the differential equation 𝑦^′′(𝑡)+𝛾𝑡²𝑦(𝑡)=0 (5) is conditionally oscillatory with the oscillation constant Γ=1/4. The corresponding difference equation Δ²𝑦_𝑘+𝛾()𝑘𝑦𝑘+1^𝑘+1=0 (6) is conditionally oscillatory with the oscillation constant Γ=1/4 as well (see [24]). In [25], the previous Kneser result is generalized for the equation 𝑟(𝑡)𝑦^′(𝑡)^′+𝛾𝑠(𝑡)𝑡²𝑦(𝑡)=0, (7) where 𝑟,𝑠∶ℝ→ℝ are positive periodic continuous functions. We can also mention [26] and more generally [27, 28]. For the discrete linear equation Δ𝑟_𝑘Δ𝑦_𝑘+𝛾𝑠_𝑘()𝑘𝑦𝑘+1^𝑘+1=0, (8) we refer to [29], where the corresponding result is obtained. Nevertheless, the first discrete version of the result of [25] was proved in [30]. The oscillation constant from [30] coincides with the one from [29] if the considered coefficients are asymptotically constant. In fact, the conditional oscillation of (8) with almost periodic coefficients is proved in [29]. The oscillation constant for the periodic half-linear equation 𝑟(𝑡)Φ𝑦^′(𝑡)^′+𝛾𝑠(𝑡)𝑡^𝑝Φ𝑦(𝑡)=0 (9) is derived in [31] (see also [32]) and the oscillation constant for (9) with asymptotically almost periodic coefficients is obtained in [20]. Our aim is to prove the discrete counterpart of the main result of [20]; that is, we find the oscillation constant for (4) with asymptotically almost periodic coefficients. We add that this result is new in the periodic case and in the linear case as well.

The paper is organized as follows. In Section 2, we recall elements of the theory of almost periodic and asymptotically almost periodic sequences, which we will use later. In Section 3, we mention basic definitions from the oscillation theory and we derive the adapted Riccati equation. Using the prepared Riccati technique, we prove the announced result in Section 4. Then, we state its corollaries and we illustrate them by examples. We also show a way how the main result can be applied for equations with coefficients which change their signs.

Henceforth, let 𝑝>1 be arbitrarily given and let 𝑞∈ℝ be such that 1𝑝+1𝑞=1,i.e.,𝑝+𝑞=𝑝𝑞. (10) Thus, for Φ(𝑥)=|𝑥|^𝑝−1sgn𝑥, we have Φ⁻¹(𝑥)=|𝑥|^𝑞−1sgn𝑥. For the reader’s convenience, we recall the generalized power function (sometimes called the falling factorial power) 𝑘^(𝛼)=Γ(𝑘+1)Γ()()𝑘+1−𝛼,𝛼∈−1,𝑘,𝑘∈ℕ, (11) where Γ(𝑥)=^∞₀𝑒^−𝑠𝑠^𝑥−1𝑑𝑠,𝑥>1, (12) and the identities Δ𝑘^(𝛼+1)=(𝛼+1)𝑘^(𝛼),𝑘^(𝛼+1)𝑘^(𝛼)()=𝑘−𝛼,𝛼∈−1,𝑘,𝑘∈ℕ, (13) which we will often apply later. We also refer to [22, Chapter 2].

2. Asymptotic Almost Periodicity

We begin with the notion of almost periodicity. The fundamental results of the theory of almost periodic sequences can be found, for example, in [33, Section I.6].

Definition 1. A sequence {𝑓_𝑘}_𝑘∈ℤ⊂ℝ is called almost periodic if, for every 𝜀>0, there exists an integer 𝑝(𝜀) with the property that any set consisting of 𝑝(𝜀) consecutive integers contains at least one number 𝑙 for which 𝑓_𝑘+𝑙−𝑓_𝑘<𝜀,𝑘∈ℤ. (14)

Definition 2. A continuous function 𝑓∶ℝ→ℝ is called almost periodic if, for every 𝜀>0, there exists a number 𝑝(𝜀)>0 such that any interval of length 𝑝(𝜀) of the real line contains at least one point 𝑠 satisfying 𝑓(𝑡+𝑠)−𝑓(𝑡)<𝜀,𝑡∈ℝ. (15)

Theorem 3. A necessary and sufficient condition for a sequence {𝑓_𝑘}_𝑘∈ℤ⊂ℝ to be almost periodic is the existence of an almost periodic function 𝑔∶ℝ→ℝ with the property that 𝑓_𝑘=𝑔(𝑘) for all 𝑘∈ℤ.

Proof. See [33, Theorem 1.27].

Theorem 4. Any almost periodic sequence is bounded.

Proof. The statement of the theorem follows from Theorem 3 and from the boundedness of any almost periodic function on the real line (see, e.g., [33, Theorem 1.2]).

Theorem 5. A sequence {𝑓_𝑘}_𝑘∈ℤ⊂ℝ is almost periodic if and only if any sequence of the form {{𝑓_{𝑘+ℎ(𝑛)}}_𝑘∈ℤ}_𝑛∈ℕ, where ℎ(𝑛)∈ℤ, has a uniformly convergent subsequence with respect to 𝑘.

Proof. See [33, Theorem 1.26].

The following result is a consequence of Theorems 4 and 5.

Corollary 6. The sum and the product of two almost periodic sequences are almost periodic as well.

Theorem 7. If {𝑓_𝑘}_𝑘∈ℤ is almost periodic, then the limit lim_𝑘→∞1𝑘^{𝑗+𝑘−1}_𝑖=𝑗𝑓_𝑖 (16) is finite and exists uniformly with respect to 𝑗∈ℤ.

Proof. See [33, Theorem 1.28].

Theorem 8. If {𝑓_𝑘}_𝑘∈ℤ⊂ℝ is almost periodic and 𝐹∶ℝ→ℝ is a uniformly continuous function on the set {𝑓_𝑘;𝑘∈ℤ}, then the composition {𝐹(𝑓_𝑘)}_𝑘∈ℤ is almost periodic.

Proof. The theorem follows, for example, from [34, Theorem 2.4].

As a direct generalization of almost periodicity, we consider the asymptotic almost periodicity. We refer to [35], where asymptotically almost periodic sequences were introduced with their basic characteristics.

Definition 9. We say that a sequence {𝑓_𝑘}_𝑘∈ℕ⊂ℝ is asymptotically almost periodic if it can be represented in the form 𝑓_𝑘=𝑓¹_𝑘+𝑓²_𝑘, 𝑘∈ℕ, where {𝑓¹_𝑘}_𝑘∈ℤ is almost periodic and {𝑓²_𝑘}_𝑘∈ℕ has the property that lim_𝑘→∞𝑓²_𝑘=0. The sequence {𝑓¹_𝑘} is called the almost periodic part of {𝑓_𝑘}.

Considering Theorem 7, we immediately obtain the following.

Corollary 10. Let an asymptotically almost periodic sequence {𝑓_𝑘}_𝑘∈ℕ be given, and let {𝑓¹_𝑘}_𝑘∈ℤ be its almost periodic part. The limit 𝑀𝑓_𝑘∶=lim_𝑘→∞1𝑘^{𝑗+𝑘−1}_𝑖=𝑗𝑓_𝑖=lim_𝑘→∞1𝑘^{𝑗+𝑘−1}_𝑖=𝑗𝑓¹_𝑖 (17) is finite and exists uniformly with respect to 𝑗∈ℕ.

Definition 11. The number 𝑀({𝑓_𝑘}) introduced in (17) is called the mean value of an asymptotically almost periodic sequence {𝑓_𝑘}_𝑘∈ℕ.

Remark 12. According to Definition 1 and (17), we have 𝑀({𝑓_𝑘})>0 for any positive asymptotically almost periodic sequence {𝑓_𝑘}_𝑘∈ℕ with the property that limsup_𝑘→∞𝑓_𝑘>0.

Corollary 13. If {𝑓_𝑘}_𝑘∈ℕ is an asymptotically almost periodic sequence satisfying inf𝑓_𝑘;𝑘∈ℕ>0, (18) then the sequence {𝑔_𝑘}_𝑘∈ℕ, defined by the formula 𝑔_𝑘=𝑓^𝛼_𝑘, is asymptotically almost periodic for arbitrarily given 𝛼∈ℝ.

Proof. Let 𝑓_𝑘=𝑓¹_𝑘+𝑓²_𝑘, 𝑘∈ℕ, where {𝑓¹_𝑘}_𝑘∈ℤ is the almost periodic part of {𝑓_𝑘}. Considering Definition 1, we have inf𝑓¹_𝑘;𝑘∈ℤ=inf𝑓¹_𝑘;𝑘∈ℤ,𝑘≥𝑎,𝑎∈ℝ, (19) and therefore (see (18)) inf𝑓¹_𝑘;𝑘∈ℤ≥inf𝑓_𝑘;𝑘∈ℕ>0. (20) If we put 𝑔¹_𝑘∶=[𝑓¹_𝑘]^𝛼, 𝑘∈ℤ, then Theorems 4 and 8 imply the almost periodicity of {𝑔¹_𝑘}_𝑘∈ℤ. Since lim_𝑘→∞𝑔_𝑘−𝑔¹_𝑘=lim_𝑘→∞𝑓¹_𝑘+𝑓²_𝑘^𝛼−𝑓¹_𝑘^𝛼=0, (21) we know that {𝑔_𝑘} can be expressed as the sum of the almost periodic sequence {𝑔¹_𝑘} and the sequence {𝑔_𝑘−𝑔¹_𝑘} vanishing at infinity. It means that {𝑔_𝑘} is asymptotically almost periodic.

3. Half-Linear Riccati Equation

In this section, we recall the oscillation behaviour of half-linear difference equations and we introduce the (adapted) half-linear Riccati equation. For more details, we refer to [15, Chapter 3] or [19, Chapter 8] (with references cited therein). We remark that the (adapted) Riccati technique is an essential tool to prove the following given Theorem 17.

Let us consider the equation Δ𝑟_𝑘ΦΔ𝑥_𝑘+𝑝_𝑘Φ𝑥_𝑘+1=0, (22) where 𝑟_𝑘,𝑝_𝑘>0 for all considered 𝑘 and the sequence {𝑟_𝑘} is bounded. We say that an interval (𝑙,𝑙+1], 𝑙∈ℕ, contains the generalized zero of a solution {𝑥_𝑘} of (22) if 𝑥_𝑙≠0 and 𝑥_𝑙𝑥_𝑙+1≤0. Equation (22) is said to be disconjugate on a set {𝑙,𝑙+1,…,𝑙+𝑛} provided any solution of (22) has at most one generalized zero on (𝑙,𝑙+𝑛+1] and the solution {𝑥_𝑘} given by the initial value 𝑥_𝑙=0 has no generalized zero on (𝑙,𝑙+𝑛+1]. Otherwise, (22) is said to be conjugate on {𝑙,𝑙+1,…,𝑙+𝑛}.

The Sturm type separation theorem is valid for (22) (see [1, Theorem 3]). Thus, the oscillation of an arbitrary solution of (22) implies the oscillation of all solutions of (22). It means that we can categorize (22) as oscillatory or nonoscillatory.

Definition 14. Equation (22) is called nonoscillatory if there exists 𝑛∈ℕ with the property that (22) is disconjugate on {𝑛,𝑛+1,…,𝑛+𝑖} for all 𝑖∈ℕ. In the opposite case, (22) is called oscillatory.

Now, we turn our attention to the half-linear Riccati equation. Using the substitution 𝑤_𝑘=𝑟_𝑘ΦΔ𝑥_𝑘𝑥_𝑘, (23) we transform (22) into Δ𝑤_𝑘+𝑝_𝑘+𝑤_𝑘𝑟1−^𝑘ΦΦ⁻¹𝑟_𝑘+Φ⁻¹𝑤_𝑘=0. (24) According to [15, Lemma 3.2.6, (I₈)], if 𝑤_𝑘+𝑟_𝑘>0, then 𝑤_𝑘𝑟1−^𝑘ΦΦ⁻¹𝑟_𝑘+Φ⁻¹𝑤_𝑘=𝑝−1𝑤_𝑘^𝑞𝛽_𝑘^𝑝−2ΦΦ⁻¹𝑟_𝑘+Φ⁻¹𝑤_𝑘, (25) where 𝛽_𝑘 is between Φ⁻¹(𝑟_𝑘) and Φ⁻¹(𝑟_𝑘)+Φ⁻¹(𝑤_𝑘); that is, for 𝑤_𝑘+𝑟_𝑘>0, we can rewrite (24) in the following form: Δ𝑤_𝑘+𝑝_𝑘+𝑝−1𝑤_𝑘^𝑞𝛽_𝑘^𝑝−2ΦΦ⁻¹𝑟_𝑘+Φ⁻¹𝑤_𝑘=0. (26)

Next, putting 𝜁_𝑘=−𝑘^(𝑝−1)𝑤_𝑘, (27) we obtain Δ𝜁_𝑘=−Δ𝑘^(𝑝−1)𝑤_𝑘=−𝑝−1𝑘^(𝑝−2)𝑤_𝑘−(𝑘+1)^(𝑝−1)Δ𝑤_𝑘, (28) and then, applying (26), we obtain the adapted Riccati equation Δ𝜁_𝑘=1×𝜁𝑘−𝑝+2𝑝−1_𝑘+(𝑘+1)^(𝑝)𝑝_𝑘+(𝑘+1)𝑝−1𝛽_𝑘^𝑝−2𝜁_𝑘^𝑞𝑘^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑘+Φ⁻¹−𝜁_𝑘/𝑘^(𝑝−1), (29) where 𝛽_𝑘 is between Φ⁻¹(𝑟_𝑘) and Φ⁻¹(𝑟_𝑘)+Φ⁻¹(−𝜁_𝑘/𝑘^(𝑝−1)).

4. Conditional Oscillation

To prove the announced theorem, we need the following two lemmas.

Lemma 15. If (22) is nonoscillatory, then there exists a negative solution {𝜁_𝑘}_𝑘≥𝑘0 of (29) such that lim_𝑘→∞𝜁_𝑘𝑘^(𝑝−1)=0. (30)

Proof. Applying [1, Theorem 1] (see also [15, Theorem 3.3.4]), we know that the nonoscillation of (22) implies the existence of a solution {𝑤_𝑘}_𝑘≥𝑘0 of (26) satisfying 𝑤_𝑘+𝑟_𝑘>0, 𝑘≥𝑘₀. Considering [36, Lemma 1, (v) and Theorem 1], we obtain lim_𝑘→∞𝑤_𝑘=0. Since the sequence {𝑤_𝑘} is decreasing (see directly (26)), we have 𝑤_𝑘>0 for all 𝑘≥𝑘₀. Thus, there exists a negative solution {𝜁_𝑘}_𝑘≥𝑘0 of (29). Finally, (27) gives lim_𝑘→∞𝜁_𝑘𝑘^(𝑝−1)=−lim_𝑘→∞𝑤_𝑘=0. (31)

Lemma 16. If there exists a negative solution {𝜁_𝑘}_𝑘≥𝑘0 of (29), then (22) is nonoscillatory.

Proof. The statement of the lemma follows from [1, Theorem 1].

Now we can formulate and prove the main result.

Theorem 17. Let 𝛾∈ℝ⁺ be given, and let {𝑟_𝑘}_𝑘∈ℕ and {𝑠_𝑘}_𝑘∈ℕ be arbitrary positive asymptotically almost periodic sequences such that inf𝑟_𝑘;𝑘∈ℕ>0,limsup_𝑘→∞𝑠_𝑘>0. (32) Let Γ∶=𝑞^−𝑝𝑀𝑟_𝑘^{1−𝑞1−𝑝}𝑀𝑠_𝑘⁻¹. (33) Consider the equation Δ𝑟_𝑘ΦΔ𝑥_𝑘+𝛾𝑠_𝑘(𝑘+1)^(𝑝)Φ𝑥_𝑘+1=0. (34) If 𝛾>Γ, then (34) is oscillatory. If 𝛾<Γ, then (34) is nonoscillatory.

Proof. We put (see Theorem 4) 𝑟⁻∶=inf𝑟_𝑘,𝑟;𝑘∈ℕ₊∶=sup𝑟_𝑘,𝑠;𝑘∈ℕ₊∶=sup𝑠_𝑘;𝑘∈ℕ. (35) For an arbitrarily given number 𝛾≠Γ, we will consider an integer 𝛼≥2 such that 1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖>𝑀𝑠_𝑘2>0,𝑘∈ℕ,Γ−𝑞^−𝑝1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑟_𝑖^{1−𝑞1−𝑝}1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖⁻¹<Γ−𝛾2,𝑘∈ℕ. (36)(37) That is, 𝛾−𝑞^−𝑝1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑟_𝑖^{1−𝑞1−𝑝}1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖⁻¹>𝛾−Γ2,𝑘∈ℕ. (38) From Remark 12 and Corollaries 10 and 13, we know that such a number 𝛼 exists.
At first, we consider the oscillatory part of the theorem. By contradiction, we suppose that 𝛾>Γ is given and that (34) is nonoscillatory. Lemma 15 says that there exists a negative solution {𝜁_𝑘}_𝑘≥𝑘0 of the equation Δ𝜁_𝑘=1×𝜁𝑘−𝑝+2𝑝−1_𝑘+𝛾𝑠_𝑘+(𝑘+1)𝑝−1𝛽_𝑘^𝑝−2𝜁_𝑘^𝑞𝑘^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑘+Φ⁻¹−𝜁_𝑘/𝑘^(𝑝−1), (39) where Φ⁻¹𝑟_𝑘≤𝛽_𝑘≤Φ⁻¹𝑟_𝑘+Φ⁻¹−𝜁_𝑘𝑘^(𝑝−1),𝑘≥𝑘₀. (40) Note that (39) is the adapted Riccati equation associated with (34) and that (30) holds. Particularly, from (30), we know lim_𝑘→∞ΦΦ⁻¹𝑟_𝑘+Φ⁻¹−𝜁_𝑘/𝑘^(𝑝−1)𝑟_𝑘=1 (41) and (see (40)) lim_𝑘→∞𝛽_𝑘Φ⁻¹𝑟_𝑘=lim_𝑘→∞𝛽_𝑘𝑟_𝑘^𝑞−1=1. (42) Using (consider, e.g., the Stirling formula) lim_𝑘→∞𝑘^𝑎𝑘^(𝑎)=lim_𝑘→∞𝑘^𝑎Γ(𝑘+1−𝑎)Γ(𝑘+1)=1,𝑎>0, (43) we also obtain lim_𝑘→∞𝑘+1𝑘^{(𝑝−1)𝑞−1}=lim_𝑘→∞𝑘+1𝑘^{𝑝−1𝑞−1}=lim_𝑘→∞𝑘+1𝑘=1. (44) Combining (39), (41), (42), and (44), we can find an integer 𝑘≥𝑘₀ with the property that Δ𝜁_𝑘≥1𝜁𝑘−𝑝+2𝑝−1_𝑘+12𝑝−1𝑟_𝑘^{[𝑞−1][𝑝−2]}𝜁_𝑘^𝑞𝑟_𝑘=𝑝−1𝜁𝑘−𝑝+2^𝑘+12𝑟_𝑘^1−𝑞𝜁_𝑘^𝑞≥𝑝−1𝜁𝑘−𝑝+2^𝑘+𝜁_𝑘^𝑞2[𝑟⁺]^𝑞−1 (45) and that Δ𝜁_𝑘≤1𝑘−𝑝+2𝛾𝑠⁺𝑟+2𝑝−1_𝑘^{[𝑞−1][𝑝−2]}𝜁_𝑘^𝑞𝑟_𝑘=1𝑘−𝑝+2𝛾𝑠⁺+2𝑝−1𝜁_𝑘^𝑞𝑟_𝑘^𝑞−1≤1×𝑘−𝑝+2𝛾𝑠⁺+2𝑝−1𝜁_𝑘^𝑞[𝑟⁻]^𝑞−1, (46) whenever 𝑘≥𝑘. Let 𝜁_𝑘<−2^1/(𝑞−1)𝑟⁺ for some 𝑘≥𝑘. In this case, we estimate 𝜁_𝑘^𝑞=𝜁_𝑘⋅𝜁_𝑘^𝑞−1>𝜁_𝑘⋅2^1/(𝑞−1)𝑟^+𝑞−1=𝜁_𝑘⋅2𝑟^+𝑞−1, (47) and hence, by (45), we get Δ𝜁_𝑘>0. Thus, there exists 𝐿>0 satisfying 𝜁_𝑘∈(−𝐿,0),𝑘≥𝑘₀. (48) Therefore (see (45) and (46)), we have Δ𝜁_𝑘<𝐾𝑘−𝑝+2,𝑘≥𝑘, (49) where 𝐾∶=𝑝−1𝐿+𝛾𝑠⁺+2𝑝−1𝐿^𝑞[𝑟⁻]^𝑞−1. (50) Particularly, (49) implies 𝜁_𝑘+𝑖−𝜁_𝑘+𝑗<(𝛼−1)𝐾{}𝑘−𝑝+2,𝑖,𝑗∈0,…,𝛼−1,𝑘≥𝑘. (51)
We define 𝜉_𝑘1∶=𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝜁_𝑖,𝑘≥𝑘₀,𝑋_𝑘𝑝∶=𝑝−1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑟_𝑖^{1−𝑞−1/𝑞},𝑌_𝑘∶=𝜉_𝑘𝑝𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑟_𝑖^{1−𝑞1/𝑞},𝑘≥𝑘₀. (52) From (48), we can see 𝜉_𝑘∈(−𝐿,0),𝑘≥𝑘₀. (53)
For 𝑘≥𝑘₀, we compute (see (39)) Δ𝜉_𝑘=1𝛼^{𝑘+𝛼−1}_𝑖=𝑘Δ𝜁_𝑖=1𝛼^{𝑘+𝛼−1}_𝑖=𝑘1×𝜁𝑖−𝑝+2𝑝−1_𝑖+𝛾𝑠_𝑖+(𝑖+1)𝑝−1𝛽_𝑖^𝑝−2𝜁_𝑖^𝑞𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)≥1×𝜉𝑘+𝛼−𝑝+1𝑝−1_𝑘+𝛾𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖+1𝛼^{𝑘+𝛼−1}_𝑖=𝑘(𝑖+1)𝑝−1𝛽_𝑖^𝑝−2𝜁_𝑖^𝑞𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)=1×𝜉𝑘+𝛼−𝑝+1𝑝−1_𝑘+𝑋^𝑝_𝑘𝑝+𝑌^𝑞_𝑘𝑞+𝛾𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖−𝑋^𝑝_𝑘𝑝+1𝛼^{𝑘+𝛼−1}_𝑖=𝑘(𝑖+1)𝑝−1𝛽_𝑖^𝑝−2𝜁_𝑖^𝑞𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)−𝑌^𝑞_𝑘𝑞 (54) if Δ𝜉_𝑘>0, or Δ𝜉_𝑘≥1×𝜉𝑘−𝑝+2𝑝−1_𝑘+𝑋^𝑝_𝑘𝑝+𝑌^𝑞_𝑘𝑞+𝛾𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖−𝑋^𝑝_𝑘𝑝+1𝛼^{𝑘+𝛼−1}_𝑖=𝑘(𝑖+1)𝑝−1𝛽_𝑖^𝑝−2𝜁_𝑖^𝑞𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)−𝑌^𝑞_𝑘𝑞 (55) if Δ𝜉_𝑘<0.
Since 𝑋_𝑘𝑌_𝑘=𝜉_𝑘𝑝−1=−𝜉_𝑘𝑝−1,𝑘≥𝑘₀, (56) it holds 𝑝−1𝜉_𝑘+𝑋^𝑝_𝑘𝑝+𝑌^𝑞_𝑘𝑞=𝑋^𝑝_𝑘𝑝+𝑌^𝑞_𝑘𝑞−𝑋_𝑘𝑌_𝑘≥0,𝑘≥𝑘₀, (57) which follows from the Young inequality.
Using (36), (37), and (38), we have 𝛾𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖−𝑋^𝑝_𝑘𝑝=𝛾𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖−𝑝−1^𝑝𝑝𝑝𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑟_𝑖^{1−𝑞−𝑝/𝑞}=1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖𝛾−𝑝−1𝑝^𝑝×1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑟_𝑖^{1−𝑞−𝑝/𝑞}×1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖⁻¹=1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖𝛾−𝑞^−𝑝1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑟_𝑖^{1−𝑞1−𝑝}×1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖⁻¹>𝑀𝑠_𝑘2𝛾−Γ2,𝑘≥𝑘₀. (58) That is, 𝛾𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑠_𝑖−𝑋^𝑝_𝑘𝑝>𝑀𝑠_𝑘𝛾−Γ4>0,𝑘≥𝑘₀. (59)
It remains to estimate the following: 1𝛼^{𝑘+𝛼−1}_𝑖=𝑘(𝑖+1)𝑝−1𝛽_𝑖^𝑝−2𝜁_𝑖^𝑞𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)−𝑌^𝑞_𝑘𝑞=1𝛼^{𝑘+𝛼−1}_𝑖=𝑘(𝑖+1)𝑝−1𝛽_𝑖^𝑝−2𝜁_𝑖^𝑞𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)−𝜉_𝑘^𝑞𝑞𝑝𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝑟_𝑖^1−𝑞=𝑝−1𝛼×^{𝑘+𝛼−1}_𝑖=𝑘𝜁_𝑖^𝑞(𝑖+1)𝛽_𝑖^𝑝−2𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)−𝜉_𝑘^𝑞𝑟_𝑖^1−𝑞,𝑘≥𝑘₀. (60) Applying (41), (42), and (44), we can express 1𝛼^{𝑘+𝛼−1}_𝑖=𝑘(𝑖+1)𝑝−1𝛽_𝑖^𝑝−2𝜁_𝑖^𝑞𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)−𝑌^𝑞_𝑘𝑞=𝑝−1𝛼^{𝑘+𝛼−1}_𝑖=𝑘𝜁_𝑖^𝑞𝑟_𝑖^1−𝑞+ℎ_𝑖−𝜉_𝑘^𝑞𝑟_𝑖^1−𝑞,𝑘≥𝑘₀, (61) where the sequence of ℎ_𝑖 has the property that lim_𝑖→∞ℎ_𝑖=0. From (48), it follows lim_𝑖→∞𝜁_𝑖^𝑞ℎ_𝑖=0, (62) and, from (51), we obtain 𝜁_𝑖−𝜉_𝑘<(𝛼−1)𝐾{}.𝑘−𝑝+2,𝑘≥𝑘,𝑖∈𝑘,…,𝑘+𝛼−1 (63) Since the function 𝑦=|𝑥|^𝑞 is continuously differentiable on [−𝐿,0], there exists 𝑁>0 for which 𝜉^𝑞−𝜁^𝑞≤𝑁𝜉−𝜁,𝜉,𝜁∈[−𝐿,0]. (64) Now it suffices to consider (32), (48), (53), (61), (62), (63), and (64) which imply the existence of an integer 𝑘≥𝑘 such that 1𝛼^{𝑘+𝛼−1}_𝑖=𝑘(𝑖+1)𝑝−1𝛽_𝑖^𝑝−2𝜁_𝑖^𝑞𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)−𝑌^𝑞_𝑘𝑞<𝑀𝑠_𝑘𝛾−Γ8,𝑘≥𝑘. (65)
Altogether (see (54), (57), (59), and (65)), we have Δ𝜉_𝑘>0+𝑀𝑠_𝑘𝛾−Γ/4−𝑀𝑠_𝑘𝛾−Γ/8=𝑀𝑠𝑘+𝛼−𝑝+1_𝑘𝛾−Γ8𝑘+𝛼−𝑝+1,𝑘≥𝑘. (66) Summing (66) from 𝑘 to an integer 𝑙≥𝑘, we obtain 𝜉_𝑙+1𝑘+𝑀𝑠>𝜉^𝑘𝛾−Γ8^𝑙_𝑘=𝑘1.𝑘+𝛼−𝑝+1 (67) Thus (see also (53)), it is valid that liminf_𝑙→∞𝜉_𝑙+1𝑀𝑠≥−𝐿+^𝑘𝛾−Γ8^∞_𝑘=𝑘1𝑘+𝛼−𝑝+1=∞. (68) Particularly (consider (52)), 𝜁_𝑘 has to be positive for infinitely many 𝑘. This contradiction proves that (34) is oscillatory if 𝛾>Γ.
Let 𝛾<Γ, and let 𝑘(1)∈ℕ be given. In this part of the proof, our objective is to show that there exists a negative solution {𝜁_𝑘} of (39) for all 𝑘≥𝑘(1), where 𝜁_𝑘(1)𝑞=−𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝}. (69) Evidently, it is true that Δ𝜁_𝑘≥1𝜁𝑘−𝑝+2𝑝−1_𝑘,Δ𝜁_𝑘≤1×𝑘−𝑝+2𝛾𝑠⁺+(𝑘+1)𝑝−1𝛽_𝑘^𝑝−2𝜁_𝑘^𝑞𝑘^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑘+Φ⁻¹−𝜁_𝑘/𝑘^(𝑝−1) (70) for all 𝑘≥𝑘(1) for which {𝜁_𝑘} is negative. Thus (consider (41), (42), and (44)), we can assume that 𝑘(1) is so large that 2𝜁_𝑘(1)<𝜁_𝑘(1)+𝑖<0,𝑖∈{1,…,𝛼}. (71) In addition, as in the first part of the proof (see (49)), we can estimate Δ𝜁_𝑘(1)+𝑖<𝐴𝑘(1){},−𝑝+2,𝑖∈0,…,𝛼−2 (72) for some 𝐴>0. From (72), it follows 𝜁_𝑘(1)+𝑖−𝜁_𝑘(1)+𝑗<𝐴(𝛼−1)𝑘(1){}.−𝑝+2,𝑖,𝑗∈0,…,𝛼−1 (73)
If we denote 𝜉_𝑘(1)1∶=𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝜁_𝑖, (74) then 𝜁_𝑘(1)+𝑖−𝜉_𝑘(1)<𝐴(𝛼−1)𝑘(1){}.−𝑝+2,𝑖∈0,…,𝛼−1 (75) Again (see (54)), we estimate Δ𝜉_𝑘(1)≤1𝑘(1)×𝜉−𝑝+2𝑝−1_𝑘(1)+𝑋^𝑝_𝑘(1)𝑝+𝑌^𝑞_𝑘(1)𝑞+𝛾𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝑠_𝑖−𝑋^𝑝_𝑘(1)𝑝+1𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘(𝑖+1)𝑝−1𝛽_𝑖^𝑝−2𝜁_𝑖^𝑞𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)−𝑌^𝑞_𝑘(1)𝑞 (76) if Δ𝜉_𝑘(1)>0, or Δ𝜉_𝑘(1)≤1𝑘(1)×𝜉+𝛼−𝑝+1𝑝−1_𝑘(1)+𝑋^𝑝_𝑘(1)𝑝+𝑌^𝑞_𝑘(1)𝑞+𝛾𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝑠_𝑖−𝑋^𝑝_𝑘(1)𝑝+1𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘(𝑖+1)𝑝−1𝛽_𝑖^𝑝−2𝜁_𝑖^𝑞𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)−𝑌^𝑞_𝑘(1)𝑞 (77) if Δ𝜉_𝑘(1)<0, where 𝑋_𝑘(1)𝑝∶=𝑝−1𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝑟_𝑖^{1−𝑞−1/𝑞},𝑌_𝑘(1)∶=𝜉_𝑘(1)𝑝𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝑟_𝑖^{1−𝑞1/𝑞}. (78)
If ^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝑟_𝑖^1−𝑞=^𝛼_𝑖=1𝑟_𝑖^1−𝑞,𝜁_𝑘(1)=𝜁_𝑘(1)+𝑗,𝑗∈{1,…,𝛼−1}, (79) then it is valid that 𝑝−1𝜉_𝑘(1)+𝑋^𝑝_𝑘(1)𝑝+𝑌^𝑞_𝑘(1)𝑞=𝑝−1𝜉_𝑘(1)+𝑝−1^𝑝𝑝𝑝𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞−𝑝/𝑞}+𝜉_𝑘(1)^𝑞𝑞𝑝𝛼^𝛼_𝑖=1𝑟_𝑖^1−𝑞𝑞=−𝑝−1𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝}+𝑝−1𝑝^𝑝1𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞−𝑝/𝑞}+(𝑞/𝛼)∑^𝛼_𝑖=1𝑟_𝑖^{1−𝑞𝑞[1−𝑝]}𝑞×𝑝𝛼^𝛼_𝑖=1𝑟_𝑖^1−𝑞=−𝑝−1𝑞^1−𝑝1𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝}+𝑝−1𝑝^𝑝1𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝}+𝑞^−𝑝−1𝑝1𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝}. (80) That is, 𝑝−1𝜉_𝑘(1)+𝑋^𝑝_𝑘(1)𝑝+𝑌^𝑞_𝑘(1)𝑞=1𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝}−𝑝−1𝑞^1−𝑝+𝑝−1𝑝^𝑝+𝑞^−𝑝−1𝑝=1𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝}−𝑝𝑞^1−𝑝+𝑞^1−𝑝+1𝑞^𝑝+𝑞^−𝑝−1𝑝=1𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝}𝑞^−𝑝𝑝−𝑝𝑞+𝑞+1+𝑞=1𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝}𝑞^−𝑝−𝑝𝑞+𝑞+1+𝑝−1=0. (81) For any 𝜀>0, we can find 𝛼∈ℕ with the property that ^{𝑘+𝛼−1}_𝑖=𝑘𝑟_𝑖^1−𝑞−^𝛼_𝑖=1𝑟_𝑖^1−𝑞<𝜀,𝑘∈ℕ. (82) Thus, for a given number 𝛼, considering (73) and (79), we can assume that 𝑘(1) is so large that 𝑝−1𝜉_𝑘(1)+𝑋^𝑝_𝑘(1)𝑝+𝑌^𝑞_𝑘(1)𝑞<𝑀𝑠_𝑘Γ−𝛾.16 (83)
It holds (see (36), (37), and (38)) 𝛾𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝑠_𝑖−𝑋^𝑝_𝑘(1)𝑝=𝛾𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝑠_𝑖−𝑝−1^𝑝𝑝𝑝𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝑟_𝑖^{1−𝑞−𝑝/𝑞}=1𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝑠_𝑖𝛾−𝑞^−𝑝1𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝑟_𝑖^{1−𝑞1−𝑝}×1𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝑠_𝑖⁻¹<𝑀𝑠_𝑘2𝛾−Γ2. (84) That is, 𝛾𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘𝑠_𝑖−𝑋^𝑝_𝑘(1)𝑝<𝑀𝑠_𝑘𝛾−Γ4<0. (85)
As in the first part of the proof (compare (48) with (71) and (63) with (75)), we can show that lim_{𝑘(1)→∞}1𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘(𝑖+1)𝑝−1𝛽_𝑖^𝑝−2𝜁_𝑖^𝑞𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)−𝑌^𝑞_𝑘(1)𝑞=0. (86) Let 𝑘(1) be so large that 1𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘(𝑖+1)𝑝−1𝛽_𝑖^𝑝−2𝜁_𝑖^𝑞𝑖^{(𝑝−1)𝑞−1}ΦΦ⁻¹𝑟_𝑖+Φ⁻¹−𝜁_𝑖/𝑖^(𝑝−1)−𝑌^𝑞_𝑘(1)𝑞<𝑀𝑠_𝑘Γ−𝛾.16 (87)
Finally (consider (77), (83), (85), and (87)), we have Δ𝜉_𝑘(1)<1𝑘(1)×𝑀𝑠+𝛼−𝑝+1_𝑘Γ−𝛾+𝑀𝑠16^𝑘𝛾−Γ4+𝑀𝑠_𝑘Γ−𝛾=𝑀𝑠16_𝑘𝛾−Γ8𝑘(1)+𝛼−𝑝+1<0. (88) Hence, Δ𝜉_𝑘(1)=1𝛼^{𝑘(1)+𝛼−1}_(1)𝑖=𝑘Δ𝜁_𝑖=𝜁_𝑘(1)+𝛼−𝜁_𝑘(1)𝛼<0. (89) That is, 𝜁_𝑘(1)+𝛼<𝜁_𝑘(1). (90) Particularly, if 𝜁_𝑘𝑞=−𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝}forsome𝑘≥𝑘(1), (91) then (71) and (90) give 𝜁_𝑘,𝜁_𝑘+1,…,𝜁_{𝑘+𝛼−1}<0,𝜁_𝑘+𝛼<𝜁_𝑘. (92)
Since (73) remains true if 𝑘(1) is replaced by an arbitrary integer 𝑘>𝑘(1), to complete the proof, it suffices to find a number 𝜗>0 such that 𝜁_𝑘∈𝑞−𝜗−𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝},−𝑞𝛼^𝛼_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝}forsome𝑘≥𝑘(1) (93) implies 1𝛼^{𝑘+𝛼−1}_𝑖=𝑘Δ𝜁_𝑖<0. (94) Indeed, in this case, we have (92) and, consequently, we have that 𝜁_𝑘<0 for all 𝑘>𝑘(1). Then, Lemma 16 says that (34) is nonoscillatory. Of course, such a number 𝜗>0 exists because the concrete initial value is not applied to obtain (77), (85), and (87) and because (83) can be proved for a sufficiently small perturbation of the initial value (the left side of (83) continuously depends on the initial value (69)).

Remark 18. The statement of Theorem 17 remains true for 𝛾∈ℝ. It suffices to consider the nonoscillatory equation Δ𝑟⁻ΦΔ𝑥_𝑘+0⋅Φ𝑥_𝑘+1=0, (95) where 𝑟⁻ is defined in (35). We refer to [1, Theorem 2].

Remark 19. We conjecture that, for 𝛾=Γ in the general case, it is not possible to decide whether (34) is oscillatory or nonoscillatory. Our conjecture is based on the existence of positive almost periodic sequences {𝑓_𝑘}_𝑘∈ℤ, {𝑔_𝑘}_𝑘∈ℤ satisfying 1𝑛^𝑛_𝑘=1𝑓_𝑘<𝑀𝑓_𝑘,1𝑛^𝑛_𝑘=1𝑔_𝑘>𝑀𝑔_𝑘,𝑛∈ℕ. (96) Such sequences can be constructed applying, for example, [34, Theorem 3.5].

Example 20. Let 𝑎>0, 𝑏>1/2, and 𝑐,𝑑≠0. We can use Theorem 17 for the following equation: Δ𝑘²1+𝑘²([][])Φcos𝑎𝑘sin𝑎𝑘+𝑏Δ𝑥_𝑘+𝛾^𝑘√𝑘−1+|sin[𝑐𝑘]|+|cos[𝑑𝑘]|(𝑘+1)^(3/2)Φ𝑥_𝑘+1=0 (97) with asymptotically almost periodic coefficients (consider Definition 2, Theorem 3, and Corollaries 6 and 13). It holds (𝑝=3/2, i.e., 𝑞=3) 𝑀𝑟_𝑘⁻²[][]1=𝑀cos𝑎𝑘sin𝑎𝑘+𝑏+𝑘²²=18+𝑏²,𝑀𝑠_𝑘=𝑀√^𝑘|[+|[=4𝑘−1+sin𝑐𝑘]|cos𝑑𝑘]|𝜋. (98) Thus, the equation is oscillatory for 6𝛾38𝑏²+1>√2𝜋 (99) and nonoscillatory for 6𝛾38𝑏²+1<√2𝜋. (100)

Theorem 17 gives a new result also for 𝑝=2 (i.e., for the linear equations with asymptotically almost periodic coefficients). This situation is illustrated by the next example. Note that, in the below given corollaries, we explicitly mention two other new results which follow from Theorem 17.

Example 21. For any positive continuous functions 𝑓∶[−2,2]→ℝ and 𝑔∶[−1,1]→ℝ, the equation Δ𝑓()+()sin1−𝑘−1^𝑘√𝑘Δ𝑥_𝑘()+𝛾𝑘𝑔cos𝑘+1(𝑘+1)𝑘²𝑥_𝑘+1=0 (101) has the form of (34) for 𝑝=𝑞=2. Considering [29, Example 3.4], we have 𝑀𝑟_𝑘⁻¹=12𝜋^𝜋_−𝜋𝑑𝑡𝑓(),𝑀𝑠sin𝑡_𝑘=12𝜋^𝜋_−𝜋𝑔(cos𝑡)𝑑𝑡. (102) Hence, the oscillation constant is 1Γ=𝜋^2𝜋_−𝜋𝑑𝑡𝑓(sin𝑡)^𝜋_−𝜋𝑔(cos𝑡)𝑑𝑡⁻¹. (103)

Corollary 22. Consider (34), where {𝑟_𝑘}_𝑘∈ℕ and {𝑠_𝑘}_𝑘∈ℕ are positive periodic sequences with period 𝑛. Denote Γ∶=𝑞^−𝑝1𝑛^𝑛_𝑖=1𝑟_𝑖^{1−𝑞1−𝑝}1𝑛^𝑛_𝑖=1𝑠_𝑖⁻¹. (104) Then, (34) is oscillatory for 𝛾>Γ and nonoscillatory for 𝛾<Γ.

Example 23. We apply Corollary 22 for the equation Δ()𝜋sin2𝑘−1Φ18Δ𝑥_𝑘+𝛾(𝑘+1)^(𝑝)1+cos2𝑘𝜋𝑚Φ𝑥_𝑘+1=0, (105) where 𝑚≥3 is an odd integer. By the following computations: 𝑀𝑟_𝑘^1−𝑞=19⁹_𝑖=1𝑟_𝑖^1−𝑞=19⁹_𝑖=1sin(2𝑖−1)𝜋18^1−𝑞=192^𝑞+2sin^1−𝑞𝜋18+2cos^1−𝑞𝜋9+2cos^1−𝑞2𝜋9,𝑀𝑠+1_𝑘=1𝑚^𝑚_𝑖=1𝑠_𝑖=1𝑚^𝑚_𝑖=11+cos2𝑖𝜋𝑚=1, (106) we get the oscillation constant Γ=9^𝑝−1𝑞^−𝑝2^𝑞+2sin^1−𝑞𝜋18+2cos^1−𝑞𝜋9+2cos^1−𝑞2𝜋9+1^1−𝑝. (107)

For the general nonzero sequence {𝑟_𝑘}_𝑘∈ℕ in (34), we have to use the full definition of the generalized zero. An interval (𝑙,𝑙+1], 𝑙∈ℕ, contains the generalized zero of a solution {𝑥_𝑘} of (34) if 𝑥_𝑙≠0 and 𝑟_𝑙𝑥_𝑙𝑥_𝑙+1≤0. This definition is necessary for the correct formulation of Corollary 24.

Corollary 24. Consider (34), where {𝑟_𝑘}_𝑘∈ℕ and {𝑠_𝑘}_𝑘∈ℕ are nonzero asymptotically almost periodic sequences satisfying inf𝑟_𝑘;𝑘∈ℕ>0,limsup_𝑘→∞𝑠_𝑘>0. (108) Let Γ∶=𝑞^−𝑝𝑀𝑟_𝑘^{1−𝑞1−𝑝}𝑀𝑠_𝑘⁻¹. (109)(i)If 𝑟_𝑘>0 for all 𝑘 and 𝛾<Γ, then (34) is nonoscillatory. (ii)If 𝑠_𝑘>0 for all 𝑘 and 𝛾>Γ, then (34) is oscillatory.

Proof. Since {|𝑓_𝑘|}_𝑘∈ℕ is asymptotically almost periodic if {𝑓_𝑘}_𝑘∈ℕ is asymptotically almost periodic, it suffices to use the half-linear Sturm type comparison theorem (see [1, Theorem 2]) and Theorem 17.

Example 25. For an arbitrarily given 𝑎≥1, Corollary 24 says that the equation Δcos𝑘𝑘+𝑎ΦΔ𝑥_𝑘+𝛾sin𝑘(𝑘+1)^(𝑝)Φ𝑥_𝑘+1=0 (110) is nonoscillatory if 𝛾<𝑞^−𝑝𝑎^{1−𝑞1−𝑝}[𝑀({|sin𝑘|})]⁻¹=𝑞^−𝑝𝑎𝜋2 (111) and that the equation Δ(−1)^𝑘cos𝑘𝑘+𝑎ΦΔ𝑥_𝑘+𝛾|sin𝑘|(𝑘+1)^(𝑝)Φ𝑥_𝑘+1=0 (112) is oscillatory if 𝛾>𝑞^−𝑝𝑎^{1−𝑞1−𝑝}[𝑀({|sin𝑘|})]⁻¹=𝑞^−𝑝𝑎𝜋2. (113)

Acknowledgments

The authors thank the referees for their remarks and suggestions which improved the final version of the paper. The first author is supported by the Programme of “Employment of Best Young Scientists for International Cooperation Empowerment” (grant number CZ.1.07/2.3.00/30.0037) co-financed from European Social Fund and the state budget of the Czech Republic. The second author is supported by the Czech Science Foundation under Grant P201/10/1032.