Abstract

Consider the second-order linear delay differential equation ğ‘¥î…žî…ž(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))=0, 𝑡≥𝑡0, where 𝑝∈𝐶([𝑡0,∞),ℝ+), 𝜏∈𝐶([𝑡0,∞),ℝ), 𝜏(𝑡) is nondecreasing, 𝜏(𝑡)≤𝑡 for 𝑡≥𝑡0 and limğ‘¡â†’âˆžğœ(𝑡)=∞, the (discrete analogue) second-order difference equation Δ2𝑥(𝑛)+𝑝(𝑛)𝑥(𝜏(𝑛))=0, where Δ𝑥(𝑛)=𝑥(𝑛+1)−𝑥(𝑛), Δ2=Δ∘Δ, 𝑝∶ℕ→ℝ+, 𝜏∶ℕ→ℕ, 𝜏(𝑛)≤𝑛−1, and limğ‘›â†’âˆžğœ(𝑛)=+∞, and the second-order functional equation 𝑥(𝑔(𝑡))=𝑃(𝑡)𝑥(𝑡)+𝑄(𝑡)𝑥(𝑔2(𝑡)), 𝑡≥𝑡0, where the functions 𝑃, 𝑄∈𝐶([𝑡0,∞),ℝ+), 𝑔∈𝐶([𝑡0,∞),ℝ), 𝑔(𝑡)≢𝑡 for 𝑡≥𝑡0, limğ‘¡â†’âˆžğ‘”(𝑡)=∞, and 𝑔2 denotes the 2th iterate of the function 𝑔, that is, 𝑔0(𝑡)=𝑡, 𝑔2(𝑡)=𝑔(𝑔(𝑡)), 𝑡≥𝑡0. The most interesting oscillation criteria for the second-order linear delay differential equation, the second-order difference equation and the second-order functional equation, especially in the case where liminfğ‘¡â†’âˆžâˆ«ğ‘¡ğœ(𝑡)𝜏(𝑠)𝑝(𝑠)𝑑𝑠≤1/𝑒 and limsupğ‘¡â†’âˆžâˆ«ğ‘¡ğœ(𝑡)𝜏(𝑠)𝑝(𝑠)𝑑𝑠<1 for the second-order linear delay differential equation, and 0<liminfğ‘¡â†’âˆž{𝑄(𝑡)𝑃(𝑔(𝑡))}≤1/4 and limsupğ‘¡â†’âˆž{𝑄(𝑡)𝑃(𝑔(𝑡))}<1, for the second-order functional equation, are presented.

1. Introduction

The problem of establishing sufficient conditions for the oscillation of all solutions to the second-order delay differential equation ğ‘¥î…žî…ž(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))=0,𝑡≥𝑡0,(1) where 𝑝∈𝐶([𝑡0,∞),ℝ+) (here ℝ+=[0,∞)), 𝜏∈𝐶([𝑡0,∞),ℝ), 𝜏(𝑡) is nondecreasing, 𝜏(𝑡)≤𝑡 for 𝑡≥𝑡0, and limğ‘¡â†’âˆžğœ(𝑡)=∞, has been the subject of many investigations; see, for example, [1–21] and the references cited therein.

By a solution of (1) we understand a continuously differentiable function defined on [𝜏(𝑇0),∞) for some 𝑇0≥𝑡0 and such that (1) is satisfied for 𝑡≥𝑇0. Such a solution is called oscillatory if it has arbitrarily large zeros, and otherwise it is called nonoscillatory.

The oscillation theory of the (discrete analogue) second-order difference equation Δ2𝑥(𝑛)+𝑝(𝑛)𝑥(𝜏(𝑛))=0,((1)) where Δ𝑥(𝑛)=𝑥(𝑛+1)−𝑥(𝑛), Δ2=Δ∘Δ, 𝑝∶ℕ→ℝ+, 𝜏∶ℕ→ℕ, 𝜏(𝑛)≤𝑛−1, and limğ‘›â†’âˆžğœ(𝑛)=+∞, has also attracted growing attention in the recent few years. The reader is referred to [22–26] and the references cited therein.

By a solution of (1) we mean a sequence 𝑥(𝑛) which is defined for 𝑛≥min{𝜏(𝑛)∶𝑛≥0} and which satisfies (1) for all 𝑛≥0. A solution 𝑥 of (1) is said to be oscillatory if the terms 𝑥 of the solution are neither eventually positive nor eventually negative. Otherwise the solution is called nonoscillatory.

The oscillation theory of second-order functional equations of the form 𝑔𝑥(𝑔(𝑡))=𝑃(𝑡)𝑥(𝑡)+𝑄(𝑡)𝑥2(𝑡),𝑡≥𝑡0,((1)) where 𝑃,𝑄∈𝐶([𝑡0,∞),ℝ+), 𝑔∈𝐶([𝑡0,∞),ℝ) are given real-valued functions, 𝑥 is an unknown real-valued function, 𝑔(𝑡)≢𝑡 for 𝑡≥𝑡0, limğ‘¡â†’âˆžğ‘”(𝑡)=∞, and 𝑔2 denotes the 2nd iterate of the function 𝑔, that is, 𝑔0(𝑡)=𝑡,𝑔2(𝑡)=𝑔(𝑔(𝑡)),𝑡≥𝑡0,(1.1) has also been developed during the last decade. We refer to [27–35] and the references cited therein.

By a solution of (1) we mean a real-valued function 𝑥∶[𝑡0,∞)→ℝ such that sup{|𝑥(𝑠)|∶𝑠≥𝑡∗}>0 for any 𝑡∗≥𝑡0 and 𝑥 satisfies (1) on [𝑡0,∞).

In this paper our purpose is to present the state of the art on the oscillation of all solutions to (1), (1), and (1), especially in the case where liminfğ‘¡â†’âˆžî€œğ‘¡ğœ(𝑡)1𝜏(𝑠)𝑝(𝑠)𝑑𝑠≤𝑒,limsupğ‘¡â†’âˆžî€œğ‘¡ğœ(𝑡)𝜏(𝑠)𝑝(𝑠)𝑑𝑠<1(1.2) for (1), and 0<liminfğ‘¡â†’âˆž1{𝑄(𝑡)𝑃(𝑔(𝑡))}≤4,limsupğ‘¡â†’âˆž{𝑄(𝑡)𝑃(𝑔(𝑡))}<1(1.3) for (1).

2. Oscillation Criteria for (1)

In this section we study the second-order delay equation (1). For the case of ordinary differential equations, that is, when 𝜏(𝑡)≡𝑡, the history of the problem began as early as in 1836 by the work of Sturm [16] and was continued in 1893 by Kneser [8]. Essential contribution to the subject was made by E. Hille, A. Wintner, Ph. Hartman, W. Leighton, Z. Nehari, and others (see the monograph by Swanson [17] and the references cited therein). In particular, in 1948 Hille [7] obtained the following well-known oscillation criteria. Let limsupğ‘¡â†’âˆžğ‘¡î€œğ‘¡+âˆžğ‘(𝑠)𝑑𝑠>1(2.1) or liminfğ‘¡â†’âˆžğ‘¡î€œğ‘¡+âˆžğ‘1(𝑠)𝑑𝑠>4,(2.2) the conditions being assumed to be satisfied if the integral diverges. Then (1) with 𝜏(𝑡)≡𝑡 is oscillatory.

For the delay differential equation (1), earlier oscillation results can be found in the monographs by Myshkis [14] and Norkin [15]. In 1968 Waltman [18] and in 1970 Bradley [1] proved that (1) is oscillatory if +âˆžğ‘(𝑡)𝑑𝑡=+∞.(2.3) Proceeding in the direction of generalization of Hille's criteria, in 1971 Wong [20] showed that if 𝜏(𝑡)≥𝛼𝑡 for 𝑡≥0 with 0<𝛼≤1, then the condition liminfğ‘¡â†’âˆžğ‘¡î€œğ‘¡+âˆžğ‘1(𝑠)𝑑𝑠>4𝛼(2.4) is sufficient for the oscillation of (1). In 1973 Erbe [2] generalized this condition to liminfğ‘¡â†’âˆžğ‘¡î€œğ‘¡+âˆžğœ(𝑠)𝑠𝑝1(𝑠)𝑑𝑠>4(2.5) without any additional restriction on 𝜏. In 1987 Yan [21] obtained some general criteria improving the previous ones.

An oscillation criterion of different type is given in 1986 by Koplatadze [9] and in 1988 by Wei [19], where it is proved that (1) is oscillatory if limsupğ‘¡â†’âˆžî€œğ‘¡ğœ(𝑡)𝜏(𝑠)𝑝(𝑠)𝑑𝑠>1(𝐶1) or liminfğ‘¡â†’âˆžî€œğ‘¡ğœ(𝑡)1𝜏(𝑠)𝑝(𝑠)𝑑𝑠>𝑒.(𝐶2) The conditions (𝐶1) and (𝐶2) are analogous to the oscillation conditions (see [36]) 𝐴∶=limsupğ‘¡â†’âˆžî€œğ‘¡ğœ(𝑡)𝑝(𝑠)𝑑𝑠>1,(𝐻1)𝛼∶=liminfğ‘¡â†’âˆžî€œğ‘¡ğœ(𝑡)1𝑝(𝑠)𝑑𝑠>𝑒,(𝐻2) respectively, for the first-order delay equation ğ‘¥î…ž(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))=0.(2.6)

The essential difference between (2.1), (2.2), and (𝐶1), (𝐶2) is that the first two can guarantee oscillation for ordinary differential equations as well, while the last two work only for delay equations. Unlike first-order differential equations, where the oscillatory character is due to the delay only (see [36]), equation (1) can be oscillatory without any delay at all, that is, in the case 𝜏(𝑡)≡𝑡. Figuratively speaking, two factors contribute to the oscillatory character of (1): the presence of the delay and the second-order nature of the equation. The conditions (2.1), (2.2), and (𝐶1), (𝐶2) illustrate the role of these factors taken separately.

In 1999 Koplatadze et al. [11] derived the following.

Theorem 2.1 (see [11]). Let limsupğ‘¡â†’âˆžî‚»î€œğœ(𝑡)𝑡+âˆžî€œğ‘(𝑠)𝑑𝑠+𝑡𝜏(𝑡)[]𝜏(𝑠)𝑝(𝑠)𝑑𝑠+𝜏(𝑡)−10𝜏(𝑡)𝑠𝜏(𝑠)𝑝(𝑠)𝑑𝑠>1.(2.7) Then (1) is oscillatory.

The following corollaries being more convenient for applications can be deduced from this theorem.

Corollary 2.2 (see [11]). Let limsupğ‘¡â†’âˆžî€œğœ(𝑡)𝑡+âˆžğ‘(𝑠)𝑑𝑠+liminfğ‘¡â†’âˆžğ‘¡âˆ’1𝑡0𝑠𝜏(𝑠)𝑝(𝑠)𝑑𝑠>1(2.8) or liminfğ‘¡â†’âˆžî€œğœ(𝑡)𝑡+âˆžğ‘(𝑠)𝑑𝑠+limsupğ‘¡â†’âˆžğ‘¡âˆ’1𝑡0𝑠𝜏(𝑠)𝑝(𝑠)𝑑𝑠>1.(2.9) Then (1) is oscillatory.

Corollary 2.3 (see [11]). Let limsupğ‘¡â†’âˆžğœî€œ(𝑡)𝑡+âˆžğ‘(𝑠)𝑑𝑠>1(2.10) or limsupğ‘¡â†’âˆžğ‘¡âˆ’1𝑡0𝑠𝜏(𝑠)𝑝(𝑠)𝑑𝑠>1.(2.11) Then (1) is oscillatory.

In the case of ordinary differential equations (𝜏(𝑡)≡𝑡) the following theorem was given in [11].

Theorem 2.4 (see [11]). Let 𝜏(𝑡)≡𝑡 and limsupğ‘¡â†’âˆžî‚»ğ‘¡î€œğ‘¡+âˆžğ‘(𝑠)𝑑𝑠+𝑡−1𝑡0𝑠2𝑝(𝑠)𝑑𝑠>1.(2.12) Then (1) is oscillatory.

In what follows it will be assumed that the condition +âˆžğœ(𝑠)𝑝(𝑠)𝑑𝑠=+∞(2.13) is fulfilled. As it follows from Lemma 4.1 in [10], this condition is necessary for (1) to be oscillatory. The study being devoted to the problem of oscillation of (1), the condition (2.13) does not affect the generality.

Here oscillation results are obtained for (1) by reducing it to a first-order equation. Since for the latter the oscillation is due solely to the delay, the criteria hold for delay equations only and do not work in the ordinary case.

Theorem 2.5 (see [12]). Let (2.13) be fulfilled and let the differential inequality ğ‘¥î…žî‚µî€œ(𝑡)+𝜏(𝑡)+𝑇𝜏(𝑡)𝜉𝜏(𝜉)𝑝(𝜉)𝑑𝜉𝑝(𝑡)𝑥(𝜏(𝑡))≤0(2.14) have no eventually positive solution. Then (1) is oscillatory.

Theorem 2.5 reduces the question of oscillation of (1) to that of the absence of eventually positive solutions of the differential inequality ğ‘¥î…žî‚µî€œ(𝑡)+𝜏(𝑡)+𝑇𝜏(𝑡)𝜉𝜏(𝜉)𝑝(𝜉)𝑑𝜉𝑝(𝑡)𝑥(𝜏(𝑡))≤0.(2.15) So oscillation results for first-order delay differential equations can be applied since the oscillation of the equation ğ‘¢î…ž(𝑡)+𝑔(𝑡)𝑢(𝛿(𝑡))=0(2.16) is equivalent to the absence of eventually positive solutions of the inequality ğ‘¢î…ž(𝑡)+𝑔(𝑡)𝑢(𝛿(𝑡))≤0.(2.17) This fact is a simple consequence of the following comparison theorem deriving the oscillation of (2.16) from the oscillation of the equation ğ‘£î…ž(𝑡)+ℎ(𝑡)𝑣(ğœŽ(𝑡))=0.(2.18)

We assume that 𝑔,â„Žâˆ¶ğ‘…+→𝑅+ are locally integrable, 𝛿,ğœŽâˆ¶ğ‘…+→𝑅 are continuous, 𝛿(𝑡)≤𝑡, ğœŽ(𝑡)≤𝑡 for 𝑡∈𝑅+, and 𝛿(𝑡)→+∞, ğœŽ(𝑡)→+∞ as 𝑡→+∞.

Theorem 2.6 (see [12]). Let 𝑔(𝑡)≥ℎ(𝑡),𝛿(𝑡)â‰¤ğœŽ(𝑡),for𝑡∈𝑅+,(2.19) and let (2.18) be oscillatory. Then (2.16) is also oscillatory.

Corollary 2.7 (see [12]). Let (2.16) be oscillatory. Then the inequality (2.17) has no eventually positive solution.

Turning to applications of Theorem 2.5, we will use it together with the criteria (𝐻1) and (𝐻2) to get the following.

Theorem 2.8 (see [12]). Let 𝐾∶=limsupğ‘¡â†’âˆžî€œğ‘¡ğœ(𝑡)𝜏(𝑠)+0𝜏(𝑠)𝜉𝜏(𝜉)𝑝(𝜉)𝑑𝜉𝑝(𝑠)𝑑𝑠>1(𝐶3) or 𝑘∶=liminfğ‘¡â†’âˆžî€œğ‘¡ğœ(𝑡)𝜏(𝑠)+0𝜏(𝑠)1𝜉𝜏(𝜉)𝑝(𝜉)𝑑𝜉𝑝(𝑠)𝑑𝑠>𝑒.(𝐶4) Then (1) is oscillatory.

To apply Theorem 2.5 it suffices to note that (i) equation (2.13) is fulfilled since otherwise 𝑘=𝐾=0; (ii) since 𝜏(𝑡)→+∞ as 𝑡→+∞, the relations (𝐶4) and (𝐶4) imply the same relations with 0 changed by any 𝑇≥0.

Remark 2.9 (see [12]). Theorem 2.8 improves the criteria (𝐶1), (𝐶2) by Koplatadze [9] and Wei [19] mentioned above. This is directly seen from (𝐶3), (𝐶4) and can be easily checked if we take 𝜏(𝑡)≡𝑡−𝜏0 and 𝑝(𝑡)≡𝑝0/(𝑡−𝜏0) for 𝑡≥2𝜏0, where the constants 𝜏0>0 and 𝑝0>0 satisfy 𝜏0𝑝0<1𝑒.(2.20) In this case neither of (𝐶1), (𝐶2) is applicable for (1) while both (𝐶3), (𝐶4) give the positive conclusion about its oscillation. Note also that this is exactly the case where the oscillation is due to the delay since the corresponding equation without delay is nonoscillatory.

Remark 2.10 (see [12]). The criteria (𝐶3), (𝐶4) look like (𝐻1), (𝐻2) but there is an essential difference between them pointed out in the introduction. The condition (𝐻2) is close to the necessary one, since according to [9] if 𝐴≤1/𝑒, then (2.16) is nonoscillatory. On the other hand, for an oscillatory equation (1) without delay we have 𝑘=𝐾=0. Nevertheless, the constant 1/𝑒 in Theorem 2.8 is also the best possible in the sense that for any 𝜀∈(0,1/𝑒] it cannot be replaced by 1/𝑒−𝜀 without affecting the validity of the theorem. This is illustrated by the following.

Example 2.11 (see [12]). Let 𝜀∈(0,1/𝑒], 1−𝑒𝜀<𝛽<1, 𝜏(𝑡)≡𝛼𝑡, and 𝑝(𝑡)≡𝛽(1−𝛽)𝛼−𝛽𝑡−2, where 𝛼=𝑒1/(𝛽−1). Then (𝐶4) is fulfilled with 1/e replaced by 1/𝑒−𝜀. Nevertheless (1) has a nonoscillatory solution, namely, 𝑢(𝑡)≡𝑡𝛽. Indeed, denoting 𝑐=𝛽(1−𝛽)𝛼−𝛽, we see that the expression under the limit sign in (𝐶4) is constant and equals 𝛼𝑐|ln𝛼|(1+𝛼𝑐)=(𝛽/𝑒)(1+(𝛽(1−𝛽))/𝑒)>𝛽/𝑒>1/𝑒−𝜀.

Note that there is a gap between the conditions (𝐶3), (𝐶4) when 0≤𝑘≤1/𝑒, 𝑘<𝐾. In the case of first-order equations (cf., [36–48]), using results in this direction, one can derive various sufficient conditions for the oscillation of (1). According to Remark 2.9, neither of them can be optimal in the above sense but, nevertheless, they are of interest since they cannot be derived from other known results in the literature. We combine Theorem 2.5 with Corollary 1 [40] to obtain the following theorem.

Theorem 2.12 (see [12]). Let 𝐾 and 𝑘 be defined by (𝐶3), (𝐶4), 0≤𝑘≤1/𝑒, and 1𝐾>𝑘+𝜆−√(𝑘)1−𝑘−1−2𝑘−𝑘22,(𝐶9) where 𝜆(𝑘) is the smaller root of the equation 𝜆=exp(𝑘𝜆). Then (1) is oscillatory.

Note that the condition (𝐶9) is analogous to the condition (𝐶9) in [40].

3. Oscillation Criteria for (1)

In this section we study the second-order difference equation (1), where Δ𝑥(𝑛)=𝑥(𝑛+1)−𝑥(𝑛), Δ2=Δ∘Δ, 𝑝∶ℕ→𝑅+, 𝜏∶ℕ→ℕ, 𝜏(𝑛)≤𝑛−1, and limğ‘›â†’âˆžğœ(𝑛)=+∞.

In 1994, Wyrwińska [26] proved that all solutions of (1) are oscillatory if limsupğ‘›â†’âˆžîƒ¯ğ‘›î“ğ‘–=𝜏(𝑛)[][]𝜏(𝑖)−2𝑝(𝑖)+𝜏(𝑛)−2âˆžî“ğ‘–=𝑛+1𝑝(𝑖)>1,((𝐶1)) while, in 1997, Agarwal et al. [22] proved that, in the special case of the second-order difference equation with constant delay Δ2𝑥(𝑛)+𝑝(𝑛)𝑥(𝑛−𝑘)=0,𝑘≥1,((1)î…žğ‘) all solutions are oscillatory if liminfğ‘›â†’âˆžğ‘›âˆ’1𝑖=𝑛−𝑘𝑘(𝑖−𝑘)𝑝(𝑖)>2𝑘+1𝑘+1.(3.1)

In 2001, Grzegorczyk and Werbowski [23] studied (1î…žğ‘) and proved that under the following conditionslimsupğ‘›â†’âˆžîƒ¯ğ‘›î“ğ‘–=𝑛−𝑘(𝑖−𝑛+𝑘+1)𝑝(𝑖)+(𝑛−𝑘−2)+𝑛−𝑘−1𝑖=𝑛1(𝑖−𝑘)2×𝑝(𝑖)âˆžî“ğ‘–=𝑛+1𝑝(𝑖)>1,forsome𝑛1>𝑛0,(3.2)

or liminfğ‘›â†’âˆžğ‘›âˆ’1𝑖=𝑛−𝑘𝑘(𝑖−𝑘−1)𝑝(𝑖)>𝑘+1𝑘+1((𝐶2)) all solutions of (1î…žğ‘) are oscillatory. Observe that the last condition (𝐶2) may be seen as the discrete analogue of the condition (𝐶2).

In 2001 Koplatadze [24] studied the oscillatory behaviour of all solutions to (1) with variable delay and established the following.

Theorem 3.1 (see [24]). Assume that 1inf1−𝜆liminfğ‘›â†’âˆžğ‘›ğ‘›âˆ’ğœ†î“ğ‘–=1𝑖𝑝(𝑖)𝜏𝜆(𝑖)∶𝜆∈(0,1)>1,liminfğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑖=1𝑖𝑝(𝑖)𝜏(𝑖)>0.(3.3) Then all solutions of (1) oscillate.

Corollary 3.2 (see [24]). Let 𝛼>0 and liminfğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑖=1𝑖2𝛼𝑝(𝑖)>max−𝜆[].𝜆(1−𝜆)∶𝜆∈0,1(3.4) Then all solutions of the equation Δ2[]1𝑥(𝑛)+𝑝(𝑛)𝑥(𝛼𝑛)=0,𝑛≥max1,𝛼,𝑛∈𝑁(3.5) oscillate.

Corollary 3.3 (see [24]). Let 𝑛0 be an integer and liminfğ‘›â†’âˆžğ‘›ğ‘›âˆ’1𝑖=1𝑖21𝑝(𝑖)>4.(3.6) Then all solutions of the equation Δ2𝑥(𝑛)+𝑝(𝑛)𝑥𝑛−𝑛0=0,𝑛≥max1,𝑛0+1,𝑛∈𝑁(3.7) oscillate.

In 2002 Koplatadze et al. [25] studied (1) and established the following.

Theorem 3.4 (see [25]). Assume that liminfğ‘›â†’âˆžğœ(𝑛)𝑛=𝛼∈(0,∞),(3.8)liminfğ‘›â†’âˆžğ‘›âˆžî“ğ‘–=𝑛𝛼𝑝(𝑖)>max−𝜆[].𝜆(1−𝜆)∶𝜆∈0,1(3.9) Then all solutions of (1) oscillate.

In the case where 𝛼=1, the following discrete analogue of Hille's well-known oscillation theorem for ordinary differential equations (see (2.2)) is derived.

Theorem 3.5 (see [25]). Let 𝑛0 be an integer and liminfğ‘›â†’âˆžğ‘›âˆžî“ğ‘–=𝑛1𝑝(𝑖)>4.(3.10) Then all solutions of the equation Δ2𝑥(𝑛)+𝑝(𝑛)𝑥𝑛−𝑛0=0,𝑛≥𝑛0,(3.11) oscillate.

Remark 3.6 (see [25]). As in case of ordinary differential equations, the constant 1/4 in (3.10) is optimal in the sense that the strict inequality cannot be replaced by the nonstrict one. More than that, the same is true for the condition (3.9) as well. To ascertain this, denote by 𝑐 the right-hand side of (3.9) and by 𝜆0 the point where the maximum is achieved. The sequence 𝑥(𝑛)=𝑛𝜆0 obviously is a nonoscillatory solution of the equation Δ2[])𝑥(𝑛)+𝑝(𝑛)𝑥(𝛼𝑛=0,(3.12) where 𝑝(𝑛)=−Δ2(𝑛𝜆0)/[𝛼𝑛]𝜆0 and [𝛼] denotes the integer part of 𝛼. It can be easily calculated that 𝑐𝑝(𝑛)=−𝑛21+𝑜𝑛2asğ‘›âŸ¶âˆž.(3.13) Hence for arbitrary 𝜀>0, 𝑝(𝑛)≥(𝑐−𝜀)/𝑛2 for 𝑛∈ℕ𝑛0 with 𝑛0∈ℕ sufficiently large. Using the inequality âˆ‘âˆžğ‘–=𝑛𝑖2≥𝑛−1 and the arbitrariness of 𝜀, we obtain liminfğ‘›â†’âˆžğ‘›âˆžî“ğ‘–=𝑛𝑝(𝑖)≥𝑐.(3.14) This limit cannot be greater than 𝑐 by Theorem 3.4. Therefore it equals 𝑐 and (3.9) is violated.

4. Oscillation Criteria for (1)

In this section we study the functional equation (1).

In 1993 Domshlak [27] studied the oscillatory behaviour of equations of this type. In 1994, Golda and Werbowski [28] proved that all solutions of (1) oscillate if 𝐀∶=limsupğ‘¡â†’âˆž{𝑄(𝑡)𝑃(𝑔(𝑡))}>1((𝐶1)) or a∶=liminfğ‘¡â†’âˆž1{𝑄(𝑡)𝑃(𝑔(𝑡))}>4.((𝐶2)) In the same paper they also improved condition (𝐶1) to limsupğ‘¡â†’âˆžîƒ¯ğ‘„(𝑡)𝑃(𝑔(𝑡))+𝑘𝑖𝑖=0𝑗=0𝑄𝑔𝑗+1𝑃𝑔(𝑡)𝑗+2(𝑡)>1,(4.1) where 𝑘≥0 is some integer.

In 1995, Nowakowska and Werbowski [29] extended condition (𝐶1) to higher-order linear functional equations. In 1996, Shen [30], in 1997, Zhang et al. [35], and, in 1998, Zhang et al. [34] studied functional equations with variable coefficients and constant delay, while in 1999, Yan and Zhang [33] considered a system with constant coefficients.

It should be noted that conditions (𝐶1) and (𝐶2) may be seen as the analogues of the oscillation conditions (𝐶1) and (𝐶2) for (1) and (𝐶1) and (𝐶2) for (1).

As far as the lower bound 1/4 in the condition (𝐶2) is concerned, as it was pointed out in [28], it cannot be replaced by a smaller number. Recently, in [32], this fact was generalized by proving that 1𝑄(𝑡)𝑃(𝑔(𝑡))≤4,forlarge𝑡,((𝑁1)) implies that (1) has a nonoscillatory solution.

It is obvious that there is a gap between the conditions (𝐶1) and (𝐶2) when the limit limğ‘¡â†’âˆž{𝑄(𝑡)𝑃(𝑔(𝑡))} does not exist. How to fill this gap is an interesting problem. Here we should mention that condition (4.1) is an attempt in this direction. In fact, from condition (4.1) we can obtain (see [31]) that all solutions of (1) oscillate if 0â‰¤ğ‘Žâ‰¤1/4 and 𝐀>1−2ğ‘Ž.1âˆ’ğ‘Ž(4.2)

In 2002, Shen and Stavroulakis [31] proved the following.

Theorem 4.1 (see [31]). Assume that 0â‰¤ğ‘Žâ‰¤1/4 and that for some integer𝑘≥0limsupğ‘¡â†’âˆžîƒ¯ğ‘Žğ‘„(𝑡)𝑃(𝑔(𝑡))+𝑘𝑖=0ğ‘Žğ‘–ğ‘–î‘ğ‘—=0𝑄𝑔𝑗+1𝑃𝑔(𝑡)𝑗+2(𝑡)>1,(4.3) where âˆšğ‘Ž=((1+1−4ğ‘Ž)/2)−1. Then all solutions of (1) oscillate.

Corollary 4.2 (see [31]). Assume that 0â‰¤ğ‘Žâ‰¤1/4 and √𝐀>1+1−4ğ‘Ž22.(4.4) Then all solutions of (1) oscillate.

Remark 4.3 (see [31]). It is to be noted that as ğ‘Žâ†’0 the condition (4.3) reduces to the condition (4.1) and the conditions (4.4) and (4.2) reduce to the condition (𝐶1). However the improvement is clear as 0<ğ‘Žâ‰¤1/4 because 1>1−2ğ‘Ž>√1âˆ’ğ‘Ž1+1−4ğ‘Ž22.(4.5) It is interesting to observe that when ğ‘Žâ†’1/4 condition (4.4) reduces to 𝐀>1/4,(4.6) which cannot be improved in the sense that the lower bound 1/4 cannot be replaced by a smaller number.

Example 4.4 (see [31]). Consider the equation 𝑥𝑡−2sin2𝑡1=𝑥(𝑡)+4+ğ‘žcos2𝑡𝑥𝑡−2sin2𝑡−2sin2𝑡−2sin2𝑡,(4.7) where 𝑔(𝑡)=𝑡−2sin2𝑡, 𝑃(𝑡)≡1, 𝑄(𝑡)=1/4+ğ‘žcos2𝑡, and ğ‘ž>0 is a constant. It is easy to see that ğ‘Ž=liminfğ‘¡â†’âˆžî‚€14+ğ‘žcos2𝑡=14,𝐀=limsupğ‘¡â†’âˆžî‚€14+ğ‘žcos2𝑡=141+ğ‘ž>4.(4.8) Thus, by Corollary 4.2 all solutions of (4.7) oscillate. However, the condition (𝐶1) is satisfied only for ğ‘ž>3/4, while the condition (4.2) is satisfied for (much smaller) ğ‘ž>5/12.

Acknowledgment

This work was done while L.K. Kikina was visiting the Department of Mathematics, University of Ioannina, in the framework of a postdoctoral scholarship awarded by IKY (State Scholarships Foundation), Athens, Greece.