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International Journal of Differential Equations
VolumeΒ 2010Β (2010), Article IDΒ 598068, 14 pages
http://dx.doi.org/10.1155/2010/598068
Review Article

Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations

1Department of Mathematics, University of Gjirokastra, 6002 Gjirokastra, Albania
2Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 2 December 2009; Accepted 9 January 2010

Academic Editor: LeonidΒ Berezansky

Copyright Β© 2010 L. K. Kikina and I. P. Stavroulakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Consider the second-order linear delay differential equation π‘₯ξ…žξ…ž(𝑑)+𝑝(𝑑)π‘₯(𝜏(𝑑))=0, 𝑑β‰₯𝑑0, where π‘βˆˆπΆ([𝑑0,∞),ℝ+), 𝜏∈𝐢([𝑑0,∞),ℝ), 𝜏(𝑑) is nondecreasing, 𝜏(𝑑)≀𝑑 for 𝑑β‰₯𝑑0 and limπ‘‘β†’βˆžπœ(𝑑)=∞, the (discrete analogue) second-order difference equation Ξ”2π‘₯(𝑛)+𝑝(𝑛)π‘₯(𝜏(𝑛))=0, where Ξ”π‘₯(𝑛)=π‘₯(𝑛+1)βˆ’π‘₯(𝑛), Ξ”2=Ξ”βˆ˜Ξ”, π‘βˆΆβ„•β†’β„+, πœβˆΆβ„•β†’β„•, 𝜏(𝑛)β‰€π‘›βˆ’1, and limπ‘›β†’βˆžπœ(𝑛)=+∞, and the second-order functional equation π‘₯(𝑔(𝑑))=𝑃(𝑑)π‘₯(𝑑)+𝑄(𝑑)π‘₯(𝑔2(𝑑)), 𝑑β‰₯𝑑0, where the functions 𝑃, π‘„βˆˆπΆ([𝑑0,∞),ℝ+), π‘”βˆˆπΆ([𝑑0,∞),ℝ), 𝑔(𝑑)≒𝑑 for 𝑑β‰₯𝑑0, limπ‘‘β†’βˆžπ‘”(𝑑)=∞, and 𝑔2 denotes the 2th iterate of the function 𝑔, that is, 𝑔0(𝑑)=𝑑, 𝑔2(𝑑)=𝑔(𝑔(𝑑)), 𝑑β‰₯𝑑0. 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 liminfπ‘‘β†’βˆžβˆ«π‘‘πœ(𝑑)𝜏(𝑠)𝑝(𝑠)𝑑𝑠≀1/𝑒 and limsupπ‘‘β†’βˆžβˆ«π‘‘πœ(𝑑)𝜏(𝑠)𝑝(𝑠)𝑑𝑠<1 for the second-order linear delay differential equation, and 0<liminfπ‘‘β†’βˆž{𝑄(𝑑)𝑃(𝑔(𝑑))}≀1/4 and limsupπ‘‘β†’βˆž{𝑄(𝑑)𝑃(𝑔(𝑑))}<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 π‘₯ξ…žξ…ž(𝑑)+𝑝(𝑑)π‘₯(𝜏(𝑑))=0,𝑑β‰₯𝑑0,(1) where π‘βˆˆπΆ([𝑑0,∞),ℝ+) (here ℝ+=[0,∞)), 𝜏∈𝐢([𝑑0,∞),ℝ), 𝜏(𝑑) is nondecreasing, 𝜏(𝑑)≀𝑑 for 𝑑β‰₯𝑑0, and limπ‘‘β†’βˆžπœ(𝑑)=∞, 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 𝑇0β‰₯𝑑0 and such that (1) is satisfied for 𝑑β‰₯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 Ξ”2π‘₯(𝑛)+𝑝(𝑛)π‘₯(𝜏(𝑛))=0,((1)ξ…ž) where Ξ”π‘₯(𝑛)=π‘₯(𝑛+1)βˆ’π‘₯(𝑛), Ξ”2=Ξ”βˆ˜Ξ”, π‘βˆΆβ„•β†’β„+, πœβˆΆβ„•β†’β„•, 𝜏(𝑛)β‰€π‘›βˆ’1, and limπ‘›β†’βˆžπ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 π‘₯(𝑛) which is defined for 𝑛β‰₯min{𝜏(𝑛)βˆΆπ‘›β‰₯0} and which satisfies (1ξ…ž) for all 𝑛β‰₯0. A solution π‘₯ of (1ξ…ž) is said to be oscillatory if the terms π‘₯ 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 𝑔π‘₯(𝑔(𝑑))=𝑃(𝑑)π‘₯(𝑑)+𝑄(𝑑)π‘₯2ξ€Έ(𝑑),𝑑β‰₯𝑑0,((1)ξ…žξ…ž) where 𝑃,π‘„βˆˆπΆ([𝑑0,∞),ℝ+), π‘”βˆˆπΆ([𝑑0,∞),ℝ) are given real-valued functions, π‘₯ is an unknown real-valued function, 𝑔(𝑑)≒𝑑 for 𝑑β‰₯𝑑0, limπ‘‘β†’βˆžπ‘”(𝑑)=∞, and 𝑔2 denotes the 2nd iterate of the function 𝑔, that is, 𝑔0(𝑑)=𝑑,𝑔2(𝑑)=𝑔(𝑔(𝑑)),𝑑β‰₯𝑑0,(1.1) has also been developed during the last decade. We refer to [27–35] and the references cited therein.

By a solution of (1ξ…žξ…ž) we mean a real-valued function π‘₯∢[𝑑0,∞)→ℝ such that sup{|π‘₯(𝑠)|βˆΆπ‘ β‰₯π‘‘βˆ—}>0 for any π‘‘βˆ—β‰₯𝑑0 and π‘₯ satisfies (1ξ…žξ…ž) on [𝑑0,∞).

In this paper our purpose is to present the state of the art on the oscillation of all solutions to (1), (1ξ…ž), and (1ξ…žξ…ž), especially in the case where liminfπ‘‘β†’βˆžξ€œπ‘‘πœ(𝑑)1𝜏(𝑠)𝑝(𝑠)𝑑𝑠≀𝑒,limsupπ‘‘β†’βˆžξ€œπ‘‘πœ(𝑑)𝜏(𝑠)𝑝(𝑠)𝑑𝑠<1(1.2) for (1), and 0<liminfπ‘‘β†’βˆž1{𝑄(𝑑)𝑃(𝑔(𝑑))}≀4,limsupπ‘‘β†’βˆž{𝑄(𝑑)𝑃(𝑔(𝑑))}<1(1.3) for (1ξ…žξ…ž).

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(𝑑)≑𝑑, 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 limsupπ‘‘β†’βˆžπ‘‘ξ€œπ‘‘+βˆžπ‘(𝑠)𝑑𝑠>1(2.1) or liminfπ‘‘β†’βˆžπ‘‘ξ€œπ‘‘+βˆžπ‘1(𝑠)𝑑𝑠>4,(2.2) the conditions being assumed to be satisfied if the integral diverges. Then (1) with 𝜏(𝑑)≑𝑑 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 ξ€œ+βˆžπ‘(𝑑)𝑑𝑑=+∞.(2.3) Proceeding in the direction of generalization of Hille's criteria, in 1971 Wong [20] showed that if 𝜏(𝑑)β‰₯𝛼𝑑 for 𝑑β‰₯0 with 0<𝛼≀1, then the condition liminfπ‘‘β†’βˆžπ‘‘ξ€œπ‘‘+βˆžπ‘1(𝑠)𝑑𝑠>4𝛼(2.4) is sufficient for the oscillation of (1). In 1973 Erbe [2] generalized this condition to liminfπ‘‘β†’βˆžπ‘‘ξ€œπ‘‘+∞𝜏(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 limsupπ‘‘β†’βˆžξ€œπ‘‘πœ(𝑑)𝜏(𝑠)𝑝(𝑠)𝑑𝑠>1(𝐢1) or liminfπ‘‘β†’βˆžξ€œπ‘‘πœ(𝑑)1𝜏(𝑠)𝑝(𝑠)𝑑𝑠>𝑒.(𝐢2) The conditions (𝐢1) and (𝐢2) are analogous to the oscillation conditions (see [36]) 𝐴∢=limsupπ‘‘β†’βˆžξ€œπ‘‘πœ(𝑑)𝑝(𝑠)𝑑𝑠>1,(𝐻1)π›ΌβˆΆ=liminfπ‘‘β†’βˆžξ€œπ‘‘πœ(𝑑)1𝑝(𝑠)𝑑𝑠>𝑒,(𝐻2) respectively, for the first-order delay equation π‘₯ξ…ž(𝑑)+𝑝(𝑑)π‘₯(𝜏(𝑑))=0.(2.6)

The essential difference between (2.1), (2.2), and (𝐢1), (𝐢2) 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(𝑑)≑𝑑. 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 (𝐢1), (𝐢2) illustrate the role of these factors taken separately.

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

Theorem 2.1 (see [11]). Let limsupπ‘‘β†’βˆžξ‚»ξ€œπœ(𝑑)𝑑+βˆžξ€œπ‘(𝑠)𝑑𝑠+π‘‘πœ(𝑑)[]𝜏(𝑠)𝑝(𝑠)𝑑𝑠+𝜏(𝑑)βˆ’1ξ€œ0𝜏(𝑑)ξ‚Όπ‘ πœ(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 limsupπ‘‘β†’βˆžξ€œπœ(𝑑)𝑑+βˆžπ‘(𝑠)𝑑𝑠+liminfπ‘‘β†’βˆžπ‘‘βˆ’1ξ€œπ‘‘0π‘ πœ(𝑠)𝑝(𝑠)𝑑𝑠>1(2.8) or liminfπ‘‘β†’βˆžξ€œπœ(𝑑)𝑑+βˆžπ‘(𝑠)𝑑𝑠+limsupπ‘‘β†’βˆžπ‘‘βˆ’1ξ€œπ‘‘0π‘ πœ(𝑠)𝑝(𝑠)𝑑𝑠>1.(2.9) Then (1) is oscillatory.

Corollary 2.3 (see [11]). Let limsupπ‘‘β†’βˆžπœξ€œ(𝑑)𝑑+βˆžπ‘(𝑠)𝑑𝑠>1(2.10) or limsupπ‘‘β†’βˆžπ‘‘βˆ’1ξ€œπ‘‘0π‘ πœ(𝑠)𝑝(𝑠)𝑑𝑠>1.(2.11) Then (1) is oscillatory.

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

Theorem 2.4 (see [11]). Let 𝜏(𝑑)≑𝑑 and limsupπ‘‘β†’βˆžξ‚»π‘‘ξ€œπ‘‘+βˆžπ‘(𝑠)𝑑𝑠+π‘‘βˆ’1ξ€œπ‘‘0𝑠2𝑝(𝑠)𝑑𝑠>1.(2.12) Then (1) is oscillatory.

In what follows it will be assumed that the condition ξ€œ+∞𝜏(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 π‘₯ξ…žξ‚΅ξ€œ(𝑑)+𝜏(𝑑)+π‘‡πœ(𝑑)ξ‚Άπœ‰πœ(πœ‰)𝑝(πœ‰)π‘‘πœ‰π‘(𝑑)π‘₯(𝜏(𝑑))≀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 π‘₯ξ…žξ‚΅ξ€œ(𝑑)+𝜏(𝑑)+π‘‡πœ(𝑑)ξ‚Άπœ‰πœ(πœ‰)𝑝(πœ‰)π‘‘πœ‰π‘(𝑑)π‘₯(𝜏(𝑑))≀0.(2.15) So oscillation results for first-order delay differential equations can be applied since the oscillation of the equation π‘’ξ…ž(𝑑)+𝑔(𝑑)𝑒(𝛿(𝑑))=0(2.16) is equivalent to the absence of eventually positive solutions of the inequality π‘’ξ…ž(𝑑)+𝑔(𝑑)𝑒(𝛿(𝑑))≀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 π‘£ξ…ž(𝑑)+β„Ž(𝑑)𝑣(𝜎(𝑑))=0.(2.18)

We assume that 𝑔,β„ŽβˆΆπ‘…+→𝑅+ are locally integrable, 𝛿,πœŽβˆΆπ‘…+→𝑅 are continuous, 𝛿(𝑑)≀𝑑, 𝜎(𝑑)≀𝑑 for π‘‘βˆˆπ‘…+, and 𝛿(𝑑)β†’+∞, 𝜎(𝑑)β†’+∞ as 𝑑→+∞.

Theorem 2.6 (see [12]). Let 𝑔(𝑑)β‰₯β„Ž(𝑑),𝛿(𝑑)β‰€πœŽ(𝑑),forπ‘‘βˆˆπ‘…+,(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∢=limsupπ‘‘β†’βˆžξ€œπ‘‘πœ(𝑑)ξ‚΅ξ€œπœ(𝑠)+0𝜏(𝑠)ξ‚Άπœ‰πœ(πœ‰)𝑝(πœ‰)π‘‘πœ‰π‘(𝑠)𝑑𝑠>1(𝐢3) or π‘˜βˆΆ=liminfπ‘‘β†’βˆžξ€œπ‘‘πœ(𝑑)ξ‚΅ξ€œπœ(𝑠)+0𝜏(𝑠)ξ‚Ά1πœ‰πœ(πœ‰)𝑝(πœ‰)π‘‘πœ‰π‘(𝑠)𝑑𝑠>𝑒.(𝐢4) Then (1) is oscillatory.

To apply Theorem 2.5 it suffices to note that (i) equation (2.13) is fulfilled since otherwise π‘˜=𝐾=0; (ii) since 𝜏(𝑑)β†’+∞ as 𝑑→+∞, the relations (𝐢4) and (𝐢4) imply the same relations with 0 changed by any 𝑇β‰₯0.

Remark 2.9 (see [12]). Theorem 2.8 improves the criteria (𝐢1), (𝐢2) by Koplatadze [9] and Wei [19] mentioned above. This is directly seen from (𝐢3), (𝐢4) and can be easily checked if we take 𝜏(𝑑)β‰‘π‘‘βˆ’πœ0 and 𝑝(𝑑)≑𝑝0/(π‘‘βˆ’πœ0) for 𝑑β‰₯2𝜏0, where the constants 𝜏0>0 and 𝑝0>0 satisfy 𝜏0𝑝0<1𝑒.(2.20) In this case neither of (𝐢1), (𝐢2) is applicable for (1) while both (𝐢3), (𝐢4) 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 (𝐢3), (𝐢4) 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 π‘˜=𝐾=0. Nevertheless, the constant 1/𝑒 in Theorem 2.8 is also the best possible in the sense that for any πœ€βˆˆ(0,1/𝑒] it cannot be replaced by 1/π‘’βˆ’πœ€ without affecting the validity of the theorem. This is illustrated by the following.

Example 2.11 (see [12]). Let πœ€βˆˆ(0,1/𝑒], 1βˆ’π‘’πœ€<𝛽<1, 𝜏(𝑑)≑𝛼𝑑, and 𝑝(𝑑)≑𝛽(1βˆ’π›½)π›Όβˆ’π›½π‘‘βˆ’2, where 𝛼=𝑒1/(π›½βˆ’1). Then (𝐢4) is fulfilled with 1/e replaced by 1/π‘’βˆ’πœ€. Nevertheless (1) has a nonoscillatory solution, namely, 𝑒(𝑑)≑𝑑𝛽. Indeed, denoting 𝑐=𝛽(1βˆ’π›½)π›Όβˆ’π›½, we see that the expression under the limit sign in (𝐢4) is constant and equals 𝛼𝑐|ln𝛼|(1+𝛼𝑐)=(𝛽/𝑒)(1+(𝛽(1βˆ’π›½))/𝑒)>𝛽/𝑒>1/π‘’βˆ’πœ€.

Note that there is a gap between the conditions (𝐢3), (𝐢4) when 0β‰€π‘˜β‰€1/𝑒, π‘˜<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 π‘˜ be defined by (𝐢3), (𝐢4), 0β‰€π‘˜β‰€1/𝑒, and 1𝐾>π‘˜+πœ†βˆ’βˆš(π‘˜)1βˆ’π‘˜βˆ’1βˆ’2π‘˜βˆ’π‘˜22,(𝐢9) where πœ†(π‘˜) is the smaller root of the equation πœ†=exp(π‘˜πœ†). Then (1) is oscillatory.

Note that the condition (𝐢9) is analogous to the condition (𝐢9) in [40].

3. Oscillation Criteria for (1ξ…ž)

In this section we study the second-order difference equation (1ξ…ž), where Ξ”π‘₯(𝑛)=π‘₯(𝑛+1)βˆ’π‘₯(𝑛), Ξ”2=Ξ”βˆ˜Ξ”, π‘βˆΆβ„•β†’π‘…+, πœβˆΆβ„•β†’β„•, 𝜏(𝑛)β‰€π‘›βˆ’1, and limπ‘›β†’βˆžπœ(𝑛)=+∞.

In 1994, WyrwiΕ„ska [26] proved that all solutions of (1ξ…ž) are oscillatory if limsupπ‘›β†’βˆžξƒ―π‘›ξ“π‘–=𝜏(𝑛)[][]𝜏(𝑖)βˆ’2𝑝(𝑖)+𝜏(𝑛)βˆ’2βˆžξ“π‘–=𝑛+1𝑝(𝑖)>1,((𝐢1)ξ…ž) while, in 1997, Agarwal et al. [22] proved that, in the special case of the second-order difference equation with constant delay Ξ”2π‘₯(𝑛)+𝑝(𝑛)π‘₯(π‘›βˆ’π‘˜)=0,π‘˜β‰₯1,((1)ξ…žπ‘) all solutions are oscillatory if liminfπ‘›β†’βˆžπ‘›βˆ’1𝑖=π‘›βˆ’π‘˜ξ‚€π‘˜(π‘–βˆ’π‘˜)𝑝(𝑖)>2ξ‚π‘˜+1π‘˜+1.(3.1)

In 2001, Grzegorczyk and Werbowski [23] studied (1ξ…žπ‘) and proved that under the following conditionslimsupπ‘›β†’βˆžξƒ―π‘›ξ“π‘–=π‘›βˆ’π‘˜ξƒ¬(π‘–βˆ’π‘›+π‘˜+1)𝑝(𝑖)+(π‘›βˆ’π‘˜βˆ’2)+π‘›βˆ’π‘˜βˆ’1𝑖=𝑛1(π‘–βˆ’π‘˜)2×𝑝(𝑖)βˆžξ“π‘–=𝑛+1𝑝(𝑖)>1,forsome𝑛1>𝑛0,(3.2)

or liminfπ‘›β†’βˆžπ‘›βˆ’1𝑖=π‘›βˆ’π‘˜ξ‚€π‘˜(π‘–βˆ’π‘˜βˆ’1)𝑝(𝑖)>ξ‚π‘˜+1π‘˜+1((𝐢2)ξ…ž) all solutions of (1ξ…žπ‘) are oscillatory. Observe that the last condition (𝐢2ξ…ž) may be seen as the discrete analogue of the condition (𝐢2).

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

Theorem 3.1 (see [24]). Assume that ξƒ―1inf1βˆ’πœ†liminfπ‘›β†’βˆžπ‘›π‘›βˆ’πœ†ξ“π‘–=1𝑖𝑝(𝑖)πœπœ†ξƒ°(𝑖)βˆΆπœ†βˆˆ(0,1)>1,liminfπ‘›β†’βˆžπ‘›π‘›βˆ’1𝑖=1𝑖𝑝(𝑖)𝜏(𝑖)>0.(3.3) Then all solutions of (1ξ…ž) oscillate.

Corollary 3.2 (see [24]). Let 𝛼>0 and liminfπ‘›β†’βˆžπ‘›π‘›βˆ’1𝑖=1𝑖2𝛼𝑝(𝑖)>maxβˆ’πœ†[]ξ€Ύ.πœ†(1βˆ’πœ†)βˆΆπœ†βˆˆ0,1(3.4) Then all solutions of the equation Ξ”2[]1π‘₯(𝑛)+𝑝(𝑛)π‘₯(𝛼𝑛)=0,𝑛β‰₯max1,𝛼,π‘›βˆˆπ‘(3.5) oscillate.

Corollary 3.3 (see [24]). Let 𝑛0 be an integer and liminfπ‘›β†’βˆžπ‘›π‘›βˆ’1𝑖=1𝑖21𝑝(𝑖)>4.(3.6) Then all solutions of the equation Ξ”2ξ€·π‘₯(𝑛)+𝑝(𝑛)π‘₯π‘›βˆ’π‘›0ξ€Έξ€½=0,𝑛β‰₯max1,𝑛0ξ€Ύ+1,π‘›βˆˆπ‘(3.7) oscillate.

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

Theorem 3.4 (see [25]). Assume that liminfπ‘›β†’βˆžπœ(𝑛)𝑛=π›Όβˆˆ(0,∞),(3.8)liminfπ‘›β†’βˆžπ‘›βˆžξ“π‘–=𝑛𝛼𝑝(𝑖)>maxβˆ’πœ†[]ξ€Ύ.πœ†(1βˆ’πœ†)βˆΆπœ†βˆˆ0,1(3.9) Then all solutions of (1ξ…ž) 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 liminfπ‘›β†’βˆžπ‘›βˆžξ“π‘–=𝑛1𝑝(𝑖)>4.(3.10) Then all solutions of the equation Ξ”2ξ€·π‘₯(𝑛)+𝑝(𝑛)π‘₯π‘›βˆ’π‘›0ξ€Έ=0,𝑛β‰₯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 πœ†0 the point where the maximum is achieved. The sequence π‘₯(𝑛)=π‘›πœ†0 obviously is a nonoscillatory solution of the equation Ξ”2[])π‘₯(𝑛)+𝑝(𝑛)π‘₯(𝛼𝑛=0,(3.12) where 𝑝(𝑛)=βˆ’Ξ”2(π‘›πœ†0)/[𝛼𝑛]πœ†0 and [𝛼] denotes the integer part of 𝛼. It can be easily calculated that 𝑐𝑝(𝑛)=βˆ’π‘›2ξ‚€1+π‘œπ‘›2asπ‘›βŸΆβˆž.(3.13) Hence for arbitrary πœ€>0, 𝑝(𝑛)β‰₯(π‘βˆ’πœ€)/𝑛2 for π‘›βˆˆβ„•π‘›0 with 𝑛0βˆˆβ„• sufficiently large. Using the inequality βˆ‘βˆžπ‘–=𝑛𝑖2β‰₯π‘›βˆ’1 and the arbitrariness of πœ€, we obtain liminfπ‘›β†’βˆžπ‘›βˆžξ“π‘–=𝑛𝑝(𝑖)β‰₯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 (1ξ…žξ…ž)

In this section we study the functional equation (1ξ…žξ…ž).

In 1993 Domshlak [27] studied the oscillatory behaviour of equations of this type. In 1994, Golda and Werbowski [28] proved that all solutions of (1ξ…žξ…ž) oscillate if π€βˆΆ=limsupπ‘‘β†’βˆž{𝑄(𝑑)𝑃(𝑔(𝑑))}>1((𝐢1)ξ…žξ…ž) or a∢=liminfπ‘‘β†’βˆž1{𝑄(𝑑)𝑃(𝑔(𝑑))}>4.((𝐢2)ξ…žξ…ž) In the same paper they also improved condition (𝐢1ξ…žξ…ž) to limsupπ‘‘β†’βˆžξƒ―π‘„(𝑑)𝑃(𝑔(𝑑))+π‘˜ξ“π‘–π‘–=0𝑗=0𝑄𝑔𝑗+1𝑃𝑔(𝑑)𝑗+2ξ€Έξƒ°(𝑑)>1,(4.1) where π‘˜β‰₯0 is some integer.

In 1995, Nowakowska and Werbowski [29] extended condition (𝐢1ξ…žξ…ž) 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 (𝐢1ξ…žξ…ž) and (𝐢2ξ…žξ…ž) may be seen as the analogues of the oscillation conditions (𝐢1) and (𝐢2) for (1) and (𝐢1ξ…ž) and (𝐢2ξ…ž) for (1ξ…ž).

As far as the lower bound 1/4 in the condition (𝐢2ξ…žξ…ž) 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(𝑑)𝑃(𝑔(𝑑))≀4,forlarge𝑑,((𝑁1)ξ…ž) implies that (1ξ…žξ…ž) has a nonoscillatory solution.

It is obvious that there is a gap between the conditions (𝐢1ξ…žξ…ž) and (𝐢2ξ…žξ…ž) when the limit limπ‘‘β†’βˆž{𝑄(𝑑)𝑃(𝑔(𝑑))} 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 (1ξ…žξ…ž) oscillate if 0β‰€π‘Žβ‰€1/4 and 𝐀>1βˆ’2π‘Ž.1βˆ’π‘Ž(4.2)

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

Theorem 4.1 (see [31]). Assume that 0β‰€π‘Žβ‰€1/4 and that for some integerπ‘˜β‰₯0limsupπ‘‘β†’βˆžξƒ―π‘Žπ‘„(𝑑)𝑃(𝑔(𝑑))+π‘˜ξ“π‘–=0π‘Žπ‘–π‘–ξ‘π‘—=0𝑄𝑔𝑗+1𝑃𝑔(𝑑)𝑗+2ξ€Έξƒ°(𝑑)>1,(4.3) where βˆšπ‘Ž=((1+1βˆ’4π‘Ž)/2)βˆ’1. Then all solutions of (1ξ…žξ…ž) oscillate.

Corollary 4.2 (see [31]). Assume that 0β‰€π‘Žβ‰€1/4 and ξƒ©βˆšπ€>1+1βˆ’4π‘Ž2ξƒͺ2.(4.4) Then all solutions of (1ξ…žξ…ž) oscillate.

Remark 4.3 (see [31]). It is to be noted that as π‘Žβ†’0 the condition (4.3) reduces to the condition (4.1) and the conditions (4.4) and (4.2) reduce to the condition (𝐢1ξ…žξ…ž). However the improvement is clear as 0<π‘Žβ‰€1/4 because 1>1βˆ’2π‘Ž>ξƒ©βˆš1βˆ’π‘Ž1+1βˆ’4π‘Ž2ξƒͺ2.(4.5) It is interesting to observe that when π‘Žβ†’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 π‘₯ξ€·π‘‘βˆ’2sin2𝑑1=π‘₯(𝑑)+4+π‘žcos2𝑑π‘₯ξ€·π‘‘βˆ’2sin2π‘‘βˆ’2sin2ξ€·π‘‘βˆ’2sin2𝑑,ξ€Έξ€Έ(4.7) where 𝑔(𝑑)=π‘‘βˆ’2sin2𝑑, 𝑃(𝑑)≑1, 𝑄(𝑑)=1/4+π‘žcos2𝑑, and π‘ž>0 is a constant. It is easy to see that π‘Ž=liminfπ‘‘β†’βˆžξ‚€14+π‘žcos2𝑑=14,𝐀=limsupπ‘‘β†’βˆžξ‚€14+π‘žcos2𝑑=141+π‘ž>4.(4.8) Thus, by Corollary 4.2 all solutions of (4.7) oscillate. However, the condition (𝐢1ξ…žξ…ž) is satisfied only for π‘ž>3/4, while the condition (4.2) is satisfied for (much smaller) π‘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.

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