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International Journal of Differential Equations
VolumeΒ 2011Β (2011), Article IDΒ 863801, 15 pages
http://dx.doi.org/10.1155/2011/863801
Research Article

Oscillation of Second-Order Nonlinear Delay Dynamic Equations on Time Scales

Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

Received 3 May 2011; Accepted 6 June 2011

Academic Editor: ElenaΒ Braverman

Copyright Β© 2011 H. A. Agwa et al. 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

In this work, we use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation (𝑝(𝑑)(π‘₯Ξ”(𝑑))𝛾)Ξ”+π‘ž(𝑑)𝑓(π‘₯(𝜏(𝑑)))=0, on a time scale 𝕋, where 𝛾 is the quotient of odd positive integers and p(t) and q(t) are positive right-dense continuous (rd-continuous) functions on 𝕋. Our results improve and extend some results established by Sun et al. 2009. Also our results unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our main results.

1. Introduction

The theory of time scales was introduced by Hilger [1] in order to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus. Many authors have expounded on various aspects of this new theory, see [2–4]. A time scale 𝕋 is a nonempty closed subset of the real numbers, If the time scale equals the real numbers or integer numbers, it represents the classical theories of the differential and difference equations. Many other interesting time scales exist and give rise to many applications. The new theory of the so-called β€œdynamic equation” not only unify the theories of differential equations and difference equations, but also extends these classical cases to the so-called π‘ž-difference equations (when 𝕋=π‘žβ„•0∢={π‘žπ‘‘βˆΆπ‘‘βˆˆβ„•0 for π‘ž>1} or 𝕋=π‘žβ„€=π‘žβ„€βˆͺ{0}) which have important applications in quantum theory (see [5]). Also it can be applied on different types of time scales like 𝕋=β„Žβ„€,𝕋=β„•20, and the space of the harmonic numbers 𝕋=𝕋𝑛. In the last two decades, there has been increasing interest in obtaining sufficient conditions for oscillation (nonoscillation) of the solutions of different classes of dynamic equations on time scales, see [6–9]. In this paper, we deal with the oscillation behavior of all solutions of the second-order nonlinear delay dynamic equationξ€·ξ€·π‘₯𝑝(𝑑)Ξ”ξ€Έ(𝑑)𝛾Δ+π‘ž(𝑑)𝑓(π‘₯(𝜏(𝑑)))=0,π‘‘βˆˆπ•‹,𝑑β‰₯𝑑0,(1.1) subject to the hypotheses (H1)𝕋 is a time scale which is unbounded above, and 𝑑0βˆˆπ•‹ with 𝑑0>0. We define the time scale interval [𝑑0,∞)𝕋 by [𝑑0,∞)𝕋=[𝑑0⋂𝕋,∞).(H2)𝛾 is the quotient of odd positive integers.(H3)𝑝 and π‘ž are positive rd-continuous functions on an arbitrary time scale 𝕋, andξ€œβˆžπ‘‘0Ξ”(𝑑)𝑝1/𝛾(𝑑)=∞(1.2)(H4)πœβˆΆπ•‹β†’π•‹ is a strictly increasing and differentiable function such that 𝜏(𝑑)≀𝑑, limπ‘‘β†’βˆžπœ(𝑑)=∞.(H5)π‘“βˆˆπΆ(ℝ,ℝ) is a continuous function such that for some positive constant 𝐿, it satisfies 𝑓(π‘₯)/π‘₯𝛾β‰₯𝐿 for all π‘₯β‰ 0.

By a solution of (1.1), we mean that a nontrivial real valued function π‘₯ satisfies (1.1) for π‘‘βˆˆπ•‹. A solution π‘₯ of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. (1.1) is said to be oscillatory if all of its solutions are oscillatory. We concentrate our study to those solutions of (1.1) which are not identically vanishing eventually.

It is easy to see that (1.1) can be transformed into a half linear dynamic equationξ€·ξ€·π‘₯𝑝(𝑑)Ξ”ξ€Έ(𝑑)𝛾Δ+π‘ž(𝑑)π‘₯𝛾(𝑑)=0,π‘‘βˆˆπ•‹,𝑑β‰₯𝑑0,(1.3) where 𝑓(π‘₯)=π‘₯𝛾, 𝜏(𝑑)=𝑑. If 𝛾=1, then (1.1) is transformed into the equation𝑝(𝑑)π‘₯Ξ”ξ€Έ(𝑑)Ξ”+π‘ž(𝑑)𝑓(π‘₯(𝜏(𝑑)))=0,π‘‘βˆˆπ•‹,𝑑β‰₯𝑑0.(1.4) If 𝑝(𝑑)=1, then (1.4) has the formπ‘₯ΔΔ(𝑑)+π‘ž(𝑑)𝑓(π‘₯(𝜏(𝑑)))=0,π‘‘βˆˆπ•‹,𝑑β‰₯𝑑0.(1.5) If 𝑓(π‘₯)=π‘₯, then (1.5) becomesπ‘₯ΔΔ(𝑑)+π‘ž(𝑑)π‘₯(𝜏(𝑑))=0,π‘‘βˆˆπ•‹,𝑑β‰₯𝑑0.(1.6) Recently, Zhang et al. [10] have considered the nonlinear delay (1.1) and established some sufficient conditions for oscillation of (1.1) when 𝛾β‰₯1. Also Grace et al. [11] introduced some new sufficient conditions for oscillation of the half linear dynamic equation (1.3). In 2009, Sun et al. [12] extended and improved the results of [6, 13, 14] to (1.1) when 𝛾β‰₯1, but their results can not be applied for 0<𝛾<1. In 2008, Hassan [15] considered the half linear dynamic equation (1.3) and established some sufficient conditions for oscillation of (1.3). In 2007, Erbe et al. [13] considered the nonlinear delay dynamic equation (1.4) and obtained some new oscillation criteria which improve the results of Şahiner [14]. In 2005, Agarwal et al. [6] studied the linear delay dynamic equation (1.6), also Şahiner [14] considered the nonlinear delay dynamic equation (1.5) and gave some sufficient conditions for oscillation of (1.6) and (1.5). In this work, we give some new oscillation criteria of (1.1) by using the generalized Riccati transformation and the inequality technique. Our results are general cases for some results of [12, 15].

This paper is organized as follows. In Section 2, we present some preliminaries on time scales. In Section 3, we give several lemmas. In Section 4, we establish some new sufficient conditions for oscillation of (1.1). Finally, in Section 5, we present some examples to illustrate our results.

2. Some Preliminaries on Time Scales

A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. On any time scale 𝕋, we define the forward and backward jump operators by𝜎(𝑑)=inf{π‘ βˆˆπ•‹,𝑠>𝑑},𝜌(𝑑)=sup{π‘ βˆˆπ•‹,𝑠<𝑑}.(2.1) A point π‘‘βˆˆπ•‹, 𝑑>inf 𝕋 is said to be left dense if 𝜌(𝑑)=𝑑, right dense if 𝑑<sup𝕋 and 𝜎(𝑑)=𝑑, left scattered if 𝜌(𝑑)<𝑑, and right scattered if 𝜎(𝑑)>𝑑. The graininess function πœ‡ for a time scale 𝕋 is defined by πœ‡(𝑑)=𝜎(𝑑)βˆ’π‘‘.

A function π‘“βˆΆπ•‹β†’β„ is called rd-continuous provided that it is continuous at right-dense points of 𝕋, and its left-sided limits exist (finite) at left-dense points of 𝕋. The set of rd-continuous functions is denoted by 𝐢rd(𝕋,ℝ). By 𝐢1rd(𝕋,ℝ), we mean the set of functions whose delta derivative belongs to 𝐢rd(𝕋,ℝ).

For a function π‘“βˆΆπ•‹β†’β„ (the range ℝ of 𝑓 may be actually replaced with any Banach space), the delta derivative 𝑓Δ is defined by 𝑓Δ(𝑑)=𝑓(𝜎(𝑑))βˆ’π‘“(𝑑)𝜎(𝑑)βˆ’π‘‘,(2.2) provided that 𝑓 is continuous at 𝑑, and 𝑑 is right scattered. If 𝑑 is not right scattered, then the derivative is defined by 𝑓Δ(𝑑)=lim𝑠→𝑑+𝑓(𝜎(𝑑))βˆ’π‘“(𝑑)π‘‘βˆ’π‘ =lim𝑠→𝑑+𝑓(𝑑)βˆ’π‘“(𝑠)π‘‘βˆ’π‘ ,(2.3) provided that this limit exists.

A function π‘“βˆΆ[π‘Ž,𝑏]→ℝ is said to be differentiable if its derivative exists. The derivative 𝑓Δ and the shift π‘“πœŽ of a function 𝑓 are related by the equation π‘“πœŽ=𝑓(𝜎(𝑑))=𝑓(𝑑)+πœ‡(𝑑)𝑓Δ(𝑑).(2.4) The derivative rules of the product 𝑓𝑔 and the quotient 𝑓/𝑔 (where π‘”π‘”πœŽβ‰ 0) of two differentiable functions 𝑓 and 𝑔 are given by(𝑓⋅𝑔)Ξ”(𝑑)=𝑓Δ(𝑑)𝑔(𝑑)+π‘“πœŽ(𝑑)𝑔Δ(𝑑)=𝑓(𝑑)𝑔Δ(𝑑)+𝑓Δ(𝑑)π‘”πœŽξ‚΅π‘“(𝑑),𝑔Δ𝑓(𝑑)=Ξ”(𝑑)𝑔(𝑑)βˆ’π‘“(𝑑)𝑔Δ(𝑑)𝑔(𝑑)π‘”πœŽ.(𝑑)(2.5) An integration by parts formula readsξ€œπ‘π‘Žπ‘“(𝑑)𝑔Δ[](𝑑)Δ𝑑=𝑓(𝑑)𝑔(𝑑)π‘π‘Žβˆ’ξ€œπ‘π‘Žπ‘“Ξ”(𝑑)π‘”πœŽ(𝑑)Δ𝑑(2.6) or ξ€œπ‘π‘Žπ‘“πœŽ(𝑑)𝑔Δ[](𝑑)Δ𝑑=𝑓(𝑑)𝑔(𝑑)π‘π‘Žβˆ’ξ€œπ‘π‘Žπ‘“Ξ”(𝑑)𝑔(𝑑)Δ𝑑(2.7) and the infinite integral is defined by ξ€œβˆžπ‘π‘“(𝑠)Δ𝑠=limπ‘‘β†’βˆžξ€œπ‘‘π‘π‘“(𝑠)Δ𝑠.(2.8) Note that in case 𝕋=ℝ, we have𝜎(𝑑)=𝜌(𝑑)=𝑑,πœ‡(𝑑)=0,𝑓Δ(𝑑)=π‘“ξ…žξ€œ(𝑑),π‘π‘Žξ€œπ‘“(𝑑)Δ𝑑=π‘π‘Žπ‘“(𝑑)𝑑𝑑,(2.9)

and in case 𝕋=β„€, we have𝜎(𝑑)=𝑑+1,𝜌(𝑑)=π‘‘βˆ’1,πœ‡(𝑑)=1,π‘“Ξ”ξ€œ(𝑑)=Δ𝑓(𝑑)=𝑓(𝑑+1)βˆ’π‘“(𝑑)ifπ‘Ž<𝑏,π‘π‘Žπ‘“(𝑑)Δ𝑑=π‘βˆ’1𝑑=π‘Žπ‘“(𝑑).(2.10)

Throughout this paper, we use𝑑+(𝑑)∢=max{0,𝑑(𝑑)},π‘‘βˆ’ξ‚»π›Ό(𝑑)∢=max{0,βˆ’π‘‘(𝑑)},𝛽(𝑑)∢=𝛼(𝑑)0<𝛾≀1𝛾(𝑑)𝛾>1,(2.11) where𝛼(𝑑)∢=𝑅(𝑑)𝑅(𝑑)+πœ‡(𝑑),𝑅(𝑑)∢=𝑝1/𝛾(ξ€œπ‘‘)𝑑𝑑0Δ𝑠𝑝1/𝛾(𝑠),for𝑑β‰₯𝑑0.(2.12)

3. Several Lemmas

In this section, we present some lemmas that we need in the proofs of our results in Section 4.

Lemma 1 (Bohner and Peterson [3, Theorem 1.90]).  If x(t) is delta differentiable and eventually positive or negative, then ((π‘₯(𝑑))𝛾)Ξ”ξ€œ=𝛾10[]β„Žπ‘₯(𝜎(𝑑))+(1βˆ’β„Ž)π‘₯(𝑑)π›Ύβˆ’1π‘₯Ξ”(𝑑)π‘‘β„Ž.(3.1)

Lemma 2 (Hardy et al. [16, Theorem 41]). If 𝐴 and 𝐡 are nonnegative real numbers, then πœ†π΄π΅πœ†βˆ’1βˆ’π΄πœ†β‰€(πœ†βˆ’1)π΅πœ†,πœ†>1,(3.2) where the equality holds if and only if 𝐴=𝐡.

Lemma 3. If (H1)–(H3) and (1.2) hold and (1.1) has a positive solution π‘₯ on [𝑑0,∞)𝕋, then ξ€·ξ€·π‘₯𝑝(𝑑)Ξ”ξ€Έ(𝑑)𝛾Δ<0,π‘₯Ξ”(𝑑)>0,π‘₯(𝑑)π‘₯𝜎(𝑑𝑑)>𝛼(𝑑),forπ‘‘βˆˆ0ξ€Έ,∞.(3.3)

Proof. The proof is similar to the proof of Lemma 2.1 in [15] and, hence, is omitted.

4. Main Results

Theorem 1. Assume that (H1)–(H5), (1.2), Lemma 3 hold and 𝜏∈C1π‘Ÿπ‘‘([t0,∞)𝕋,𝕋), 𝜏([t0,∞)𝕋)=[t0,∞)𝕋. Furthermore, assume that there exists a positive Ξ”-differentiable function 𝛿(t) such that limsupπ‘‘β†’βˆžξ€œπ‘‘π‘‘0βŽ‘βŽ’βŽ’βŽ£πΏπ›Όπ›Ύ(𝜏(𝑠))π‘ž(𝑠)π›ΏπœŽπ›Ώ(𝑠)βˆ’π‘(𝜏(𝑠))ξ€·ξ€·Ξ”ξ€Έ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(s))π›ΏπœŽ(𝑠)πœΞ”ξ€Έ(𝑠)π›ΎβŽ€βŽ₯βŽ₯βŽ¦Ξ”π‘ =∞.(4.1) Then every solution of (1.1) is oscillatory on [𝑑0,∞)𝕋.

Proof. Assume that (1.1) has a nonoscillatory solution on [𝑑0,∞)𝕋. Then, without loss of generality, we assume that π‘₯(𝑑)>0,π‘₯(𝜏(𝑑))>0 for all π‘‘βˆˆ[𝑑1,∞)𝕋,𝑑1∈[𝑑0,∞)𝕋, and there is π‘‡βˆˆ[𝑑0,∞)𝕋 such that π‘₯(𝑑) satisfies the conclusion of Lemma 3 on [𝑇,∞)𝕋. Consider the generalized Riccati substitution ξ‚΅π‘₯𝑀(𝑑)=𝛿(𝑑)𝑝(𝑑)Ξ”(𝑑)ξ‚Άπ‘₯(𝜏(𝑑))𝛾.(4.2) Using the delta derivative rules of the product and quotient of two functions, we have 𝑀Δ(𝑑)=𝛿Δπ‘₯(𝑑)𝑝(𝑑)Ξ”ξ€Έ(𝑑)𝛾(π‘₯(𝜏(𝑑)))𝛾+π›ΏπœŽξƒ©ξ€·π‘₯(𝑑)𝑝(𝑑)Ξ”ξ€Έ(𝑑)𝛾(π‘₯(𝜏(𝑑)))𝛾ξƒͺΞ”=𝛿Δ(𝑑)𝛿𝛿(𝑑)𝑀(𝑑)+πœŽξ€·ξ€·π‘₯(𝑑)𝑝(𝑑)Ξ”ξ€Έ(𝑑)𝛾Δ(π‘₯(𝜏(𝜎(𝑑))))π›Ύβˆ’π›ΏπœŽξ€·π‘₯(𝑑)𝑝(𝑑)Ξ”ξ€Έ(𝑑)𝛾((π‘₯(𝜏(𝑑)))𝛾)Ξ”(π‘₯(𝜏(𝑑)))𝛾(π‘₯(𝜏(𝜎(𝑑))))𝛾,(4.3) using the fact 𝑓(π‘₯)/π‘₯𝛾β‰₯𝐿 and π‘₯(𝑑)/π‘₯𝜎(𝑑)>𝛼(𝑑), we have 𝑀Δ(𝛿𝑑)≀Δ(𝑑)𝛿(𝑑)𝑀(𝑑)βˆ’πΏπ›Όπ›Ύ(𝜏(𝑑))π›ΏπœŽ(𝛿𝑑)π‘ž(𝑑)βˆ’πœŽξ€·π‘₯(𝑑)𝑝(𝑑)Ξ”ξ€Έ(𝑑)𝛾((π‘₯(𝜏(𝑑)))𝛾)Ξ”(π‘₯(𝜏(𝑑)))𝛾(π‘₯(𝜏(𝜎(𝑑))))𝛾.(4.4) If 0<𝛾≀1, then using the chain rule and the fact that π‘₯(𝑑) is strictly increasing on [𝑇,∞)𝕋, we obtain ((π‘₯(𝜏(𝑑)))𝛾)Ξ”ξ€œ=𝛾10ξ€Ίπ‘₯(𝜏(𝑑))+β„Žπœ‡(𝜏(𝑑))(π‘₯(𝜏(𝑑)))Ξ”ξ€»π›Ύβˆ’1π‘‘β„Ž(π‘₯(𝜏(𝑑)))Ξ”ξ€œ=𝛾10[(1βˆ’β„Ž)π‘₯(𝜏(𝑑))+β„Žπ‘₯𝜎](𝜏(𝑑))π›Ύβˆ’1π‘‘β„Ž(π‘₯(𝜏(𝑑)))Ξ”β‰₯𝛾(π‘₯𝜎(𝜏(𝑑)))π›Ύβˆ’1(π‘₯(𝜏(𝑑)))Ξ”πœΞ”(𝑑)(4.5) which implies 𝑀Δ𝛿(𝑑)≀Δ(𝑑)𝑀𝛿(𝑑)(𝑑)βˆ’πΏπ›Όπ›Ύ(𝜏(𝑑))π›ΏπœŽ(𝑑)π‘ž(𝑑)βˆ’π›Ύπ›ΏπœŽξ€·π‘₯(𝑑)𝑝(𝑑)Ξ”ξ€Έ(𝑑)𝛾(π‘₯𝜎(𝜏(𝑑)))π›Ύβˆ’1(π‘₯(𝜏(𝑑)))Ξ”πœΞ”(𝑑)(π‘₯(𝜏(𝑑)))𝛾(π‘₯(𝜏(𝜎(𝑑))))𝛾≀𝛿Δ(𝑑)𝛿(𝑑)𝑀(𝑑)βˆ’πΏπ›Όπ›Ύ(𝜏(𝑑))π›ΏπœŽ(𝑑)π‘ž(𝑑)βˆ’π›Ύπ›ΏπœŽ(𝑑)(π‘₯(𝜏(𝑑)))Δ𝛼(𝜏(𝑑))πœΞ”(𝑑)𝛿(𝑑)π‘₯(𝜏(𝑑))𝑀(𝑑),(4.6) since (𝑝(𝑑)(π‘₯Ξ”(𝑑))𝛾)Ξ”<0, then by integrating from 𝑑 to 𝜏(𝑑), we get (π‘₯(𝜏(𝑑)))Ξ”>(𝑝(𝑑))1/𝛾((π‘πœ(𝑑)))1/𝛾π‘₯Ξ”(𝑑)(4.7)𝑀Δ𝛿(𝑑)≀Δ(𝑑)𝛿(𝑑)𝑀(𝑑)βˆ’πΏπ›Όπ›Ύ(𝜏(𝑑))π›ΏπœŽ(𝑑)q(t)βˆ’π›Ύπ›ΏπœŽ(𝑑)(𝑝(𝑑))1/𝛾π‘₯Ξ”(𝑑)𝛼(𝜏(𝑑))πœΞ”(𝑑)𝛿(𝑑)π‘₯(𝜏(𝑑))(𝑝(𝜏(𝑑)))1/𝛾𝑀(𝑑),(4.8) that is, 𝑀Δ≀𝛿(𝑑)Ξ”(𝑑)𝛿(𝑑)𝑀(𝑑)βˆ’πΏπ›Όπ›Ύ(𝜏(𝑑))π›ΏπœŽ(𝑑)π‘ž(𝑑)βˆ’π›Ύπ›ΏπœŽ(𝑑)𝛼(𝜏(𝑑))πœΞ”(𝑑)𝛿(𝛾+1)/𝛾(𝑑)(𝑝(𝜏(𝑑)))1/𝛾𝑀(𝛾+1)/𝛾(𝑑).(4.9)
If 𝛾>1, then using the chain rule and the fact that π‘₯(𝑑) is strictly increasing on [𝑇,∞)𝕋, we obtain ((π‘₯(𝜏(𝑑)))𝛾)Ξ”β‰₯(π‘₯(𝜏(𝑑)))π›Ύβˆ’1(π‘₯(𝜏(𝑑)))Ξ”πœΞ”(𝑑).(4.10) From (4.4), (4.7), and (4.10), we have 𝑀Δ𝛿(𝑑)≀Δ(𝑑)𝛿(𝑑)𝑀(𝑑)βˆ’πΏπ›Όπ›Ύ(𝜏(𝑑))π›ΏπœŽ(𝑑)π‘ž(𝑑)βˆ’π›Ύπ›ΏπœŽ(𝑑)𝛼𝛾(𝜏(𝑑))πœΞ”(𝑑)𝛿(𝛾+1)/𝛾(𝑑)(𝑝(𝜏(𝑑)))1/𝛾𝑀(𝛾+1)/𝛾(𝑑).(4.11) By (4.9), (4.11), and the definition of 𝛽(𝑑), we have for 𝛾>0𝑀Δ𝛿(𝑑)≀Δ(𝑑)+𝛿(𝑑)𝑀(𝑑)βˆ’πΏπ›Όπ›Ύ(𝜏(𝑑))π›ΏπœŽ(𝑑)π‘ž(𝑑)βˆ’π›Ύπ›ΏπœŽ(𝑑)𝛽(𝜏(𝑑))πœΞ”(𝑑)π›Ώπœ†(𝑑)π‘πœ†βˆ’1𝑀(𝜏(𝑑))πœ†(𝑑),(4.12) where πœ†=(𝛾+1)/𝛾. Defining 𝐴β‰₯0 and 𝐡β‰₯0 by π΄πœ†=π›Ύπ›ΏπœŽ(𝑑)𝛽(𝜏(𝑑))πœΞ”(𝑑)π›Ώπœ†(𝑑)π‘πœ†βˆ’1𝑀(𝜏(𝑑))πœ†(𝑑),π΅πœ†βˆ’1=𝑝(πœ†βˆ’1)/πœ†ξ€·π›Ώ(𝜏(𝑑))Ξ”ξ€Έ(𝑑)+πœ†ξ€·π›Ύπ›ΏπœŽ(𝑑)𝛽(𝜏(𝑑))πœΞ”ξ€Έ(𝑑)1/πœ†,(4.13) then using Lemma 2, we get 𝛿Δ(𝑑)+𝛿(𝑑)𝑀(𝑑)βˆ’π›Ύπ›ΏπœŽ(𝑑)𝛽(𝜏(𝑑))πœΞ”(𝑑)π›Ώπœ†(𝑑)π‘πœ†βˆ’1𝑀(𝜏(𝑑))πœ†π›Ώ(𝑑)≀𝑝(𝜏(𝑑))ξ€·ξ€·Ξ”ξ€Έ(𝑑)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑑))π›ΏπœŽ(𝑑)πœΞ”ξ€Έ(𝑑)𝛾.(4.14) From this last inequality and (4.12), we get 𝑀Δ(𝑑)β‰€βˆ’πΏπ›Όπ›Ύ(𝜏(𝑑))π›ΏπœŽπ›Ώ(𝑑)π‘ž(𝑑)+𝑝(𝜏(𝑑))ξ€·ξ€·Ξ”ξ€Έ(𝑑)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑑))π›ΏπœŽ(𝑑)πœΞ”ξ€Έ(𝑑)𝛾.(4.15) Integrating both sides from 𝑇 to 𝑑, we get ξ€œπ‘‘π‘‡βŽ‘βŽ’βŽ’βŽ£πΏπ›Όπ›Ύ(𝜏(𝑠))π›ΏπœŽπ›Ώ(𝑠)π‘ž(𝑠)βˆ’π‘(𝜏(𝑠))ξ€·ξ€·Ξ”ξ€Έ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))π›ΏπœŽ(𝑠)πœΞ”ξ€Έ(𝑠)π›ΎβŽ€βŽ₯βŽ₯βŽ¦Ξ”π‘ β‰€π‘€(𝑇)βˆ’π‘€(𝑑)≀𝑀(𝑇),(4.16) which contradicts the assumption (4.1). This contradiction completes the proof.

Theorem 2. Assume that (H1)–(H5), (1.2), Lemma 3 hold and 𝜏∈C1π‘Ÿπ‘‘([t0,∞)𝕋,𝕋), 𝜏([t0,∞)𝕋)=[t0,∞)𝕋. Furthermore, assume that there exist functions H,h∈Cπ‘Ÿπ‘‘(𝔻,ℝ) (where 𝔻≑{(t,s)∢tβ‰₯sβ‰₯t0}) such that 𝐻(𝑑,𝑑)=0,𝑑β‰₯𝑑0,𝐻(𝑑,𝑠)>0,𝑑>𝑠β‰₯𝑑0,(4.17) and 𝐻 has a nonpositive continuous Ξ”-partial derivative with respect to the second variable 𝐻Δ𝑠(𝑑,𝑠) which satisfies 𝐻Δ𝑠𝛿(𝜎(𝑑),𝑠)+𝐻(𝜎(𝑑),𝜎(𝑠))Ξ”(𝑑)𝛿(𝑑)=βˆ’β„Ž(𝑑,𝑠)𝛿(𝑑)(𝐻(𝜎(𝑑),𝜎(𝑠)))𝛾/(𝛾+1),(4.18)limsupπ‘‘β†’βˆž1π»ξ€·πœŽ(𝑑),𝑑0ξ€Έξ€œπ‘‘πœŽ(𝑑)0𝐾(𝑑,𝑠)Δ𝑠=∞,(4.19) where 𝛿(𝑑) is positive Ξ”-differentiable function and 𝐾(𝑑,𝑠)=𝐻(𝜎(𝑑),𝜎(𝑠))𝐿𝛼𝛾(𝜏(𝑠))π‘ž(𝑠)π›ΏπœŽξ€·β„Ž(𝑠)βˆ’π‘(𝜏(𝑠))βˆ’ξ€Έ(𝑑,𝑠)𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))π›ΏπœŽ(𝑠)πœΞ”ξ€Έ(𝑠)𝛾.(4.20) Then every solution of (1.1) is oscillatory on [𝑑0,∞)𝕋.

Proof. Assume that (1.1) has a nonoscillatory solution on [𝑑0,∞)𝕋. Then, without loss of generality, we assume that π‘₯(𝑑)>0,π‘₯(𝜏(𝑑))>0 for all π‘‘βˆˆ[𝑑1,∞)𝕋,𝑑1∈[𝑑0,∞)𝕋, and there is π‘‡βˆˆ[𝑑0,∞)𝕋 such that π‘₯(𝑑) satisfies the conclusion of Lemma 3 on [𝑇,∞)𝕋. Define 𝑀(𝑑) as in the proof of Theorem 1. Replacing (𝛿Δ(𝑑))+ with𝛿Δ (𝑑) in (4.12), we have 𝐿𝛼𝛾(𝜏(𝑑))π›ΏπœŽ(𝑑)π‘ž(𝑑)β‰€βˆ’π‘€Ξ”π›Ώ(𝑑)+Ξ”(𝑑)𝛿(𝑑)𝑀(𝑑)βˆ’π›Ύπ›ΏπœŽ(𝑑)𝛽(𝜏(𝑑))πœΞ”(𝑑)π›Ώπœ†(𝑑)π‘πœ†βˆ’1(π‘€πœ(𝑑))πœ†(𝑑).(4.21) Multiplying (4.21) by 𝐻(𝜎(𝑑),𝜎(𝑠)), and integrating with respect to 𝑠 from 𝑇 to 𝜎(𝑑), we get ξ€œπ‘‡πœŽ(𝑑)𝐻(𝜎(𝑑),𝜎(𝑠))𝐿𝛼𝛾(𝜏(𝑠))π›ΏπœŽξ€œ(𝑠)π‘ž(𝑠)Ξ”π‘ β‰€βˆ’π‘‡πœŽ(𝑑)𝐻(𝜎(𝑑),𝜎(𝑠))𝑀Δ+ξ€œ(𝑠)Ξ”π‘ π‘‡πœŽ(𝑑)𝛿𝐻(𝜎(𝑑),𝜎(𝑠))Ξ”(𝑠)βˆ’ξ€œπ›Ώ(𝑠)𝑀(𝑠)Ξ”π‘ π‘‡πœŽ(𝑑)𝐻(𝜎(𝑑),𝜎(𝑠))π›Ύπ›ΏπœŽ(𝑠)𝛽(𝜏(𝑠))πœΞ”(𝑠)π›Ώπœ†(𝑠)π‘πœ†βˆ’1𝑀(𝜏(𝑠))πœ†(𝑠)Δ𝑠.(4.22) Integrating by parts and using (4.17) and (4.18), we obtain ξ€œπ‘‡πœŽ(𝑑)𝐻(𝜎(𝑑),𝜎(𝑠))𝐿𝛼𝛾(𝜏(𝑠))π›ΏπœŽ+ξ€œ(𝑠)π‘ž(𝑠)Δ𝑠≀𝐻(𝜎(𝑑),𝑇)𝑀(𝑇)π‘‡πœŽ(𝑑)ξ‚Έβ„Žβˆ’(𝑑,𝑠)𝛿(𝑠)(𝐻(𝜎(𝑑),𝜎(𝑠)))1/πœ†π‘€(𝑠)βˆ’π»(𝜎(𝑑),𝜎(𝑠))π›Ύπ›ΏπœŽ(𝑠)𝛽(𝜏(𝑠))πœΞ”(𝑠)π›Ώπœ†(𝑠)π‘πœ†βˆ’1𝑀(𝜏(𝑠))πœ†ξ‚Ή(𝑠)Δ𝑠.(4.23) Defining 𝐴β‰₯0 and 𝐡β‰₯0 by π΄πœ†=𝐻(𝜎(𝑑),𝜎(𝑠))π›Ύπ›ΏπœŽ(𝑠)𝛽(𝜏(𝑠))πœΞ”(𝑠)π›Ώπœ†(𝑠)π‘πœ†βˆ’1𝑀(𝜏(𝑠))πœ†(𝑠),π΅πœ†βˆ’1=𝑝(πœ†βˆ’1)/πœ†(𝜏(𝑠))β„Žβˆ’(𝑑,𝑠)πœ†ξ€·π›Ύπ›ΏπœŽ(𝑠)𝛽(𝜏(𝑠))πœΞ”ξ€Έ(𝑠)1/πœ†,(4.24) then using Lemma 2, we get β„Žβˆ’(𝑑,𝑠)𝛿(𝑠)(𝐻(𝜎(𝑑),𝜎(𝑠)))1/πœ†π‘€(𝑠)βˆ’π»(𝜎(𝑑),𝜎(𝑠))π›Ύπ›ΏπœŽ(𝑠)𝛽(𝜏(𝑠))πœΞ”(𝑠)π›Ώπœ†(𝑠)π‘πœ†βˆ’1(π‘€πœ(𝑠))πœ†β‰€ξ€·β„Ž(𝑠)𝑝(𝜏(𝑠))βˆ’ξ€Έ(𝑑,𝑠)𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))π›ΏπœŽ(𝑠)πœΞ”ξ€Έ(𝑠)𝛾,(4.25) therefore, ξ€œπ‘‡πœŽ(𝑑)𝐻(𝜎(𝑑),𝜎(𝑠))𝐿𝛼𝛾(𝜏(𝑠))π›ΏπœŽ+ξ€œ(𝑠)π‘ž(𝑠)Δ𝑠≀𝐻(𝜎(𝑑),𝑇)𝑀(𝑇)π‘‡πœŽ(𝑑)ξ€·β„Žπ‘(𝜏(𝑠))βˆ’ξ€Έ(𝑑,𝑠)𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))π›ΏπœŽ(𝑠)πœΞ”(𝑠)𝛾Δ𝑠.(4.26) By the definition of 𝐾(𝑑,𝑠), we get ξ€œπ‘‡πœŽ(𝑑)𝐾(𝑑,𝑠)Δ𝑠≀𝐻(𝜎(𝑑),𝑇)𝑀(𝑇),(4.27) and this implies that 1ξ€œπ»(𝜎(𝑑),𝑇)π‘‡πœŽ(𝑑)𝐾(𝑑,𝑠)Δ𝑠≀𝑀(𝑇),(4.28) which contradicts the assumption (4.19). This contradiction completes the proof.

Theorem 3. Assume that (H1)–(H5), (1.2), Lemma 3 hold and 𝜏∈C1π‘Ÿπ‘‘([t0,∞)𝕋,𝕋), 𝜏([t0,∞)𝕋)=[t0,∞)𝕋. Furthermore, assume that there exists a positive Ξ”-differentiable function 𝛿(t) such that for mβ‰₯1limsupπ‘‘β†’βˆž1π‘‘π‘šξ€œπ‘‘π‘‘0(π‘‘βˆ’π‘ )π‘šβŽ‘βŽ’βŽ’βŽ£πΏπ›Όπ›Ύ(𝜏(𝑠))π‘ž(𝑠)π›ΏπœŽπ›Ώ(𝑠)βˆ’π‘(𝜏(𝑠))ξ€·ξ€·Ξ”ξ€Έ(𝑠)+𝛾+1(𝛿+1)(𝛾+1)𝛽(𝜏(𝑠))π›ΏπœŽ(𝑠)πœΞ”ξ€Έ(𝑠)π›ΎβŽ€βŽ₯βŽ₯βŽ¦Ξ”π‘ =∞.(4.29) Then every solution of (1.1) is oscillatory on [𝑑0,∞)𝕋.

Proof. Assume that (1.1) has a nonoscillatory solution on [𝑑0,∞)𝕋. Then, without loss of generality, we assume that π‘₯(𝑑)>0,π‘₯(𝜏(𝑑))>0 for all π‘‘βˆˆ[𝑑1,∞)𝕋,𝑑1∈[𝑑0,∞)𝕋, and there is π‘‡βˆˆ[𝑑0,∞)𝕋 such that π‘₯(𝑑) satisfies the conclusion of Lemma 3 on [𝑇,∞)𝕋. Proceeding as in the proof of Theorem 1, we get (4.15) from which we have 𝐿𝛼𝛾(𝜏(𝑑))π›ΏπœŽπ›Ώ(𝑑)π‘ž(𝑑)βˆ’π‘(𝜏(𝑑))ξ€·ξ€·Ξ”ξ€Έ(𝑑)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑑))π›ΏπœŽ(𝑑)πœΞ”ξ€Έ(𝑑)π›Ύβ‰€βˆ’π‘€Ξ”(𝑑),(4.30) therefore, ξ€œπ‘‘π‘‘1(π‘‘βˆ’π‘ )π‘šβŽ›βŽœβŽœβŽπΏπ›Όπ›Ύ(𝜏(𝑠))π›ΏπœŽπ›Ώ(𝑠)π‘ž(𝑠)βˆ’π‘(𝜏(𝑠))ξ€·ξ€·Ξ”ξ€Έ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))π›ΏπœŽ(𝑠)πœΞ”ξ€Έ(𝑠)π›ΎβŽžβŽŸβŽŸβŽ ξ€œΞ”π‘ .β‰€βˆ’π‘‘π‘‘1(π‘‘βˆ’π‘ )π‘šπ‘€Ξ”(𝑑)Δ𝑠.(4.31) The right hand side of the above inequality gives ξ€œπ‘‘π‘‘1(π‘‘βˆ’π‘ )π‘šπ‘€Ξ”(𝑠)Δ𝑠=(π‘‘βˆ’π‘ )π‘šπ‘€(𝑠)𝑑𝑑1βˆ’ξ€œπ‘‘π‘‘1((π‘‘βˆ’π‘ )π‘š)Δ𝑠𝑀(𝜎(𝑠))Δ𝑠.(4.32) Since ((π‘‘βˆ’π‘ )π‘š)Ξ”π‘ β‰€βˆ’π‘š(π‘‘βˆ’πœŽ(𝑠))π‘šβˆ’1≀0 for 𝑑β‰₯𝜎(𝑠), π‘šβ‰₯1, then we have ξ€œπ‘‘π‘‘1(π‘‘βˆ’π‘ )π‘šβŽ‘βŽ’βŽ’βŽ£πΏπ›Όπ›Ύ(𝜏(𝑠))π‘ž(𝑠)π›ΏπœŽπ›Ώ(𝑠)βˆ’π‘(𝜏(𝑠))ξ€·ξ€·Ξ”ξ€Έ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))π›ΏπœŽ(𝑠)πœΞ”ξ€Έ(𝑠)π›ΎβŽ€βŽ₯βŽ₯βŽ¦ξ€·Ξ”π‘ β‰€π‘‘βˆ’π‘‘1ξ€Έπ‘šπ‘€ξ€·π‘‘1ξ€Έ,(4.33) then, 1π‘‘π‘šξ€œπ‘‘π‘‘1(π‘‘βˆ’π‘ )π‘šβŽ‘βŽ’βŽ’βŽ£πΏπ›Όπ›Ύ(𝜏(𝑠))π‘ž(𝑠)π›ΏπœŽπ›Ώ(𝑠)βˆ’π‘(𝜏(𝑠))ξ€·ξ€·Ξ”ξ€Έ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))π›ΏπœŽ(𝑠)πœΞ”ξ€Έ(𝑠)π›ΎβŽ€βŽ₯βŽ₯βŽ¦ξ‚΅Ξ”π‘ β‰€π‘‘βˆ’π‘‘1π‘‘ξ‚Άπ‘šπ‘€ξ€·π‘‘1ξ€Έ(4.34) which contradicts (4.29). This contradiction completes the proof.

Theorem 4. Assume that βˆ«βˆžπ‘‘0Δ𝑑/𝑝1/𝛾(𝑑)=∞ and limsupπ‘‘β†’βˆžπΏπ‘…π›Ύ(𝜏(𝑑))ξ€œπ‘(𝜏(𝑑))βˆžπ‘‘π‘ž(𝑠)Δ𝑠>1,β„Žπ‘œπ‘™π‘‘.(4.35) Then every solution of (1.1) is oscillatory on [𝑑0,∞)𝕋.

Proof. Assume that (1.1) has a nonoscillatory solution on [𝑑0,∞)𝕋. Then, without loss of generality, we assume that π‘₯(𝑑)>0,π‘₯(𝜏(𝑑))>0 for all π‘‘βˆˆ[𝑑1,∞)𝕋,𝑑1∈[𝑑0,∞)𝕋, and there is π‘‡βˆˆ[𝑑0,∞)𝕋 such that π‘₯(𝑑) satisfies the conclusion of Lemma 3 on [𝑇,∞)𝕋. From (1.1), we have ξ€·ξ€·π‘₯𝑝(𝑑)Ξ”ξ€Έ(𝑑)𝛾Δ=βˆ’π‘ž(𝑑)𝑓(π‘₯(𝜏(𝑑)))β‰€βˆ’πΏπ‘ž(𝑑)π‘₯𝛾(𝜏(𝑑)).(4.36) Integrating last equation from 𝜏(𝑑) to ∞, we obtain ξ€œβˆžπœ(𝑑)πΏπ‘ž(𝑠)π‘₯𝛾π‘₯(𝜏(𝑠))Δ𝑠<𝑝(𝜏(𝑑))Ξ”ξ€Έ(𝜏(𝑑))π›Ύβˆ’limπ‘ β†’βˆžξ€·π‘₯𝑝(𝑠)Ξ”ξ€Έ(𝑠)𝛾.(4.37) Since 𝑝(𝑠)(π‘₯Ξ”(𝑠))𝛾 decreasing and 𝑝(𝑠)(π‘₯Ξ”(𝑠))𝛾>0, then we have 1ξ€œπ‘(𝜏(𝑑))∞𝜏(𝑑)πΏπ‘ž(𝑠)π‘₯𝛾π‘₯(𝜏(𝑠))Δ𝑠<Ξ”ξ€Έ(𝜏(𝑑))𝛾.(4.38) Since π‘₯(𝑑)>𝑅(𝑑)π‘₯Ξ”(𝑑), then π‘₯(𝜏(𝑑))>𝑅(𝜏(𝑑))π‘₯Ξ”(𝜏(𝑑)), and consequently πΏξ€œπ‘(𝜏(𝑑))∞𝜏(𝑑)π‘ž(𝑠)π‘₯𝛾(𝜏(𝑠))Δ𝑠<π‘₯(𝜏(𝑑))𝑅(𝜏(𝑑))𝛾,𝐿𝑅𝛾(𝜏(𝑑))ξ€œπ‘(𝜏(𝑑))∞𝜏(𝑑)π‘ž(𝑠)π‘₯𝛾(𝜏(𝑠))Δ𝑠<π‘₯𝛾(𝜏(𝑑)),(4.39) but 𝐿𝑅𝛾(𝜏(𝑑))ξ€œπ‘(𝜏(𝑑))βˆžπ‘‘π‘ž(𝑠)π‘₯𝛾(𝜏(𝑠))Δ𝑠<𝐿𝑅𝛾(𝜏(𝑑))ξ€œπ‘(𝜏(𝑑))∞𝜏(𝑑)π‘ž(𝑠)π‘₯𝛾(𝜏(𝑠))Δ𝑠<π‘₯𝛾(𝜏(𝑑)).(4.40) Since π‘₯(𝑑) and 𝜏(𝑑) are strictly increasing, then we get that 𝐿𝑅𝛾(𝜏(𝑑))ξ€œπ‘(𝜏(𝑑))βˆžπ‘‘π‘ž(𝑠)Δ𝑠<1,(4.41) therefore, 𝐿𝑅𝛾(𝜏(𝑑))ξ€œπ‘(𝜏(𝑑))βˆžπ‘‘π‘ž(𝑠)Δ𝑠≀1.(4.42) This contradiction completes the proof.

5. Examples

In this section, we give some examples to illustrate our main results.

Example 1. Consider the second-order nonlinear delay dynamic equation 𝑑𝛾π‘₯Ξ”ξ€Έ(𝑑)𝛾Δ+πœ†π‘‘π›Όπ›Ύπ‘₯(𝜏(𝑑))𝛾𝑑(𝜏(𝑑))=0forπ‘‘βˆˆ0ξ€Έ,βˆžπ•‹,𝑑0β‰₯0,(5.1) where πœ† is a positive constant,l and 𝛾 is the quotient of odd positive integers.
Here, 𝑝(𝑑)=π‘‘π›Ύπœ†,π‘ž(𝑑)=𝑑𝛼𝛾(𝜏(𝑑)),𝑓(π‘₯)=π‘₯𝛾,𝐿=1.(5.2) If 𝛿(𝑑)=1, then ξ€œβˆžπ‘‘0Δ𝑑(𝑝(𝑑))1/𝛾=ξ€œβˆžπ‘‘0Δ𝑑𝑑=∞,limsupπ‘‘β†’βˆžξ€œπ‘‘π‘‘0βŽ‘βŽ’βŽ’βŽ£πΏπ›Όπ›Ύ(𝜏(𝑠))π‘ž(𝑠)π›ΏπœŽπ›Ώ(𝑠)βˆ’π‘(𝜏(𝑠))ξ€·ξ€·Ξ”ξ€Έ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))π›ΏπœŽ(𝑠)πœΞ”ξ€Έ(𝑠)π›ΎβŽ€βŽ₯βŽ₯βŽ¦Ξ”π‘ =limsupπ‘‘β†’βˆžξ€œπ‘‘π‘‘0πœ†π‘ Ξ”π‘ =∞.(5.3) Therefore, by Theorem 1, every solution of (5.1) is oscillatory.

Example 2. Consider the second-order nonlinear delay dynamic equation π‘₯ξ€·ξ€·Ξ”ξ€Έ(𝑑)𝛾Δ+πœ†πœŽπ›Ύβˆ’1(𝑑)π‘‘π›Ύπœπ›Ύ(π‘₯𝑑)𝛾π‘₯(𝜏(𝑑))2𝛾𝑑(𝜏(𝑑))+1=0forπ‘‘βˆˆ0ξ€Έ,βˆžπ•‹,𝑑0β‰₯0,(5.4) where πœ† is a positive constant, and 0<𝛾≀1 is the quotient of odd positive integers, that is, 𝛼(𝑑)=𝛽(𝑑).
Here, 𝑝(𝑑)=1,π‘ž(𝑑)=πœ†πœŽπ›Ύβˆ’1(𝑑)π‘‘π›Ύπœπ›Ύ(𝑑),𝑓(π‘₯)=π‘₯𝛾π‘₯2𝛾𝑑+1,𝐿=1,𝜏(𝑑)=2.(5.5) It is clear that (1.2) holds.
Since 𝑅(𝜏(𝑑))=𝑝1/π›Ύβˆ«(𝜏(𝑑))π‘‘πœ(𝑑)0Δ𝑠/𝑝1/𝛾(𝑠)=𝜏(𝑑)βˆ’π‘‘0, then we can find 0<𝑏<1 such that 𝛼(𝜏(𝑑))=𝑅(𝜏(𝑑))=𝑅(𝜏(𝑑))+πœ‡(𝜏(𝑑))𝜏(𝑑)βˆ’π‘‘0𝜏(𝑑)βˆ’π‘‘0=𝜏+𝜎(𝜏(𝑑))βˆ’πœ(𝑑)(𝑑)βˆ’π‘‘0𝜎(𝜏(𝑑))βˆ’π‘‘0>π‘πœ(𝑑)𝜎(𝜏(𝑑)),for𝑑β‰₯𝑑𝑏>𝑑0.(5.6) If 𝛿(𝑑)=𝑑, then limsupπ‘‘β†’βˆžξ€œπ‘‘π‘‘0βŽ‘βŽ’βŽ’βŽ£πΏπ›Όπ›Ύ(𝜏(𝑠))π‘ž(𝑠)π›ΏπœŽπ›Ώ(𝑠)βˆ’π‘(𝜏(𝑠))ξ€·ξ€·Ξ”ξ€Έ(𝑠)+𝛾+1(𝛾+1)(𝛾+1)𝛽(𝜏(𝑠))π›ΏπœŽ(𝑠)πœΞ”ξ€Έ(𝑠)π›ΎβŽ€βŽ₯βŽ₯βŽ¦Ξ”π‘ >limsupπ‘‘β†’βˆžξ€œπ‘‘π‘‘0ξ‚Έπ‘π›Ύπœπ›Ύ(𝑠)πœ†πœŽπ›Ύβˆ’1(𝑠)𝜎(𝑠)πœŽπ›Ύ(𝜏(𝑠))πœπ›Ύ(𝑠)π‘ π›Ύβˆ’22π›ΎπœŽπ›Ύ(𝜏(𝑠))(𝛾+1)(𝛾+1)π‘π›Ύπ‘ π›ΎπœŽπ›Ύξ‚Ή(𝑠)Δ𝑠>limsupπ‘‘β†’βˆžξ€œπ‘‘π‘‘0ξ‚Έπ‘π›Ύπœ†πœŽπ›Ύ(𝜏(𝑠))π‘ π›ΎπœŽπ›Ύβˆ’2(𝜏(𝑠))2π›ΎπœŽπ›Ύ(𝑠)(𝛾+1)(𝛾+1)π‘π›Ύπ‘ π›ΎπœŽπ›Ύξ‚Ή=𝑏(𝑠)Δ𝑠𝛾2πœ†βˆ’2𝛾(𝛾+1)(𝛾+1)𝑏𝛾limsupπ‘‘β†’βˆžξ€œπ‘‘π‘‘01𝑠𝛾Δ𝑠=∞,(5.7) if πœ†>22𝛾/(𝑏2𝛾(𝛾+1)(𝛾+1)). Then by Theorem 1, every solution of (5.4) is oscillatory if πœ†>22𝛾/(𝑏2𝛾(𝛾+1)(𝛾+1)).

Example 3. Consider the second-order nonlinear delay dynamic equation ξ€·π‘‘π›Ύβˆ’1ξ€·π‘₯Ξ”ξ€Έ(𝑑)𝛾Δ+πœ†π‘₯π‘‘πœŽ(𝑑)𝛾𝑑(𝜏(𝑑))=0forπ‘‘βˆˆ0ξ€Έ,βˆžπ•‹,𝑑0β‰₯0,(5.8) where πœ† is a positive constant and 𝛾β‰₯1 is the quotient of odd positive integers.
Here, 𝑝(𝑑)=π‘‘π›Ύβˆ’1πœ†,π‘ž(𝑑)=π‘‘πœŽ(𝑑),𝑓(π‘₯)=π‘₯𝛾,𝐿=1.(5.9) It is clear that βˆ«βˆžπ‘‘0Δ𝑑/𝑝1/π›Ύβˆ«(𝑑)=βˆžπ‘‘0Δ𝑑/𝑑(π›Ύβˆ’1)/𝛾=∞, for 𝛾β‰₯1, (i.e., (1.2) holds) and 𝑅(𝜏(𝑑))β‰₯𝜏(𝑑)βˆ’π‘‘0β‰₯π‘˜πœ(𝑑) for 0<π‘˜<1, and 𝑑β‰₯𝑑0β‰₯1.
Then, limsupπ‘‘β†’βˆžπΏπ‘…π›Ύ(𝜏(𝑑))ξ€œπ‘(𝜏(𝑑))βˆžπ‘‘π‘ž(𝑠)Δ𝑠β‰₯limsupπ‘‘β†’βˆžπ‘˜π›Ύπœπ›Ύ(𝑑)πœπ›Ύβˆ’1ξ€œ(𝑑)βˆžπ‘‘πœ†π‘ πœŽ(𝑠)Δ𝑠=πœ†limsupπ‘‘β†’βˆžπ‘˜π›Ύξ€œπœ(𝑑)βˆžπ‘‘ξ‚€βˆ’1𝑠ΔΔ𝑠=πœ†π‘˜π›Ύπœ(𝑑)𝑑>1,(5.10) if πœ†>𝑑/π‘˜π›Ύπœ(𝑑). Then by Theorem 4, every solution of (5.8) is oscillatory if πœ†>𝑑/π‘˜π›Ύπœ(𝑑).

Remarks 1. (1) The recent results due to Hassan [15], Grace et al. [11] and Agarwal et al. [7] cannot be applied to (5.1), (5.4), and (5.8) as they deal with ordinary equations without delay.
(2) If 0<𝛾≀1, the results of Sun et al. [12] cannot be applied to (5.1) and (5.4).

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