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Discrete Dynamics in Nature and Society
VolumeΒ 2010, Article IDΒ 796256, 9 pages
http://dx.doi.org/10.1155/2010/796256
Research Article

New Oscillation Criteria for Second-Order Delay Differential Equations with Mixed Nonlinearities

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Received 10 January 2010; Accepted 17 June 2010

Academic Editor: BinggenΒ Zhang

Copyright Β© 2010 Yuzhen Bai and Lihua Liu. 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

We establish new oscillation criteria for second-order delay differential equations with mixed nonlinearities of the form βˆ‘(𝑝(𝑑)π‘₯β€²(𝑑))β€²+𝑛𝑖=1𝑝𝑖(𝑑)π‘₯(π‘‘βˆ’πœπ‘–βˆ‘)+𝑛𝑖=1π‘žπ‘–(𝑑)|π‘₯(π‘‘βˆ’πœπ‘–)|𝛼𝑖sgnπ‘₯(π‘‘βˆ’πœπ‘–)=𝑒(𝑑),𝑑β‰₯0, where 𝑝(𝑑), 𝑝𝑖(𝑑), π‘žπ‘–(𝑑), and 𝑒(𝑑) are continuous functions defined on [0,∞), and 𝑝(𝑑)>0, π‘ξ…ž(𝑑)β‰₯0, and 𝛼1>β‹―>π›Όπ‘š>1>π›Όπ‘š+1>β‹―>𝛼𝑛>0. No restriction is imposed on the potentials 𝑝𝑖(𝑑), π‘žπ‘–(𝑑), and 𝑒(𝑑) to be nonnegative. These oscillation criteria extend and improve the results given in the recent papers. An interesting example illustrating the sharpness of our results is also provided.

1. Introduction

We consider the second-order delay differential equations containing mixed nonlinearities of the form 𝑝(𝑑)π‘₯ξ…žξ€Έ(𝑑)ξ…ž+𝑛𝑖=1𝑝𝑖(𝑑)π‘₯π‘‘βˆ’πœπ‘–ξ€Έ+𝑛𝑖=1π‘žπ‘–||(𝑑)π‘₯(π‘‘βˆ’πœπ‘–)||𝛼𝑖sgnπ‘₯π‘‘βˆ’πœπ‘–ξ€Έ=𝑒(𝑑),𝑑β‰₯0.(1.1)

In what follows we assume that πœπ‘–β‰₯0, π‘βˆˆπΆ1[0,∞), 𝑝(𝑑)>0,π‘ξ…ž(𝑑)β‰₯0,𝑝𝑖,π‘žπ‘–,π‘’βˆˆπΆ[0,∞), 𝛼1>β‹―>π›Όπ‘š>1>π›Όπ‘š+1>β‹―>𝛼𝑛>0(𝑛>π‘šβ‰₯1), and ξ€œβˆž01𝑝(𝑑)𝑑𝑑=∞.(1.2)

As usual, a solution π‘₯(𝑑) of (1.1) is called oscillatory if it is defined on some ray [𝑇,∞) with 𝑇β‰₯0 and has arbitrary large zeros, otherwise, it is called nonoscillatory. Equation (1.1) is called oscillatory if all of its extendible solutions are oscillatory.

Recently, Mustafa [1] has studied the oscillatory solutions of certain forced Emden-Fowler like equations π‘₯ξ…žξ…ž||||(𝑑)+π‘Ž(𝑑)π‘₯(𝑑)πœ†sgnπ‘₯(𝑑)=𝑒(𝑑),𝑑β‰₯𝑑0β‰₯1.(1.3) Sun and Wong [2], as well as Sun and Meng [3] have established oscillation criteria for the second-order equation 𝑝(𝑑)π‘₯ξ…žξ€Έ(𝑑)ξ…ž+π‘ž(𝑑)π‘₯(𝑑)+𝑛𝑖=1π‘žπ‘–||||(𝑑)π‘₯(𝑑)𝛼𝑖sgnπ‘₯(𝑑)=𝑒(𝑑),𝑑β‰₯0.(1.4) Later in [4], Li and Chen have extended (1.4) to the delay differential equation 𝑝(𝑑)π‘₯ξ…žξ€Έ(𝑑)ξ…ž+π‘ž(𝑑)π‘₯(π‘‘βˆ’πœ)+𝑛𝑖=1π‘žπ‘–||||(𝑑)π‘₯(π‘‘βˆ’πœ)𝛼𝑖sgnπ‘₯(π‘‘βˆ’πœ)=𝑒(𝑑),𝑑β‰₯0.(1.5) As it is indicated in [2, 3], further research on the oscillation of equations of mixed type is necessary as such equations arise in mathematical modeling, for example, in the growth of bacteria population with competitive species. In this paper, we will continue in the direction to study the oscillatory properties of (1.1). We will employ the method in study of Kong in [5] and the arithmetic-geometric mean inequality (see [6]) to establish the interval oscillation criteria for the unforced (1.1) and forced (1.1), which extend and improve the known results. Our results are generalizations of the main results in [3, 4]. We also give an example to illustrate the sharpness of our main results.

2. Main Results

We need the following lemma proved in [2, 3] for our subsequent discussion.

Lemma 2.1. For any given 𝑛-tuple {𝛼1,𝛼2,…,𝛼𝑛} satisfying 𝛼1>𝛼2>β‹―>π›Όπ‘š>1>π›Όπ‘š+1>β‹―>𝛼𝑛>0, there corresponds an n-tuple {πœ‚1,πœ‚2,…,πœ‚π‘›} such that 𝑛𝑖=1π›Όπ‘–πœ‚π‘–=1,(a) which also satisfies either 𝑛𝑖=1πœ‚π‘–<1,0<πœ‚π‘–<1,(b) or 𝑛𝑖=1πœ‚π‘–=1,0<πœ‚π‘–<1.(c)

For a given set of exponents {𝛼𝑖} satisfying 𝛼1>𝛼2>β‹―>π›Όπ‘š>1>π›Όπ‘š+1>β‹―>𝛼𝑛>0, Lemma 2.1 ensures the existence of an 𝑛-tuple {πœ‚1,πœ‚2,…,πœ‚π‘›} such that either (a) and (b) hold or (a) and (c) hold. When 𝑛=2 and 𝛼1>1>𝛼2>0, in the first case, we have πœ‚1=1βˆ’π›Ό2ξ€·1βˆ’πœ‚0𝛼1βˆ’π›Ό2,πœ‚2=𝛼1ξ€·1βˆ’πœ‚0ξ€Έβˆ’1𝛼1βˆ’π›Ό2,(2.1) where πœ‚0 can be any positive number satisfying 0<πœ‚0<(𝛼1βˆ’1)/𝛼1. This will ensure that 0<πœ‚1, πœ‚2<1, and conditions (a) and (b) are satisfied. In the second case, we simply solve (a) and (c) and obtain πœ‚1=1βˆ’π›Ό2𝛼1βˆ’π›Ό2,πœ‚2=𝛼1βˆ’1𝛼1βˆ’π›Ό2.(2.2)

Following Philos [7], we say that a continuous function 𝐻(𝑑,𝑠) belongs to a function class π’Ÿπ‘Ž,𝑏, denoted by π»βˆˆπ’Ÿπ‘Ž,𝑏, if 𝐻(𝑏,𝑠)>0,𝐻(𝑠,π‘Ž)>0 for 𝑏>𝑠>π‘Ž, and 𝐻(𝑑,𝑠) has continuous partial derivatives πœ•π»(𝑑,𝑠)/πœ•π‘‘ and πœ•π»(𝑑,𝑠)/πœ•π‘  in [π‘Ž,𝑏]Γ—[π‘Ž,𝑏]. Set β„Ž1(𝑑,𝑠)=πœ•π»(𝑑,𝑠)/πœ•π‘‘2√𝐻(𝑑,𝑠),β„Ž2(𝑑,𝑠)=βˆ’πœ•π»(𝑑,𝑠)/πœ•π‘ 2√.𝐻(𝑑,𝑠)(2.3)

Based on Lemma 2.1, we have the following interval criterion for oscillation of (1.1).

Theorem 2.2. If, for any 𝑇β‰₯0, there exist π‘Ž1, 𝑏1, 𝑐1, π‘Ž2,𝑏2 and 𝑐2 such that π‘‡β‰€π‘Ž1<𝑐1<𝑏1β‰€π‘Ž2<𝑐2<𝑏2,π‘π‘–ξ€Ίπ‘Ž(𝑑)β‰₯0,π‘‘βˆˆ1βˆ’πœπ‘–,𝑏1ξ€»βˆͺξ€Ίπ‘Ž2βˆ’πœπ‘–,𝑏2ξ€»π‘ž,𝑖=1,2,…,𝑛,𝑖(ξ€Ίπ‘Žπ‘‘)β‰₯0,π‘‘βˆˆ1βˆ’πœπ‘–,𝑏1ξ€»βˆͺξ€Ίπ‘Ž2βˆ’πœπ‘–,𝑏2ξ€»π‘’ξ€Ίπ‘Ž,𝑖=1,2,…,𝑛,(𝑑)≀0,π‘‘βˆˆ1βˆ’πœπ‘–,𝑏1ξ€»ξ€Ίπ‘Ž,𝑒(𝑑)β‰₯0,π‘‘βˆˆ2βˆ’πœπ‘–,𝑏2ξ€»,(2.4) and there exist π»π‘—βˆˆπ’Ÿπ‘Žπ‘—,𝑏𝑗 such that 1𝐻𝑗𝑐𝑗,π‘Žπ‘—ξ€Έξ€œπ‘π‘—π‘Žπ‘—ξ‚€π‘„π‘—(𝑠)𝐻𝑗𝑠,π‘Žπ‘—ξ€Έβˆ’π‘(𝑠)β„Ž2𝑗1𝑠,π‘Žπ‘—ξ€Έξ‚+1𝑑𝑠𝐻𝑗𝑏𝑗,π‘π‘—ξ€Έξ€œπ‘π‘—π‘π‘—ξ‚€π‘„π‘—(𝑠)𝐻𝑗𝑏𝑗,π‘ βˆ’π‘(𝑠)β„Ž2𝑗2𝑏𝑗,𝑠𝑑𝑠>0,(2.5) for 𝑗=1,2, where β„Žπ‘—1, β„Žπ‘—2 are defined as in (2.3), πœ‚1,πœ‚2,…,πœ‚π‘› are positive constants satisfying (a) and (b) in Lemma 2.1, πœ‚0βˆ‘=1βˆ’π‘›π‘–=1πœ‚π‘–, and 𝑄𝑗(𝑑)=𝑛𝑖=1𝑝𝑖(𝑑)π‘‘βˆ’π‘Žπ‘—π‘‘βˆ’π‘Žπ‘—+πœπ‘–ξ‚Ά+ξ€·πœ‚0βˆ’1||||𝑒(𝑑)πœ‚0𝑛𝑖=1ξ€·πœ‚π‘–βˆ’1π‘žπ‘–ξ€Έ(𝑑)πœ‚π‘–ξ‚΅π‘‘βˆ’π‘Žπ‘—π‘‘βˆ’π‘Žπ‘—+πœπ‘–ξ‚Άπ›Όπ‘–πœ‚π‘–,(2.6) then (1.1) is oscillatory.

Proof. Let π‘₯(𝑑) be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that π‘₯(𝑑)>0 for all 𝑑β‰₯𝑑1βˆ’πœβ‰₯0, where 𝑑1 depends on the solution π‘₯(𝑑) and 𝜏=max{πœπ‘–}, 𝑖=1,…,𝑛. When π‘₯(𝑑) is eventually negative, the proof follows the same argument by using the interval [π‘Ž2,𝑏2] instead of [π‘Ž1,𝑏1]. Choose π‘Ž1,𝑏1β‰₯𝑑1 such that 𝑝𝑖(𝑑),π‘žπ‘–(𝑑)β‰₯0, 𝑒(𝑑)≀0 for π‘‘βˆˆ[π‘Ž1βˆ’πœπ‘–,𝑏1], and 𝑖=1,2,…,𝑛.
From (1.1), we have that π‘₯ξ…ž(𝑑)β‰₯0 for π‘‘βˆˆ[π‘Ž1βˆ’πœπ‘–,𝑏1]. If not, there exists 𝑑2∈[π‘Ž1βˆ’πœπ‘–,𝑏1] such that π‘₯ξ…ž(𝑑2)<0. Because 𝑝(𝑑)π‘₯ξ…žξ€Έ(𝑑)ξ…žβ‰€0,(2.7) we have 𝑝(𝑑)π‘₯ξ…ž(𝑑)≀𝑝(𝑑2)π‘₯ξ…ž(𝑑2). Integrating from 𝑑2 to 𝑑, we obtain 𝑑π‘₯(𝑑)≀π‘₯2𝑑+𝑝2ξ€Έπ‘₯ξ…žξ€·π‘‘2ξ€Έξ€œπœπ‘‘21𝑝(𝑠)𝑑𝑠.(2.8) Noting the assumption (1.2), we have π‘₯(𝑑)≀0 for sufficient large 𝑑. This is a contradiction with π‘₯(𝑑)>0. From (2.7) and the conditions 𝑝(𝑑)>0, π‘ξ…ž(𝑑)β‰₯0, we obtain π‘₯ξ…žξ…ž(𝑑)≀0 for π‘‘βˆˆ[π‘Ž1βˆ’πœπ‘–,𝑏1].
Employing the convexity of π‘₯(𝑑), we obtain π‘₯ξ€·π‘‘βˆ’πœπ‘–ξ€Έβ‰₯π‘₯(𝑑)π‘‘βˆ’π‘Ž1π‘‘βˆ’π‘Ž1+πœπ‘–ξ€Ίπ‘Ž,π‘‘βˆˆ1,𝑏1ξ€».(2.9)
Define π‘πœ”(𝑑)=βˆ’(𝑑)π‘₯ξ…ž(𝑑).π‘₯(𝑑)(2.10)
Recall the arithmetic-geometric mean inequality 𝑛𝑖=0πœ‚π‘–π‘’π‘–β‰₯𝑛𝑖=0π‘’πœ‚π‘–π‘–,𝑒𝑖β‰₯0,(2.11) where πœ‚0βˆ‘=1βˆ’π‘›π‘–=1πœ‚π‘– and πœ‚π‘–>0, 𝑖=1,2,…,𝑛, are chosen according to the given 𝛼1,𝛼2,…,𝛼𝑛 as in Lemma 2.1 satisfying (a) and (b). Let 𝑒0(𝑑)=πœ‚0βˆ’1||||𝑒(𝑑),𝑒𝑖(𝑑)=πœ‚π‘–βˆ’1π‘žπ‘–ξ€·π‘₯ξ€·(𝑑)π‘‘βˆ’πœπ‘–ξ€Έξ€Έπ›Όπ‘–.(2.12) We have πœ”ξ…ž(𝑑)=βˆ’(𝑝(𝑑)π‘₯(𝑑))ξ…ž+πœ”π‘₯(𝑑)2(𝑑)=βˆ‘π‘(𝑑)𝑛𝑖=1𝑝𝑖(𝑑)π‘₯π‘‘βˆ’πœπ‘–ξ€Έ+βˆ‘π‘›π‘–=1π‘žπ‘–ξ€·π‘₯ξ€·(𝑑)π‘‘βˆ’πœπ‘–ξ€Έξ€Έπ›Όπ‘–βˆ’π‘’(𝑑)+πœ”π‘₯(𝑑)2(𝑑)β‰₯𝑝(𝑑)𝑛𝑖=1𝑝𝑖(𝑑)π‘‘βˆ’π‘Ž1π‘‘βˆ’π‘Ž1+πœπ‘–ξ‚Ά+ξ€·πœ‚0βˆ’1||||𝑒(𝑑)πœ‚0βˆπ‘›π‘–=1ξ€·πœ‚π‘–βˆ’1π‘žπ‘–ξ€Έ(𝑑)πœ‚π‘–π‘₯π›Όπ‘–πœ‚π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έ+πœ”π‘₯(𝑑)2(𝑑)=𝑝(𝑑)𝑛𝑖=1𝑝𝑖(𝑑)π‘‘βˆ’π‘Ž1π‘‘βˆ’π‘Ž1+πœπ‘–ξ‚Ά+ξ€·πœ‚0βˆ’1||||𝑒(𝑑)πœ‚0βˆπ‘›π‘–=1ξ€·πœ‚π‘–βˆ’1π‘žπ‘–ξ€Έ(𝑑)πœ‚π‘–π‘₯π›Όπ‘–πœ‚π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έβˆπ‘›π‘–=1π‘₯π›Όπ‘–πœ‚π‘–+πœ”(𝑑)2(𝑑)β‰₯𝑝(𝑑)𝑛𝑖=1𝑝𝑖(𝑑)π‘‘βˆ’π‘Ž1π‘‘βˆ’π‘Ž1+πœπ‘–ξ‚Ά+ξ€·πœ‚0βˆ’1||||𝑒(𝑑)πœ‚0𝑛𝑖=1ξ€·πœ‚π‘–βˆ’1π‘žπ‘–ξ€Έ(𝑑)πœ‚π‘–ξ‚΅π‘‘βˆ’π‘Ž1π‘‘βˆ’π‘Ž1+πœπ‘–ξ‚Άπ›Όπ‘–πœ‚π‘–+πœ”2(𝑑)𝑝(𝑑)=𝑄1(πœ”π‘‘)+2(𝑑).𝑝(𝑑)(2.13)
Multiplying both sides of (2.13) by 𝐻1(𝑏1,𝑑)βˆˆπ’Ÿπ‘Ž1,𝑏1 and integrating by parts, we find that ξ€·π‘βˆ’πœ”1𝐻1𝑏1,𝑐1ξ€Έβ‰₯ξ€œπ‘1𝑐1𝑄1(𝑠)𝐻1𝑏1ξ€Έ,π‘ βˆ’π‘(𝑠)β„Ž212𝑏1,𝑠𝑑𝑠.(2.14) That is, ξ€·π‘βˆ’πœ”1ξ€Έβ‰₯1𝐻1𝑏1,𝑐1ξ€Έξ€œπ‘1𝑐1𝑄1(𝑠)𝐻1𝑏1ξ€Έ,π‘ βˆ’π‘(𝑠)β„Ž212𝑏1,𝑠𝑑𝑠.(2.15)
On the other hand, multiplying both sides of (2.13) by 𝐻1(𝑑,π‘Ž1)βˆˆπ’Ÿπ‘Ž1,𝑏1 and integrating by parts, we can easily obtain πœ”ξ€·π‘1ξ€Έβ‰₯1𝐻1𝑐1,π‘Ž1ξ€Έξ€œπ‘1π‘Ž1𝑄1(𝑠)𝐻1𝑠,π‘Ž1ξ€Έβˆ’π‘(𝑠)β„Ž211𝑠,π‘Ž1𝑑𝑠.(2.16)
Equations (2.15) and (2.16) yield 1𝐻1𝑐1,π‘Ž1ξ€Έξ€œπ‘1π‘Ž1𝑄1(𝑠)𝐻1𝑠,π‘Ž1ξ€Έβˆ’π‘(𝑠)β„Ž211𝑠,π‘Ž1+1𝑑𝑠𝐻1𝑏1,𝑐1ξ€Έξ€œπ‘1𝑐1𝑄1(𝑠)𝐻1𝑏1ξ€Έ,π‘ βˆ’π‘(𝑠)β„Ž212𝑏1,𝑠𝑑𝑠≀0,(2.17) which contradicts (2.5) for 𝑗=1. The proof of Theorem 2.2 is complete.

Remark 2.3. When 𝜏1=β‹―=πœπ‘›=0, Σ𝑛𝑖=1𝑝𝑖(𝑑)=π‘ž(𝑑), the conditions π‘ž(𝑑)β‰₯0 for π‘‘βˆˆ[π‘Ž1,𝑏1]βˆͺ[π‘Ž2,𝑏2], π‘ξ…ž(𝑑)β‰₯0 and (1.2) can be removed. Therefore, Theorem 2.2 reduces to Theorem 1 in [3].

Remark 2.4. When 𝜏1=β‹―=πœπ‘›=𝜏, Σ𝑛i=1𝑝𝑖(𝑑)=π‘ž(𝑑), Theorem 2.2 reduces to Theorem 1 in [4] for which the conditions π‘ž(𝑑)β‰₯0 for π‘‘βˆˆ[π‘Ž1βˆ’πœ,𝑏1]βˆͺ[π‘Ž2βˆ’πœ,𝑏2],π‘ξ…ž(𝑑)β‰₯0 and (1.2) are needed. There are some mistakes in the proof of Theorem 1 in [4].

The following theorem gives an oscillation criterion for the unforced (1.1).

Theorem 2.5. If, for any 𝑇β‰₯0, there exist π‘Ž, 𝑏, and 𝑐 such that π‘‡β‰€π‘Ž<𝑐<𝑏, 𝑝𝑖(𝑑)β‰₯0, and π‘žπ‘–(𝑑)β‰₯0 for π‘‘βˆˆ[π‘Žβˆ’πœπ‘–,𝑏],𝑖=1,2,…,𝑛, and there exists π»βˆˆπ’Ÿπ‘Ž,𝑏, such that 1ξ€œπ»(𝑐,π‘Ž)π‘π‘Žξ€·π»(𝑠,π‘Ž)𝑄(𝑠)βˆ’π‘(𝑠)β„Ž21ξ€Έ1(𝑠,π‘Ž)𝑑𝑠+ξ€œπ»(𝑏,𝑐)𝑏𝑐𝐻(𝑏,𝑠)𝑄(𝑠)βˆ’π‘(𝑠)β„Ž22ξ€Έ(𝑏,𝑠)𝑑𝑠>0,(2.18) where 𝑄(𝑑)=𝑛𝑖=1𝑝𝑖(𝑑)π‘‘βˆ’π‘Žπ‘‘βˆ’π‘Ž+πœπ‘–ξ‚Ά+𝑛𝑖=1ξ€·πœ‚π‘–βˆ’1π‘žπ‘–ξ€Έ(𝑑)πœ‚π‘–ξ‚΅π‘‘βˆ’π‘Žπ‘‘βˆ’π‘Ž+πœπ‘–ξ‚Άπ›Όπ‘–πœ‚π‘–,(2.19)πœ‚1,πœ‚2,…,πœ‚π‘› are positive constants satisfying (a) and (c) in Lemma 2.1, and β„Ž1, β„Ž2 are defined as in (2.3), then the unforced (1.1) is oscillatory.

Proof. Let π‘₯(𝑑) be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that π‘₯(𝑑)>0 for all 𝑑β‰₯𝑑1βˆ’πœβ‰₯0, where 𝑑1 depends on the solution π‘₯(𝑑) and 𝜏=max{πœπ‘–}, 𝑖=1,…,𝑛. Similar to the proof in Theorem 2.2, we can obtain π‘₯ξ€·π‘‘βˆ’πœπ‘–ξ€Έβ‰₯π‘₯(𝑑)π‘‘βˆ’π‘Žπ‘‘βˆ’π‘Ž+πœπ‘–[].,π‘‘βˆˆπ‘Ž,𝑏(2.20) Define π‘πœ”(𝑑)=βˆ’(𝑑)π‘₯ξ…ž(𝑑).π‘₯(𝑑)(2.21) Recall the arithmetic-geometric mean inequality 𝑛𝑖=1πœ‚π‘–π‘’π‘–β‰₯𝑛𝑖=1π‘’πœ‚π‘–π‘–,𝑒𝑖β‰₯0,(2.22) where πœ‚π‘–>0, 𝑖=1,2,…,𝑛, are chosen according to the given 𝛼1,𝛼2,…,𝛼𝑛 as in Lemma 2.1 satisfying (a) and (c). Let 𝑒𝑖=πœ‚π‘–βˆ’1π‘žπ‘–ξ€·π‘₯ξ€·(𝑑)π‘‘βˆ’πœπ‘–ξ€Έξ€Έπ›Όπ‘–.(2.23) We can obtain πœ”ξ…ž(𝑑)=βˆ’(𝑝(𝑑)π‘₯(𝑑))ξ…ž+πœ”π‘₯(𝑑)2(𝑑)=βˆ‘π‘(𝑑)𝑛𝑖=1𝑝𝑖(𝑑)π‘₯π‘‘βˆ’πœπ‘–ξ€Έ+βˆ‘π‘›π‘–=1π‘žπ‘–ξ€·π‘₯ξ€·(𝑑)π‘‘βˆ’πœπ‘–ξ€Έξ€Έπ›Όπ‘–+πœ”π‘₯(𝑑)2(𝑑)β‰₯𝑝(𝑑)𝑛𝑖=1𝑝𝑖(𝑑)π‘‘βˆ’π‘Žπ‘‘βˆ’π‘Ž+πœπ‘–ξ‚Ά+βˆπ‘›π‘–=1ξ€·πœ‚π‘–βˆ’1π‘žπ‘–ξ€Έ(𝑑)πœ‚π‘–π‘₯π›Όπ‘–πœ‚π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έ+πœ”π‘₯(𝑑)2(𝑑)=𝑝(𝑑)𝑛𝑖=1𝑝𝑖(𝑑)π‘‘βˆ’π‘Žπ‘‘βˆ’π‘Ž+πœπ‘–ξ‚Ά+βˆπ‘›π‘–=1ξ€·πœ‚π‘–βˆ’1π‘žπ‘–ξ€Έ(𝑑)πœ‚π‘–π‘₯π›Όπ‘–πœ‚π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έβˆπ‘›π‘–=1π‘₯π›Όπ‘–πœ‚π‘–+πœ”(𝑑)2(𝑑)β‰₯𝑝(𝑑)𝑛𝑖=1𝑝𝑖(t)π‘‘βˆ’π‘Žπ‘‘βˆ’π‘Ž+πœπ‘–ξ‚Ά+𝑛𝑖=1ξ€·πœ‚π‘–βˆ’1π‘žπ‘–ξ€Έ(𝑑)πœ‚π‘–ξ‚΅π‘‘βˆ’π‘Žπ‘‘βˆ’π‘Ž+πœπ‘–ξ‚Άπ›Όπ‘–πœ‚π‘–+πœ”2(𝑑)πœ”π‘(𝑑)=𝑄(𝑑)+2(𝑑).𝑝(𝑑)(2.24)
Multiplying both sides of (2.24) by 𝐻(𝑏,𝑑)βˆˆπ’Ÿπ‘Ž,𝑏 and integrating by parts, we obtain ξ€œπ‘π‘π»(𝑏,𝑑)πœ”ξ…žξ€œ(𝑑)𝑑𝑑β‰₯π‘π‘ξ€œπ»(𝑏,𝑑)𝑄(𝑑)𝑑𝑑+π‘π‘πœ”π»(𝑏,𝑑)2(𝑑)ξ€œπ‘(𝑑)𝑑𝑑,βˆ’π»(𝑏,𝑐)πœ”(𝑐)β‰₯π‘π‘ξ€œπ»(𝑏,𝑑)𝑄(𝑑)𝑑𝑑+π‘π‘ξ‚΅πœ”π»(𝑏,𝑑)2(𝑑)𝑝(𝑑)βˆ’2πœ”(𝑑)β„Ž2√(𝑏,𝑑)ξ‚Ά=ξ€œπ»(𝑏,𝑑)𝑑𝑑𝑏𝑐𝐻(𝑏,𝑑)𝑄(𝑑)βˆ’π‘(𝑑)β„Ž22ξ€Έξ€œ(𝑏,𝑑)𝑑𝑑+π‘π‘ξƒ©ξƒŽπ»(𝑏,𝑑)βˆšπ‘(𝑑)πœ”(𝑑)βˆ’π‘(𝑑)β„Ž2ξƒͺ(𝑏,𝑑)2β‰₯ξ€œπ‘‘π‘‘π‘π‘ξ€·π»(𝑏,𝑑)𝑄(𝑑)βˆ’π‘(𝑑)β„Ž22ξ€Έ(𝑏,𝑑)𝑑𝑑.(2.25) It follows that 1βˆ’πœ”(𝑐)β‰₯ξ€œπ»(𝑏,𝑐)𝑏𝑐𝐻(𝑏,𝑑)𝑄(𝑑)βˆ’π‘(𝑑)β„Ž22ξ€Έ(𝑏,𝑑)𝑑𝑑.(2.26) On the other hand, multiplying both sides of (2.24) by 𝐻(𝑑,π‘Ž)βˆˆπ’Ÿπ‘Ž,𝑏 and integrating by parts, we have 1πœ”(𝑐)β‰₯ξ€œπ»(𝑐,π‘Ž)π‘π‘Žξ€·π»(𝑑,π‘Ž)𝑄(𝑑)βˆ’π‘(𝑑)β„Ž21ξ€Έ(𝑑,π‘Ž)𝑑𝑑.(2.27) Equations (2.26) and (2.27) yield 1ξ€œπ»(𝑐,π‘Ž)π‘π‘Žξ€·π»(𝑑,π‘Ž)𝑄(𝑑)βˆ’π‘(𝑑)β„Ž21ξ€Έ1(𝑑,π‘Ž)𝑑𝑑+ξ€œπ»(𝑏,𝑐)𝑏𝑐𝐻(𝑏,𝑑)𝑄(𝑑)βˆ’π‘(𝑑)β„Ž22ξ€Έ(𝑏,𝑑)𝑑𝑑<0,(2.28) which contradicts (2.24). The proof of Theorem 2.5 is complete.

Remark 2.6. When 𝜏1=β‹―=πœπ‘›=0, Σ𝑛𝑖=1𝑝𝑖(𝑑)=π‘ž(𝑑), the conditions π‘ž(𝑑)β‰₯0 for π‘‘βˆˆ[π‘Ž,𝑏], π‘ξ…ž(𝑑)β‰₯0 and (1.2) can be removed. Therefore, Theorem 2.5 reduces to Theorem 2 in [3].

Remark 2.7. When 𝜏1=β‹―=πœπ‘›=𝜏, Σ𝑛𝑖=1𝑝𝑖(𝑑)=π‘ž(𝑑), Theorem 2.5 reduces to Theorem 2 in [4] for which the conditions π‘ž(𝑑)β‰₯0 for π‘‘βˆˆ[π‘Žβˆ’πœ,𝑏], π‘ξ…ž(𝑑)β‰₯0 and (1.2) are needed.

3. Example

In this section, we provide an example to illustrate our results.

Consider the following equation: π‘₯ξ…žξ…ž(|||π‘₯ξ‚€πœ‹π‘‘)+π‘˜sinπ‘‘π‘‘βˆ’8|||𝛼1ξ‚€πœ‹sgnπ‘₯π‘‘βˆ’8|||π‘₯ξ‚€πœ‹+𝑙cosπ‘‘π‘‘βˆ’4|||𝛼2ξ‚€πœ‹sgnπ‘₯π‘‘βˆ’4=βˆ’π‘šcos2𝑑,𝑑β‰₯0,(3.1) where π‘˜, 𝑙, π‘š are positive constants, 𝛼1>1, and 0<𝛼2<1. Here 𝑝(𝑑)=1,𝑝1(𝑑)=𝑝2(𝑑)=0,π‘ž1(𝑑)=π‘˜sin𝑑,π‘ž2𝜏(𝑑)=𝑙cos𝑑,1=πœ‹8,𝜏2=πœ‹4,𝑒(𝑑)=βˆ’π‘šcos2𝑑.(3.2)

According to the direct computation, we have 𝑄𝑗(𝑑)=π‘˜0||||cos2π‘‘πœ‚0(sin𝑑)πœ‚1(cos𝑑)πœ‚2ξ‚΅π‘‘βˆ’π‘Žπ‘—π‘‘βˆ’π‘Žπ‘—+𝜏1𝛼1πœ‚1ξ‚΅π‘‘βˆ’π‘Žπ‘—π‘‘βˆ’π‘Žπ‘—+𝜏2𝛼2πœ‚2,𝑗=1,2,(3.3) where π‘˜0=(πœ‚0βˆ’1/π‘š)πœ‚0(πœ‚1βˆ’1/π‘˜)πœ‚1(πœ‚2βˆ’1/𝑙)πœ‚2,πœ‚0 can be any positive number satisfying 0<πœ‚0<(𝛼1βˆ’1)/𝛼1, and πœ‚1, πœ‚2 satisfy (2.1). For any 𝑇β‰₯0, we can choose π‘Ž1=2π‘–πœ‹,π‘Ž2=𝑏1πœ‹=2π‘–πœ‹+4,𝑏2πœ‹=2π‘–πœ‹+2,𝑐1πœ‹=2π‘–πœ‹+8,𝑐2=2π‘–πœ‹+3πœ‹8,(3.4) for 𝑖=0,1,…, and 𝐻1(𝑑,𝑠)=𝐻2(𝑑,𝑠)=(π‘‘βˆ’π‘ )2. By simple computation, we obtain β„Žπ‘—1(𝑑,𝑠)=β„Žπ‘—2(𝑑,𝑠)=1, 𝑗=1,2. From Theorem 2.2, we have that (3.1) is oscillatory if ξ€œ2π‘–πœ‹+πœ‹/82π‘–πœ‹π‘„1(𝑠)(π‘ βˆ’2π‘–πœ‹)2ξ€œπ‘‘π‘ +2π‘–πœ‹+πœ‹/42π‘–πœ‹+πœ‹/8𝑄1ξ‚€πœ‹(𝑠)2π‘–πœ‹+4ξ‚βˆ’π‘ 2πœ‹π‘‘π‘ >4,ξ€œ2π‘–πœ‹+3πœ‹/82π‘–πœ‹+πœ‹/4𝑄2ξ‚€πœ‹(𝑠)π‘ βˆ’2π‘–πœ‹βˆ’42ξ€œπ‘‘π‘ +2π‘–πœ‹+πœ‹/22π‘–πœ‹+3πœ‹/8𝑄2ξ‚€πœ‹(𝑠)2π‘–πœ‹+2ξ‚βˆ’π‘ 2πœ‹π‘‘π‘ >4.(3.5) If 𝐻1(𝑑,𝑠)=𝐻2(𝑑,𝑠)=sin2(π‘‘βˆ’π‘ ), by simple computation, we obtain β„Žπ‘—1(𝑑,𝑠)=β„Žπ‘—2(𝑑,𝑠)=cos(π‘‘βˆ’π‘ ) for 𝑗=1,2. From Theorem 2.2, we have that (3.1) is oscillatory if ξ€œ2π‘–πœ‹+πœ‹/82π‘–πœ‹π‘„1(𝑠)(π‘ βˆ’2π‘–πœ‹)2ξ€œπ‘‘π‘ +2π‘–πœ‹+πœ‹/42π‘–πœ‹+πœ‹/8𝑄1ξ‚€πœ‹(𝑠)2π‘–πœ‹+4ξ‚βˆ’π‘ 2πœ‹π‘‘π‘ >+√1628,ξ€œ2π‘–πœ‹+3πœ‹/82π‘–πœ‹+πœ‹/4𝑄2ξ‚€πœ‹(𝑠)π‘ βˆ’2π‘–πœ‹βˆ’42ξ€œπ‘‘π‘ +2π‘–πœ‹+πœ‹/22π‘–πœ‹+3πœ‹/8𝑄2ξ‚€πœ‹(𝑠)2π‘–πœ‹+2ξ‚βˆ’π‘ 2πœ‹π‘‘π‘ >+√1628.(3.6)

Acknowledgments

The authors would like to thank the referees for their valuable comments which have led to an improvement of the presentation of this paper. This project is supported by the National Natural Science Foundation of China (10771118) and STPF of University in Shandong Province of China (J09LA04).

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