About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 857860, 15 pages
http://dx.doi.org/10.1155/2011/857860
Research Article

Properties of Third-Order Nonlinear Functional Differential Equations with Mixed Arguments

Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of KoΕ‘ice, LetnΓ‘ 9, 042 00 KoΕ‘ice, Slovakia

Received 14 December 2010; Accepted 20 January 2011

Academic Editor: JosefΒ DiblΓ­k

Copyright Β© 2011 B. BaculΓ­kovΓ‘. 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

The aim of this paper is to offer sufficient conditions for property (B) and/or the oscillation of the third-order nonlinear functional differential equation with mixed arguments [π‘Ž(𝑑)[π‘₯ξ…žξ…ž(𝑑)]𝛾]ξ…ž=π‘ž(𝑑)𝑓(π‘₯[𝜏(𝑑)])+𝑝(𝑑)β„Ž(π‘₯[𝜎(𝑑)]). Both cases βˆ«βˆžπ‘Žβˆ’1/𝛾(𝑠)d𝑠=∞ and βˆ«βˆžπ‘Žβˆ’1/𝛾(𝑠)d𝑠<∞ are considered. We deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones.

1. Introduction

We are concerned with the oscillatory and certain asymptotic behavior of all solutions of the third-order functional differential equationsξ€Ίπ‘Žξ€Ίπ‘₯(𝑑)ξ…žξ…žξ€»(𝑑)π›Ύξ€»ξ…ž[𝜏][𝜎]=π‘ž(𝑑)𝑓(π‘₯(𝑑))+𝑝(𝑑)β„Ž(π‘₯(𝑑)).(𝐸) Throughout the paper, it is assumed that π‘Ž,π‘ž,π‘βˆˆπΆ([𝑑0,∞)), 𝜏,𝜎∈𝐢1([𝑑0,∞)), 𝑓,β„ŽβˆˆπΆ((βˆ’βˆž,∞)), and (H1)𝛾 is the ratio of two positive odd integers, (H2)π‘Ž(𝑑), π‘ž(𝑑), 𝑝(𝑑) are positive, (H3)𝜏(𝑑)≀𝑑, 𝜎(𝑑)β‰₯𝑑, πœβ€²(𝑑)>0, πœŽβ€²(𝑑)>0, limπ‘‘β†’βˆžπœ(𝑑)=∞, (H4)𝑓1/𝛾(π‘₯)/π‘₯β‰₯1, π‘₯β„Ž(π‘₯)>0, 𝑓′(π‘₯)β‰₯0, and β„Žβ€²(π‘₯)β‰₯0 for π‘₯β‰ 0, (H5)βˆ’π‘“(βˆ’π‘₯𝑦)β‰₯𝑓(π‘₯𝑦)β‰₯𝑓(π‘₯)𝑓(𝑦) for π‘₯𝑦>0 and βˆ’β„Ž(βˆ’π‘₯𝑦)β‰₯β„Ž(π‘₯𝑦)β‰₯β„Ž(π‘₯)β„Ž(𝑦) for π‘₯𝑦>0.

By a solution of (𝐸), we mean a function π‘₯(𝑑)∈𝐢2([𝑇π‘₯,∞)),𝑇π‘₯β‰₯𝑑0, which has the property π‘Ž(𝑑)(π‘₯ξ…žξ…ž(𝑑))π›ΎβˆˆπΆ1([𝑇π‘₯,∞)) and satisfies (𝐸) on [𝑇π‘₯,∞). We consider only those solutions π‘₯(𝑑) of (𝐸) which satisfy sup{|π‘₯(𝑑)|βˆΆπ‘‘β‰₯𝑇}>0 for all 𝑇β‰₯𝑇π‘₯. We assume that (𝐸) possesses such a solution. A solution of (𝐸) is called oscillatory if it has arbitrarily large zeros on [𝑇π‘₯,∞), and, otherwise, it is nonoscillatory. Equation (𝐸) is said to be oscillatory if all its solutions are oscillatory.

Recently, (𝐸) and its particular cases (see [1–17]) have been intensively studied. The effort has been oriented to provide sufficient conditions for every (𝐸) to satisfylimπ‘‘β†’βˆž||||π‘₯(𝑑)=∞(1.1) or to eliminate all nonoscillatory solutions. Following [6, 8, 13, 15], we say that (𝐸) has property (B) if each of its nonoscillatory solutions satisfies (1.1).

We will discuss both casesξ€œβˆžπ‘‘0π‘Žβˆ’1/π›Ύξ€œ(𝑠)d𝑠<∞,(1.2)βˆžπ‘‘0π‘Žβˆ’1/𝛾(𝑠)d𝑠=∞.(1.3)

We will establish suitable comparison theorems that enable us to study properties of (𝐸) regardless of the fact that (1.3) or (1.2) holds. We will compare (𝐸) with the first-order advanced/delay equations, in the sense that the oscillation of these first-order equations yields property (B) or the oscillation of (𝐸).

In the paper, we are motivated by an interesting result of Grace et al. [10], where the oscillation criteria for (𝐸) are discussed. This result has been complemented by BaculΓ­kovΓ‘ et al. [5]. When studying properties of (𝐸), the authors usually reduce (𝐸) onto the corresponding differential inequalities ξ€Ίπ‘Žξ€Ίπ‘₯(𝑑)ξ…žξ…žξ€»(𝑑)π›Ύξ€»ξ…ž[𝜏]ξ€Ίξ€Ίπ‘₯β‰₯π‘ž(𝑑)𝑓(π‘₯(𝑑)),π‘Ž(𝑑)ξ…žξ…ž(𝑑)π›Ύξ€»ξ…ž[]),β‰₯𝑝(𝑑)β„Ž(π‘₯𝜎(𝑑)(𝐸𝜎) and further study only properties of these inequalities. Therefore, the criteria obtained withhold information either from delay argument 𝜏(𝑑) and the corresponding functions π‘ž(𝑑) and 𝑓(𝑒) or from advanced argument 𝜎(𝑑) and the corresponding functions 𝑝(𝑑) and β„Ž(𝑒). In the paper, we offer a technique for obtaining new criteria for property (B) and the oscillation of (𝐸) that involve both arguments 𝜏(𝑑) and 𝜎(𝑑). Consequently, our results are new even for the linear case of (𝐸) and properly complement and extend earlier ones presented in [1–17].

Remark 1.1. All functional inequalities considered in this paper are assumed to hold eventually; that is, they are satisfied for all 𝑑 large enough.

2. Main Results

The following results are elementary but useful in what comes next.

Lemma 2.1. Assume that 𝐴β‰₯0, 𝐡β‰₯0, 𝛼β‰₯1. Then, (𝐴+𝐡)𝛼β‰₯𝐴𝛼+𝐡𝛼.(2.1)

Proof. If 𝐴=0 or 𝐡=0, then (2.1) holds. For 𝐴≠0, setting π‘₯=𝐡/𝐴, condition (2.1) takes the form (1+π‘₯)𝛼β‰₯1+π‘₯𝛼, which is for π‘₯>0 evidently true.

Lemma 2.2. Assume that 𝐴β‰₯0, 𝐡β‰₯0, 0<𝛼≀1. Then, (𝐴+𝐡)𝛼β‰₯𝐴𝛼+𝐡𝛼21βˆ’π›Ό.(2.2)

Proof. We may assume that 0<𝐴<𝐡. Consider a function 𝑔(𝑒)=𝑒𝛼. Since π‘”ξ…žξ…ž(𝑒)<0 for 𝑒>0, function 𝑔(𝑒) is concave down; that is, 𝑔𝐴+𝐡2β‰₯𝑔(𝐴)+𝑔(𝐡)2(2.3) which implies (2.2).

The following result presents a useful relationship between an existence of positive solutions of the advanced differential inequality and the corresponding advanced differential equation.

Lemma 2.3. Suppose that 𝑝(𝑑), 𝜎(𝑑), and β„Ž(𝑒) satisfy (H2), (H3), and (H4), respectively. If the first-order advanced differential inequality π‘§ξ…ž(𝑑)βˆ’π‘(𝑑)β„Ž(𝑧(𝜎(𝑑)))β‰₯0(2.4) has an eventually positive solution, so does the advanced differential equation π‘§ξ…ž(𝑑)βˆ’π‘(𝑑)β„Ž(𝑧(𝜎(𝑑)))=0.(2.5)

Proof. Let 𝑧(𝑑) be a positive solution of (2.4) on [𝑑1,∞). Then, 𝑧(𝑑) satisfies the inequality 𝑑𝑧(𝑑)β‰₯𝑧1ξ€Έ+ξ€œπ‘‘π‘‘1𝑝(𝑠)β„Ž(𝑧(𝜎(𝑠)))d𝑠.(2.6) Let 𝑦1𝑦(𝑑)=𝑧(𝑑),𝑛(𝑑𝑑)=𝑧1ξ€Έ+ξ€œπ‘‘π‘‘1𝑦𝑝(𝑠)β„Žπ‘›βˆ’1(ξ€ΈπœŽ(𝑠))d𝑠,𝑛=2,3….(2.7) It follows from the definition of 𝑦𝑛(𝑑) and (𝐻4) that the sequence {𝑦𝑛} has the property 𝑧(𝑑)=𝑦1(𝑑)β‰₯𝑦2𝑑(𝑑)β‰₯β‹―β‰₯𝑧1ξ€Έ,𝑑β‰₯𝑑1.(2.8) Hence, {𝑦𝑛} converges pointwise to a function 𝑦(𝑑), where 𝑧(𝑑)β‰₯𝑦(𝑑)β‰₯𝑧(𝑑1). Let β„Žπ‘›(𝑑)=𝑝(𝑑)β„Ž(𝑦𝑛(𝜎(𝑑))), 𝑛=1,2,…, then β„Ž1(𝑑)β‰₯β„Ž2(𝑑)β‰₯β‹―β‰₯0. Since β„Ž1(𝑑) is integrable on [𝑑1,𝑑] and limπ‘›β†’βˆžβ„Žπ‘›(𝑑)=𝑝(𝑑)β„Ž(𝑦(𝜎(𝑑))), it follows by Lebesgue's dominated convergence theorem that 𝑑𝑦(𝑑)=𝑧1ξ€Έ+ξ€œπ‘‘π‘‘1𝑝(𝑠)β„Ž(𝑦(𝜎(𝑠)))d𝑠.(2.9) Thus, 𝑦(𝑑) satisfies (2.5).

We start our main results with the classification of the possible nonoscillatory solutions of (𝐸).

Lemma 2.4. Let π‘₯(𝑑) be a nonoscillatory solution of (𝐸). Then, π‘₯(𝑑) satisfies, eventually, one of the following conditions (I)π‘₯(𝑑)π‘₯ξ…ž(𝑑)>0,π‘₯(𝑑)π‘₯ξ…žξ…žξ€Ίπ‘Žξ€Ίπ‘₯(𝑑)>0,π‘₯(𝑑)(𝑑)ξ…žξ…žξ€»(𝑑)π›Ύξ€»ξ…ž>0,(2.10)(II)π‘₯(𝑑)π‘₯ξ…ž(𝑑)>0,π‘₯(𝑑)π‘₯ξ…žξ…žξ€Ίπ‘Žξ€Ίπ‘₯(𝑑)<0,π‘₯(𝑑)(𝑑)ξ…žξ…žξ€»(𝑑)π›Ύξ€»ξ…ž>0,(2.11)and if (1.2) holds, then also (III)π‘₯(𝑑)π‘₯ξ…ž(𝑑)<0,π‘₯(𝑑)π‘₯ξ…žξ…žξ€Ίπ‘Žξ€Ίπ‘₯(𝑑)>0,π‘₯(𝑑)(𝑑)ξ…žξ…žξ€»(𝑑)π›Ύξ€»ξ…ž>0.(2.12)

Proof. Let π‘₯(𝑑) be a nonoscillatory solution of (𝐸), say π‘₯(𝑑)>0 for 𝑑β‰₯𝑑0. It follows from (𝐸) that [π‘Ž(𝑑)[π‘₯ξ…žξ…ž(𝑑)]𝛾]>0, eventually. Thus, π‘₯ξ…žξ…ž(𝑑) and π‘₯β€²(𝑑) are of fixed sign for 𝑑β‰₯𝑑1, 𝑑1 large enough. At first, we assume that π‘₯ξ…žξ…ž(𝑑)<0. Then, either π‘₯ξ…ž(𝑑)>0 or π‘₯ξ…ž(𝑑)<0, eventually. But π‘₯ξ…žξ…ž(𝑑)<0 together with π‘₯β€²(𝑑)<0 imply that π‘₯(𝑑)<0. A contradiction, that is, Case (II) holds.
Now, we suppose that π‘₯ξ…žξ…ž(𝑑)>0, then either Case (I) or Case (III) holds. On the other hand, if (1.3) holds, then Case (III) implies that π‘Ž(𝑑)[π‘₯ξ…žξ…ž(𝑑)]𝛾β‰₯𝑐>0, 𝑑β‰₯𝑑1. Integrating from 𝑑1 to 𝑑, we have π‘₯ξ…ž(𝑑)βˆ’π‘₯ξ…žξ€·π‘‘1ξ€Έβ‰₯𝑐1/π›Ύξ€œπ‘‘π‘‘1π‘Žβˆ’1/𝛾(𝑠)d𝑠,(2.13) which implies that π‘₯β€²(𝑑)β†’βˆž as π‘‘β†’βˆž, and we deduce that Case (III) may occur only if (1.2) is satisfied. The proof is complete.

Remark 2.5. It follows from Lemma 2.4 that if (1.3) holds, then only Cases (I) and (II) may occur.

In the following results, we provide criteria for the elimination of Cases (I)–(III) of Lemma 2.4 to obtain property (B)/oscillation of (𝐸).

Let us denote for our further references thatξ€œπ‘ƒ(𝑑)=βˆžπ‘‘π‘Žβˆ’1/π›Ύξ‚΅ξ€œ(𝑒)βˆžπ‘’ξ‚Άπ‘(𝑠)d𝑠1/π›Ύξ€œd𝑒,(2.14)𝑄(𝑑)=βˆžπ‘‘π‘Žβˆ’1/π›Ύξƒ©ξ€œ(𝑒)βˆžπ‘’π‘žξ€·πœβˆ’1ξ€Έ(𝑠)πœξ…žξ€·πœβˆ’1ξ€Έξƒͺ(𝑠)d𝑠1/𝛾d𝑒.(2.15)

Theorem 2.6. Let 0<𝛾≀1. Assume that π‘₯(𝑑) is a nonoscillatory solution of (𝐸). If the first-order advanced differential equation π‘§ξ…ž(𝑑)βˆ’π‘ƒ(𝑑)eβˆ’βˆ«π‘‘π‘‘1𝑄(𝑠)π‘‘π‘ β„Ž1/𝛾eβˆ«π‘‘1𝜎(𝑑)𝑄(𝑠)π‘‘π‘ ξ‚β„Ž1/𝛾[](π‘§πœŽ(𝑑))=0(𝐸1) is oscillatory, then Case (II) cannot hold.

Proof. Let π‘₯(𝑑) be a nonoscillatory solution of (𝐸), satisfying Case (II) of Lemma 2.4. We may assume that π‘₯(𝑑)>0 for 𝑑β‰₯𝑑0. Integrating (𝐸) from 𝑑 to ∞, one gets ξ€Ίπ‘₯βˆ’π‘Ž(𝑑)ξ…žξ…ž(𝑑)𝛾β‰₯ξ€œβˆžπ‘‘[]ξ€œπ‘ž(𝑠)𝑓(π‘₯𝜏(𝑠))d𝑠+βˆžπ‘‘[]𝑝(𝑠)β„Ž(π‘₯𝜎(𝑠))d𝑠.(2.16) On the other hand, the substitution 𝜏(𝑠)=𝑒 gives ξ€œβˆžπ‘‘π‘ž[𝜏]ξ€œ(𝑠)𝑓(π‘₯(𝑠))d𝑠=∞𝜏(𝑑)π‘žξ€·πœβˆ’1ξ€Έ(𝑒)πœξ…žξ€·πœβˆ’1𝑓β‰₯ξ€œ(𝑒)(π‘₯(𝑒))dπ‘’βˆžπ‘‘π‘žξ€·πœβˆ’1(𝑠)πœξ…žξ€·πœβˆ’1ξ€Έ(𝑠)𝑓(π‘₯(𝑠))d𝑠.(2.17) Using (2.17) in (2.16), we find βˆ’π‘₯ξ…žξ…ž(𝑑)β‰₯π‘Žβˆ’1/π›Ύξƒ©ξ€œ(𝑑)βˆžπ‘‘π‘žξ€·πœβˆ’1ξ€Έ(𝑠)πœξ…žξ€·πœβˆ’1(ξ€Έξ€œπ‘ )𝑓(π‘₯(𝑠))d𝑠+βˆžπ‘‘[]ξƒͺ𝑝(𝑠)β„Ž(π‘₯𝜎(𝑠))d𝑠1/𝛾.(2.18) Taking into account the monotonicity of π‘₯(𝑑), it follows from Lemma 2.1 that βˆ’π‘₯ξ…žξ…žπ‘“(𝑑)β‰₯1/𝛾(π‘₯(𝑑))π‘Ž1/𝛾(ξƒ©ξ€œπ‘‘)βˆžπ‘‘π‘žξ€·πœβˆ’1ξ€Έ(𝑠)πœξ…žξ€·πœβˆ’1(ξ€Έξƒͺ𝑠)d𝑠1/𝛾+β„Ž1/𝛾[𝜎])(π‘₯(𝑑)π‘Ž1/π›Ύξ‚΅ξ€œ(𝑑)βˆžπ‘‘ξ‚Άπ‘(𝑠)d𝑠1/𝛾,(2.19) where we have used (𝐻3) and (𝐻4). An integration from 𝑑 to ∞ yields π‘₯ξ…žξ€œ(𝑑)β‰₯βˆžπ‘‘π‘“1/𝛾(π‘₯(𝑒))π‘Ž1/𝛾(ξƒ©ξ€œπ‘’)βˆžπ‘’π‘žξ€·πœβˆ’1ξ€Έ(𝑠)πœξ…žξ€·πœβˆ’1(ξ€Έξƒͺ𝑠)d𝑠1/𝛾+ξ€œdπ‘’βˆžπ‘‘β„Ž1/𝛾[𝜎])(π‘₯(𝑒)π‘Ž1/π›Ύξ‚΅ξ€œ(𝑒)βˆžπ‘’ξ‚Άπ‘(𝑠)d𝑠1/𝛾d𝑒β‰₯𝑓1/𝛾(π‘₯(𝑑))𝑄(𝑑)+β„Ž1/𝛾(π‘₯[])𝜎(𝑑)𝑃(𝑑).(2.20) Regarding (𝐻4), it follows that π‘₯(𝑑) is a positive solution of the differential inequality π‘₯ξ…ž(𝑑)βˆ’π‘„(𝑑)π‘₯(𝑑)β‰₯𝑃(𝑑)β„Ž1/𝛾[](π‘₯𝜎(𝑑)).(2.21) Applying the transformation π‘₯(𝑑)=𝑀(𝑑)π‘’βˆ«π‘‘π‘‘1𝑄(𝑠)d𝑠,(2.22) we can easily verify that 𝑀(𝑑) is a positive solution of the advanced differential inequality π‘€ξ…ž(𝑑)βˆ’π‘ƒ(𝑑)π‘’βˆ’βˆ«π‘‘π‘‘1𝑄(𝑠)dπ‘ β„Ž1/π›Ύξ‚€π‘’βˆ«π‘‘1𝜎(𝑑)𝑄(𝑠)dπ‘ ξ‚β„Ž1/𝛾[](π‘€πœŽ(𝑑))β‰₯0.(2.23) By Lemma 2.3, we conclude that the corresponding differential equation (𝐸1) has also a positive solution. A contradiction. Therefore, π‘₯(𝑑) cannot satisfy Case (II).

Remark 2.7. It follows from the proof of Theorem 2.8 that if at least one of the following conditions is satisfied: ξ€œβˆžπ‘‘0ξ€œπ‘(𝑠)d𝑠=∞,βˆžπ‘‘0π‘žξ€·πœβˆ’1ξ€Έ(𝑠)πœξ…žξ€·πœβˆ’1ξ€Έξ€œ(𝑠)d𝑠=∞,βˆžπ‘‘0π‘Žβˆ’1/π›Ύξ‚΅ξ€œ(𝑒)βˆžπ‘’ξ‚Άπ‘(𝑠)d𝑠1/π›Ύξ€œd𝑒=∞,βˆžπ‘‘0π‘Žβˆ’1/π›Ύξƒ©ξ€œ(𝑒)βˆžπ‘’π‘žξ€·πœβˆ’1ξ€Έ(𝑠)πœξ…žξ€·πœβˆ’1ξ€Έξƒͺ(𝑠)d𝑠1/𝛾d𝑒=∞,(2.24) then any nonoscillatory solution π‘₯(𝑑) of (𝐸) cannot satisfy Case (II). Therefore, we may assume that the corresponding integrals in (2.14)-(2.15) are convergent.

Now, we are prepared to provide new criteria for property (B) of (𝐸) and also the rate of divergence of all nonoscillatory solutions.

Theorem 2.8. Let (1.3) hold and 0<𝛾≀1. Assume that (𝐸1) is oscillatory. Then, (𝐸) has property (B) and, what is more, the following rate of divergence for each of its nonoscillatory solutions holds: ||||ξ€œπ‘₯(𝑑)β‰₯𝑐𝑑𝑑1π‘Žβˆ’1/𝛾(𝑠)(π‘‘βˆ’π‘ )𝑑𝑠,𝑐>0.(2.25)

Proof. Let π‘₯(𝑑) be a positive solution of (𝐸). It follows from Lemma 2.4 and Remark 2.5 that π‘₯(𝑑) satisfies either Case (I) or (II). But Theorem 2.6 implies that the Case (II) cannot hold. Therefore, π‘₯(𝑑) satisfies Case (I), which implies (1.1); that is, (𝐸) has property (B). On the other hand, there is a constant 𝑐>0 such that ξ€·π‘₯π‘Ž(𝑑)ξ…žξ…žξ€Έ(𝑑)𝛾β‰₯𝑐𝛾.(2.26) Integrating twice from 𝑑1 to 𝑑, we have ξ€œπ‘₯(𝑑)β‰₯𝑐𝑑𝑑1ξ‚΅ξ€œπ‘’π‘‘1π‘Žβˆ’1/𝛾(ξ‚Άξ€œπ‘ )d𝑠d𝑒=𝑐𝑑𝑑1π‘Žβˆ’1/𝛾(𝑠)(π‘‘βˆ’π‘ )d𝑠,(2.27) which is the desired estimate.

Employing an additional condition on the function β„Ž(π‘₯), we get easily verifiable criterion for property (B) of (𝐸).

Corollary 2.9. Let 0<𝛾≀1 and (1.3) hold. Assume that β„Ž1/𝛾(π‘₯)/π‘₯β‰₯1,|π‘₯|β‰₯1,(2.28)liminfπ‘‘β†’βˆžξ€œπ‘‘πœŽ(𝑑)𝑃(𝑒)eβˆ«π‘’πœŽ(𝑒)𝑄(𝑠)𝑑𝑠1𝑑𝑒>e.(2.29) Then, (𝐸) has property (B).

Proof. First note that (2.29) implies ξ€œβˆžπ‘‘0𝑃(𝑒)π‘’βˆ«π‘’πœŽ(𝑒)𝑄(𝑠)d𝑠d𝑒=∞.(2.30) By Theorem 2.8, it is sufficient to show that (𝐸1) is oscillatory. Assume the converse, let (𝐸1) have an eventually positive solution 𝑧(𝑑). Then, 𝑧′(𝑑)>0 and so 𝑧(𝜎(𝑑))>𝑐>0. Integrating (𝐸1) from 𝑑1 to 𝑑, we have in view of (2.28) ξ€œπ‘§(𝑑)β‰₯𝑑𝑑1𝑃(𝑒)π‘’βˆ’βˆ«π‘’π‘‘1𝑄(𝑠)dπ‘ β„Ž1/π›Ύξ‚€π‘’βˆ«π‘‘1𝜎(𝑒)𝑄(𝑠)dπ‘ ξ‚β„Ž1/𝛾(𝑧[])𝜎(𝑒)d𝑒β‰₯β„Ž1/π›Ύξ€œ(𝑐)𝑑𝑑1𝑃(𝑒)π‘’βˆ«π‘’πœŽ(𝑒)𝑄(𝑠)d𝑠d𝑒.(2.31) Using (2.30) in the previous inequalities, we get 𝑧(𝑑)β†’βˆž as π‘‘β†’βˆž. Therefore, 𝑧(𝑑)β‰₯1, eventually. Now, using (2.28) in (𝐸1), one can verify that 𝑧(𝑑) is a positive solution of the differential inequality π‘§ξ…ž(𝑑)βˆ’π‘ƒ(𝑑)π‘’βˆ«π‘‘πœŽ(𝑑)𝑄(𝑠)d𝑠𝑧(𝜎(𝑑))β‰₯0.(2.32) But, by [14, Theorem  2.4.1], condition (2.29) ensures that (2.32) has no positive solutions. This is a contradiction, and we conclude that (𝐸) has property (B).

Example 2.10. Consider the third-order nonlinear differential equation with mixed arguments 𝑑1/3ξ€·π‘₯ξ…žξ…žξ€Έ(𝑑)1/3ξ‚ξ…ž=π‘Žπ‘‘4/3π‘₯1/3𝑏(πœ†π‘‘)+𝑑4/3π‘₯𝛽(πœ”π‘‘),(𝐸π‘₯1) where π‘Ž,𝑏>0,0<πœ†<1, πœ”>1, and 𝛽β‰₯1/3 is a ratio of two positive odd integers. Since 𝑃(𝑑)=27𝑏3𝑑,𝑄(𝑑)=27π‘Ž3πœ†π‘‘,(2.33) Corollary 2.9 implies that (𝐸π‘₯1) has property (B) provided that 𝑏3πœ”27π‘Ž3πœ†1lnπœ”>.27e(2.34) Moreover, by Theorem 2.8, the rate of divergence of every nonoscillatory solution of (𝐸π‘₯1) is ||||π‘₯(𝑑)β‰₯𝑐𝑑ln𝑑,𝑐>0.(2.35) For 𝛽=1/3 and 𝛿>1 satisfying 𝛿1/3(π›Ώβˆ’1)4/3=3π‘Žπœ†π›Ώ/3+3π‘πœ”π›Ώ/3, one such solution is 𝑑𝛿.

Now, we turn our attention to the case when 𝛾β‰₯1.

Theorem 2.11. Let 𝛾β‰₯1. Assume that π‘₯(𝑑) is a nonoscillatory solution of (𝐸). If the first-order advanced differential equation π‘§ξ…ž(𝑑)βˆ’2(1βˆ’π›Ύ)/𝛾𝑃(𝑑)𝑒[βˆ’2(1βˆ’π›Ύ)/π›Ύβˆ«π‘‘π‘‘1𝑄(𝑠)𝑑𝑠]β„Ž1/𝛾e2(1βˆ’π›Ύ)/π›Ύβˆ«π‘‘1𝜎(𝑑)𝑄(𝑠)π‘‘π‘ ξ‚β„Ž1/𝛾[](π‘§πœŽ(𝑑))=0(𝐸2) is oscillatory, then Case (II) cannot hold.

Proof. Let π‘₯(𝑑) be an eventually positive solution of (𝐸), satisfying Case (II) of Lemma 2.4. Then, (2.18) holds. Lemma 2.2, in view of the monotonicity of π‘₯(𝑑), (𝐻3), and (𝐻4), implies βˆ’π‘₯ξ…žξ…žπ‘“(𝑑)β‰₯1/𝛾(π‘₯(𝑑))2(π›Ύβˆ’1)/π›Ύπ‘Ž1/π›Ύξƒ©ξ€œ(𝑑)βˆžπ‘‘π‘žξ€·πœβˆ’1ξ€Έ(𝑠)πœξ…žξ€·πœβˆ’1(ξ€Έξƒͺ𝑠)d𝑠1/𝛾+β„Ž1/𝛾[𝜎])(π‘₯(𝑑)2(π›Ύβˆ’1)/π›Ύπ‘Ž1/π›Ύξ‚΅ξ€œ(𝑑)βˆžπ‘‘ξ‚Άπ‘(𝑠)d𝑠1/𝛾.(2.36) An integration from 𝑑 to ∞ yields π‘₯ξ…žξ€œ(𝑑)β‰₯βˆžπ‘‘π‘“1/𝛾(π‘₯(𝑒))2(π›Ύβˆ’1)/π›Ύπ‘Ž1/π›Ύξƒ©ξ€œ(𝑒)βˆžπ‘’π‘žξ€·πœβˆ’1ξ€Έ(𝑠)πœξ…žξ€·πœβˆ’1(ξ€Έξƒͺ𝑠)d𝑠1/𝛾+ξ€œdπ‘’βˆžπ‘‘β„Ž1/𝛾[𝜎])(π‘₯(𝑒)2(π›Ύβˆ’1)/π›Ύπ‘Ž1/π›Ύξ‚΅ξ€œ(𝑒)βˆžπ‘’ξ‚Άπ‘(𝑠)d𝑠1/𝛾d𝑒β‰₯𝑓1/𝛾(π‘₯(𝑑))2(1βˆ’π›Ύ)/𝛾𝑄(𝑑)+β„Ž1/𝛾[](π‘₯𝜎(𝑑))2(1βˆ’π›Ύ)/𝛾𝑃(𝑑).(2.37) Noting (𝐻4), we see that π‘₯(𝑑) is a positive solution of the differential inequality π‘₯ξ…ž(𝑑)β‰₯2(1βˆ’π›Ύ)/𝛾𝑄(𝑑)π‘₯(𝑑)+2(1βˆ’π›Ύ)/𝛾𝑃(𝑑)β„Ž1/𝛾[](π‘₯𝜎(𝑑)).(2.38) Setting π‘₯(𝑑)=𝑀(𝑑)𝑒[2(1βˆ’π›Ύ)/π›Ύβˆ«π‘‘π‘‘1𝑄(𝑠)d𝑠],(2.39) one can see that 𝑀(𝑑) is a positive solution of the advanced differential inequality π‘€ξ…ž(𝑑)βˆ’2(1βˆ’π›Ύ)/𝛾𝑃(𝑑)𝑒[βˆ’2(1βˆ’π›Ύ)/π›Ύβˆ«π‘‘π‘‘1𝑄(𝑠)d𝑠]β„Ž1/𝛾𝑒2(1βˆ’π›Ύ)/π›Ύβˆ«π‘‘1𝜎(𝑑)𝑄(𝑠)dπ‘ ξ‚β„Ž1/𝛾[](π‘€πœŽ(𝑑))β‰₯0.(2.40) By Lemma 2.3, we deduce that the corresponding differential equation (𝐸2) has also a positive solution. A contradiction. Therefore, π‘₯(𝑑) cannot satisfy Case (II).

The following result is obvious.

Theorem 2.12. Let (1.3) hold and 𝛾β‰₯1. Assume that (𝐸2) is oscillatory. Then, (𝐸) has property (B) and, what is more, each of its nonoscillatory solutions satisfies (2.25).

Now, we present easily verifiable criterion for property (B) of (𝐸).

Corollary 2.13. Let (1.3) and (2.28) hold and 𝛾β‰₯1. If liminfπ‘‘β†’βˆžξ€œπ‘‘πœŽ(𝑑)𝑃(𝑒)e[2(1βˆ’π›Ύ)/π›Ύβˆ«π‘’πœŽ(𝑒)𝑄(𝑠)𝑑𝑠]2𝑑𝑒>(π›Ύβˆ’1)/𝛾e,(2.41) then (𝐸) has property (B).

Proof. The proof is similar to the proof of Corollary 2.9 and so it can be omitted.

Remark 2.14. Theorems 2.6, 2.8, 2.11, and 2.12 and Corollaries 2.9 and 2.13 provide criteria for property (B) that include both delay and advanced arguments and all coefficients and functions of (𝐸). Our results are new even for the linear case of (𝐸).

Remark 2.15. It is useful to notice that if we apply the traditional approach to (𝐸), that is, if we replace (𝐸) by the corresponding differential inequality (𝐸𝜎), then conditions (2.29) of Corollary 2.9 and (2.41) of Corollary 2.13 would take the forms liminfπ‘‘β†’βˆžξ€œπ‘‘πœŽ(𝑑)1𝑃(𝑒)d𝑒>𝑒,liminfπ‘‘β†’βˆžξ€œπ‘‘πœŽ(𝑑)2𝑃(𝑒)d𝑒>(π›Ύβˆ’1)/𝛾𝑒,(2.42) respectively, which are evidently second to (2.29) and (2.41).

Example 2.16. Consider the third-order nonlinear differential equation with mixed arguments 𝑑π‘₯ξ…žξ…žξ€Έ(𝑑)3ξ‚ξ…ž=π‘Žπ‘‘6π‘₯3𝑏(πœ†π‘‘)+𝑑6π‘₯𝛽(πœ”π‘‘),(𝐸π‘₯2) where π‘Ž,𝑏>0,0<πœ†<1, 𝛽β‰₯3 is a ratio of two positive odd integers and πœ”>1. It is easy to see that conditions (2.14) and (2.15) for (𝐸π‘₯2) reduce to 𝑏𝑃(𝑑)=1/351/3π‘‘πœ†,𝑄(𝑑)=5/3π‘Ž1/351/3𝑑,(2.43) respectively. It follows from Corollary 2.13 that (𝐸π‘₯2) has property (B) provided that 𝑏1/3ξ‚ƒπœ”πœ†5/3π‘Ž1/3/22/351/3ξ‚„2lnπœ”β‰₯2/351/3𝑒.(2.44) Moreover, (2.25) provides the following rate of divergence for every nonoscillatory solution of (𝐸π‘₯2): ||||π‘₯(𝑑)β‰₯𝑐𝑑5/3,𝑐>0.(2.45)

Now, we eliminate Case (I) of Lemma 2.4, to get the oscillation of (𝐸).

Theorem 2.17. Let π‘₯(𝑑) be a nonoscillatory solution of (𝐸). Assume that there exists a function πœ‰(𝑑)∈𝐢1([𝑑0,∞)) such that πœ‰ξ…ž(𝑑)β‰₯0,πœ‰(𝑑)<𝑑,πœ‚(𝑑)=𝜎(πœ‰(πœ‰(𝑑)))>𝑑.(2.46) If the first-order advanced differential equation π‘§ξ…žξƒ―ξ€œ(𝑑)βˆ’π‘‘πœ‰(𝑑)π‘Žβˆ’1/π›Ύξ‚΅ξ€œ(𝑒)π‘’πœ‰(𝑒)𝑝(𝑠)𝑑𝑠1/π›Ύξƒ°β„Žπ‘‘π‘’1/𝛾[](π‘§πœ‚(𝑑))=0(𝐸3) is oscillatory, then Case (I) cannot hold.

Proof. Let π‘₯(𝑑) be an eventually positive solution of (𝐸), satisfying Case (I). It follows from (𝐸) that ξ€Ίπ‘Žξ€Ίπ‘₯(𝑑)ξ…žξ…žξ€»(𝑑)π›Ύξ€»ξ…žβ‰₯𝑝[𝜎](𝑑)β„Ž(π‘₯(𝑑)).(2.47) Integrating from πœ‰(𝑑) to 𝑑, we have ξ€Ίπ‘₯π‘Ž(𝑑)ξ…žξ…žξ€»(𝑑)𝛾π‘₯βˆ’π‘Ž(πœ‰(𝑑))ξ…žξ…žξ€»(πœ‰(𝑑))𝛾β‰₯ξ€œπ‘‘πœ‰(𝑑)[])[])ξ€œπ‘(𝑠)β„Ž(π‘₯𝜎(𝑠)d𝑠β‰₯β„Ž(π‘₯𝜎(πœ‰(𝑑))π‘‘πœ‰(𝑑)𝑝(𝑠)d𝑠.(2.48) Therefore, π‘₯ξ…žξ…ž(𝑑)β‰₯β„Ž1/𝛾[𝜎](π‘₯(πœ‰(𝑑)))π‘Žβˆ’1/π›Ύξ‚΅ξ€œ(𝑑)π‘‘πœ‰(𝑑)𝑝(𝑠)d𝑠1/𝛾.(2.49) An integration from πœ‰(𝑑) to 𝑑 yields π‘₯ξ…žξ€œ(𝑑)β‰₯π‘‘πœ‰(𝑑)β„Ž1/𝛾[](π‘₯𝜎(πœ‰(𝑒)))π‘Žβˆ’1/π›Ύξ‚΅ξ€œ(𝑒)π‘’πœ‰(𝑒)𝑝(𝑠)d𝑠1/𝛾d𝑒β‰₯β„Ž1/𝛾[])ξ€œ(π‘₯πœ‚(𝑑)π‘‘πœ‰(𝑑)π‘Žβˆ’1/π›Ύξ‚΅ξ€œ(𝑒)π‘’πœ‰(𝑒)𝑝(𝑠)d𝑠1/𝛾d𝑒.(2.50) Consequently, π‘₯(𝑑) is a positive solution of the advanced differential inequality π‘₯ξ…žξƒ―ξ€œ(𝑑)βˆ’π‘‘πœ‰(𝑑)π‘Žβˆ’1/π›Ύξ‚΅ξ€œ(𝑒)π‘’πœ‰(𝑒)𝑝(𝑠)d𝑠1/π›Ύξƒ°β„Žd𝑒1/𝛾[](π‘₯πœ‚(𝑑))β‰₯0.(2.51) Hence, by Lemma 2.3, we conclude that the corresponding differential equation (𝐸3) also has a positive solution, which contradicts the oscillation of (𝐸3). Therefore, π‘₯(𝑑) cannot satisfy Case (I).

Combining Theorem 2.17 with Theorems 2.6 and 2.11, we get two criteria for the oscillation of (𝐸).

Theorem 2.18. Let (1.3) hold and 0<𝛾≀1. Assume that both of the first-order advanced equations (𝐸1) and (𝐸3) are oscillatory, then (𝐸) is oscillatory.

Proof. Assume that (𝐸) has a nonoscillatory solution. It follows from Remark 2.5 that π‘₯(𝑑) satisfies either Case (I) or (II). But both cases are excluded by the oscillation of (𝐸1) and (𝐸3).

Corollary 2.19. Let 0<𝛾≀1. Assume that (1.3), (2.28), (2.29), and (2.46) hold. If liminfπ‘‘β†’βˆžξ€œπ‘‘πœ‚(𝑑)ξƒ―ξ€œπ‘£πœ‰(𝑣)π‘Žβˆ’1/π›Ύξ‚΅ξ€œ(𝑒)π‘’πœ‰(𝑒)𝑝(𝑠)𝑑𝑠1/𝛾1𝑑𝑒𝑑𝑣>e,(2.52) then (𝐸) is oscillatory.

Proof. Conditions (2.29) and (2.52) guarantee the oscillation of (𝐸1) and (𝐸3), respectively. The assertion now follows from Theorem 2.18.

Example 2.20. We consider once more the third-order differential equation (𝐸π‘₯1) with the same restrictions as in Example 2.10. We set πœ‰(𝑑)=𝛼0𝑑, where 𝛼0√=(1+βˆšπœ”)/2πœ”. Then condition (2.52) takes the form 𝑏3ξ€·1βˆ’π›Ό0ξ€Έξ€·1βˆ’π›Ό01/3ξ€Έ3𝛼20ξ€·lnπœ”π›Ό20ξ€Έ>1,27𝑒(2.53) which by Corollary 2.19, implies the oscillation of (𝐸π‘₯1).

The following results are obvious.

Theorem 2.21. Let (1.3) hold and 𝛾β‰₯1. Assume that both of the first-order advanced equations (𝐸2) and (𝐸3) are oscillatory, then (𝐸) is oscillatory.

Corollary 2.22. Let 𝛾β‰₯1. Assume that (1.3), (2.28), (2.41), (2.46), and (2.52) hold. Then (𝐸) is oscillatory.

Example 2.23. We recall again the differential equation (𝐸π‘₯2) with the same assumptions as in Example 2.16. We set πœ‰(𝑑)=𝛼0𝑑 with 𝛼0√=(1+βˆšπœ”)/2πœ”. Then condition (2.52) reduces to 𝑏1/3ξ€·1βˆ’π›Ό0ξ€Έξ€·1βˆ’π›Ό50ξ€Έ1/3𝛼08/3ξ€·lnπœ”π›Ό20ξ€Έ>51/3𝑒,(2.54) which, by Corollary 2.22, guarantees the oscillation of (𝐸π‘₯2).

The following result is intended to exclude Case (III) of Lemma 2.4.

Theorem 2.24. Let π‘₯(𝑑) be a nonoscillatory solution of (𝐸). Assume that (1.2) holds. If the first-order delay differential equation π‘§ξ…žξ‚΅ξ€œ(𝑑)+𝑑𝑑1π‘žξ‚Ά(𝑠)𝑑𝑠1/π›Ύξ‚΅ξ€œβˆžπ‘‘aβˆ’1/𝛾𝑓(𝑠)𝑑𝑠1/𝛾[𝜏](𝑧(𝑑))=0.(𝐸4) is oscillatory, then Case (III) cannot hold.

Proof. Let π‘₯(𝑑) be a positive solution of (𝐸), satisfying Case (III) of Lemma 2.4. Using that π‘Ž(𝑑)[π‘₯ξ…žξ…ž(𝑑)]𝛾 is increasing, we find that βˆ’π‘₯ξ…žξ€œ(𝑑)β‰₯βˆžπ‘‘π‘₯ξ…žξ…žξ€œ(𝑠)d𝑠=βˆžπ‘‘ξ€·π‘Ž1/𝛾(𝑠)π‘₯ξ…žξ…žξ€Έπ‘Ž(𝑠)βˆ’1/𝛾(𝑠)d𝑠β‰₯π‘Ž(𝑑)1/𝛾π‘₯ξ…žξ…žξ€œ(𝑑)βˆžπ‘‘π‘Žβˆ’1/𝛾(𝑠)d𝑠.(2.55) Integrating the inequality [π‘Ž(𝑑)[π‘₯ξ…žξ…ž(𝑑)]𝛾]ξ…žβ‰₯π‘ž(𝑑)𝑓(π‘₯[𝜏(𝑑)]) from 𝑑1 to 𝑑, we have ξ€Ίπ‘₯π‘Ž(𝑑)ξ…žξ…ž(𝑑)𝛾β‰₯ξ€œπ‘‘π‘‘1[][])ξ€œπ‘ž(𝑠)𝑓(π‘₯𝜏(𝑠)d𝑠)β‰₯𝑓(π‘₯𝜏(𝑑)𝑑𝑑1π‘ž(𝑠)d𝑠.(2.56) Thus, π‘Ž1/𝛾(𝑑)π‘₯ξ…žξ…ž(𝑑)β‰₯𝑓1/𝛾[𝜏])ξ‚΅ξ€œ(π‘₯(𝑑)𝑑𝑑1π‘žξ‚Ά(𝑠)d𝑠1/𝛾.(2.57) Combining (2.57) with (2.55), we find 0β‰₯π‘₯ξ…žξ‚΅ξ€œ(𝑑)+𝑑𝑑1π‘žξ‚Ά(𝑠)d𝑠1/π›Ύξ‚΅ξ€œβˆžπ‘‘π‘Žβˆ’1/𝛾𝑓(𝑠)d𝑠1/𝛾[𝜏](π‘₯(𝑑)).(2.58) It follows from [16, Theorem 1] that the corresponding differential equation (𝐸4) also has a positive solution. A contradiction. For that reason, π‘₯(𝑑) cannot satisfy Case (III).

The following results are immediate.

Theorem 2.25. Let (1.2) hold and 0<𝛾≀1. Assume that both of the first-order advanced equations (𝐸1) and (𝐸4) are oscillatory, then (𝐸) has property (B).

Theorem 2.26. Let (1.2) hold and 0<𝛾≀1. Assume that all of the three first-order advanced equations (𝐸1), (𝐸3), and (𝐸4) are oscillatory, then (𝐸) is oscillatory.

Theorem 2.27. Let (1.2) hold and 𝛾β‰₯1. Assume that both of the first-order advanced equations (𝐸2) and (𝐸4) are oscillatory, then (𝐸) has property (B).

Theorem 2.28. Let (1.2) hold and 𝛾β‰₯1. Assume that all of the three first-order advanced equations (𝐸2), (𝐸3), and (𝐸4) are oscillatory, then (𝐸) is oscillatory.

3. Summary

In this paper, we have presented new comparison theorems for deducing the property (B)/oscillation of (𝐸) from the oscillation of a set of the suitable first-order delay/advanced differential equation. We were able to present such criteria for studied properties that employ all coefficients and functions included in studied equations. Our method essentially simplifies the examination of the third-order equations, and, what is more, it supports backward the research on the first-order delay/advanced differential equations. Our results here extend and complement latest ones of Grace et al. [10], Agarwal et al. [1–3], Cecchi et al. [6], Parhi and Pardi [15], and the present authors [4, 8]. The suitable illustrative examples are also provided.

References

  1. R. P. Agarwal, S.-L. Shieh, and C.-C. Yeh, β€œOscillation criteria for second-order retarded differential equations,” Mathematical and Computer Modelling, vol. 26, no. 4, pp. 1–11, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  2. R. P. Agarwal, S. R. Grace, and D. O'Regan, β€œOn the oscillation of certain functional differential equations via comparison methods,” Journal of Mathematical Analysis and Applications, vol. 286, no. 2, pp. 577–600, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  3. R. P. Agarwal, S. R. Grace, and T. Smith, β€œOscillation of certain third order functional differential equations,” Advances in Mathematical Sciences and Applications, vol. 16, no. 1, pp. 69–94, 2006. View at Zentralblatt MATH
  4. B. Baculíková and J. Džurina, β€œOscillation of third-order neutral differential equations,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 215–226, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. B. Baculíková, R. P. Agarwal, T. Li, and J. Džurina, β€œOscillation of third-order nonlinear functional differential equations with mixed arguments,” to appear in Acta Mathematica Hungarica.
  6. M. Cecchi, Z. Došlá, and M. Marini, β€œOn third order differential equations with property A and B,” Journal of Mathematical Analysis and Applications, vol. 231, no. 2, pp. 509–525, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  7. J. Džurina, β€œAsymptotic properties of third order delay differential equations,” Czechoslovak Mathematical Journal, vol. 45(120), no. 3, pp. 443–448, 1995. View at Zentralblatt MATH
  8. J. Džurina, β€œComparison theorems for functional-differential equations with advanced argument,” Unione Matematica Italiana. Bollettino, vol. 7, no. 3, pp. 461–470, 1993. View at Zentralblatt MATH
  9. L. H. Erbe, Q. Kong, and B. G. Zhang, Oscillation Theory for Functional-Differential Equations, vol. 190 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1994.
  10. S. R. Grace, R. P. Agarwal, R. Pavani, and E. Thandapani, β€œOn the oscillation of certain third order nonlinear functional differential equations,” Applied Mathematics and Computation, vol. 202, no. 1, pp. 102–112, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  11. I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Oxford Mathematical Monographs, The Clarendon Press, New York, NY, USA, 1991.
  12. T. S. Hassan, β€œOscillation of third order nonlinear delay dynamic equations on time scales,” Mathematical and Computer Modelling, vol. 49, no. 7-8, pp. 1573–1586, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  13. T. Kusano and M. Naito, β€œComparison theorems for functional-differential equations with deviating arguments,” Journal of the Mathematical Society of Japan, vol. 33, no. 3, pp. 509–532, 1981. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  14. G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, vol. 110 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1987.
  15. N. Parhi and S. Pardi, β€œOn oscillation and asymptotic property of a class of third order differential equations,” Czechoslovak Mathematical Journal, vol. 49(124), no. 1, pp. 21–33, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  16. Ch. G. Philos, β€œOn the existence of nonoscillatory solutions tending to zero at for differential equations with positive delays,” Archiv der Mathematik, vol. 36, no. 2, pp. 168–178, 1981. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  17. A. Tiryaki and M. F. Aktaş, β€œOscillation criteria of a certain class of third order nonlinear delay differential equations with damping,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 54–68, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet