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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 730128, 10 pages
http://dx.doi.org/10.1155/2011/730128
Research Article

Asymptotic Properties of Third-Order Delay Trinomial Differential Equations

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

Received 2 September 2010; Accepted 3 November 2010

Academic Editor: YuriΒ Rogovchenko

Copyright Β© 2011 J. DΕΎurina and R. KomarikovΓ‘. 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 study properties of the third-order delay trinomial differential equation ((1/π‘Ÿ(𝑑))π‘¦ξ…žξ…ž(𝑑))ξ…ž+𝑝(𝑑)π‘¦ξ…ž(𝑑)+π‘ž(𝑑)𝑦(𝜎(𝑑))=0, by transforming this equation onto the second-/third-order binomial differential equation. Using suitable comparison theorems, we establish new results on asymptotic behavior of solutions of the studied equations. Obtained criteria improve and generalize earlier ones.

1. Introduction

In this paper, we will study oscillation and asymptotic behavior of solutions of third-order delay trinomial differential equations of the form ξ‚΅1π‘¦π‘Ÿ(𝑑)ξ…žξ…žξ‚Ά(𝑑)ξ…ž+𝑝(𝑑)π‘¦ξ…ž(𝑑)+π‘ž(𝑑)𝑦(𝜎(𝑑))=0.(𝐸) Throughout the paper, we assume that π‘Ÿ(𝑑),𝑝(𝑑),π‘ž(𝑑),𝜎(𝑑)∈𝐢([𝑑0,∞)) and(i)π‘Ÿ(𝑑)>0, 𝑝(𝑑)β‰₯0, π‘ž(𝑑)>0, 𝜎(𝑑)>0,(ii)𝜎(𝑑)≀𝑑, limπ‘‘β†’βˆžπœŽ(𝑑)=∞, (iii)βˆ«π‘…(𝑑)=𝑑𝑑0π‘Ÿ(𝑠)dπ‘ β†’βˆž as π‘‘β†’βˆž.

By a solution of (𝐸), we mean a function 𝑦(𝑑)∈𝐢2([𝑇π‘₯,∞)),  𝑇π‘₯β‰₯𝑑0, that satisfies (𝐸) on [𝑇π‘₯,∞). We consider only those solutions 𝑦(𝑑) of (𝐸) which satisfy sup{|𝑦(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) itself is said to be oscillatory if all its solutions are oscillatory.

Recently, increased attention has been devoted to the oscillatory and asymptotic properties of second- and third-order differential equations (see [1–22]). Various techniques appeared for the investigation of such differential equations. Our method is based on establishing new comparison theorems, so that we reduce the examination of the third-order trinomial differential equations to the problem of the observation of binomial equations.

In earlier papers [11, 13, 16, 20], a particular case of (𝐸), namely, the ordinary differential equation (without delay) π‘¦ξ…žξ…žξ…ž(𝑑)+𝑝(𝑑)π‘¦ξ…ž(𝑑)+𝑔(𝑑)𝑦(𝑑)=0,(𝐸1) has been investigated, and sufficient conditions for all its nonoscillatory solutions 𝑦(𝑑) to satisfy𝑦(𝑑)π‘¦ξ…ž(𝑑)<0(1.1) or the stronger conditionlimπ‘‘β†’βˆžπ‘¦(𝑑)=0(1.2) are presented. It is known that (𝐸1) has always a solution satisfying (1.1). Recently, various kinds of sufficient conditions for all nonoscillatory solutions to satisfy (1.1) or (1.2) appeared. We mention here [9, 11, 13, 16, 21]. But there are only few results for differential equations with deviating argument. Some attempts have been made in [8, 10, 18, 19]. In this paper we generalize these, results and we will study conditions under which all nonoscillatory solutions of (𝐸) satisfy (1.1) and (1.2). For our further references we define as following.

Definition 1.1. We say that (𝐸) has property (𝑃0) if its every nonoscillatory solution 𝑦(𝑑) satisfies (1.1).

In this paper, we have two purposes. In the first place, we establish comparison theorems for immediately obtaining results for third-order delay equation from that of third order equation without delay. This part extends and complements earlier papers [7, 8, 10, 18].

Secondly, we present a comparison principle for deducing the desired property of (𝐸) from the oscillation of a second-order differential equation without delay. Here, we generalize results presented in [8, 9, 14, 15, 21].

Remark 1.2. All functional inequalities considered in this paper are assumed to hold eventually;0 that is, they are satisfied for all 𝑑 large enough.

2. Main Results

It will be derived that properties of (𝐸) are closely connected with the corresponding second-order differential equationξ‚΅1π‘£π‘Ÿ(𝑑)ξ…žξ‚Ά(𝑑)ξ…ž+𝑝(𝑑)𝑣(𝑑)=0(𝐸𝑣) as the following theorem says.

Theorem 2.1. Let 𝑣(𝑑) be a positive solution of (𝐸𝑣). Then (𝐸) can be written as 𝑣2(𝑑)ξ‚΅1π‘Ÿ(𝑑)𝑦𝑣(𝑑)ξ…ž(𝑑)ξ…žξ‚Άξ…ž+π‘ž(𝑑)𝑣(𝑑)𝑦(𝜎(𝑑))=0.(𝐸𝑐)

Proof. The proof follows from the fact that 1𝑣𝑣(𝑑)2(𝑑)ξ‚΅1π‘Ÿ(𝑑)𝑦𝑣(𝑑)ξ…ž(𝑑)ξ…žξ‚Άξ…ž=ξ‚΅1π‘¦π‘Ÿ(𝑑)ξ…žξ…ž(𝑑)ξ…ž+𝑝(𝑑)π‘¦ξ…ž(𝑑).(2.1)

Now, in the sequel, instead of studying properties of the trinomial equation (𝐸), we will study the behavior of the binomial equation (𝐸𝑐). For our next considerations, it is desirable for (𝐸𝑐) to be in a canonical form; that is,ξ€œβˆžξ€œπ‘£(𝑑)d𝑑=∞,(2.2)βˆžπ‘Ÿ(𝑑)𝑣2(𝑑)d𝑑=∞,(2.3) because properties of the canonical equations are nicely explored.

Now, we will study the properties of the positive solutions of (𝐸𝑣) to recognize when (2.2)-(2.3) are satisfied. The following result (see, e.g., [7, 9] or [14]) is a consequence of Sturm's comparison theorem.

Lemma 2.2. If 𝑅2(𝑑)1π‘Ÿ(𝑑)𝑝(𝑑)≀4,(2.4) then (𝐸𝑣) possesses a positive solution 𝑣(𝑑).

To be sure that (𝐸𝑣) possesses a positive solution, we will assume throughout the paper that (2.4) holds. The following result is obvious.

Lemma 2.3. If 𝑣(𝑑) is a positive solution of (𝐸𝑣), then π‘£ξ…ž(𝑑)>0, ((1/π‘Ÿ(𝑑))π‘£ξ…ž(𝑑))ξ…ž<0, and, what is more, (2.2) holds and there exists 𝑐>0 such that 𝑣(𝑑)≀𝑐𝑅(𝑑).

Now, we will show that if (𝐸𝑣) is nonoscillatory, then we always can choose a positive solution 𝑣(𝑑) of (𝐸𝑣) for which (2.3) holds.

Lemma 2.4. If 𝑣1(𝑑) is a positive solution of (𝐸𝑣) for which (2.3) is violated, then 𝑣2(𝑑)=𝑣1ξ€œ(𝑑)βˆžπ‘‘0π‘Ÿ(𝑠)𝑣21(𝑠)d𝑠(2.5) is another positive solution of (𝐸𝑣) and, for 𝑣2(𝑑), (2.3) holds.

Proof. First note that 𝑣2ξ…žξ…ž(𝑑)=𝑣1ξ…žξ…ž(ξ€œπ‘‘)𝑑𝑑0π‘Ÿ(𝑠)𝑣21(𝑠)d𝑠=βˆ’π‘(𝑑)𝑣1(ξ€œπ‘‘)𝑑𝑑0𝑣1βˆ’2(𝑠)d𝑠=βˆ’π‘(𝑑)𝑣2(𝑑).(2.6) Thus, 𝑣2(𝑑) is a positive solution of (𝐸𝑣). On the other hand, to insure that (2.3) holds for 𝑣2(𝑑), let us denote βˆ«π‘€(𝑑)=βˆžπ‘‘π‘Ÿ(𝑠)/𝑣21(𝑠)d𝑠. Then limπ‘‘β†’βˆžπ‘€(𝑑)=0 and ξ€œβˆžπ‘‘1π‘Ÿ(𝑠)𝑣22ξ€œ(𝑠)d𝑠=βˆžπ‘‘1βˆ’π‘€ξ…ž(𝑠)𝑀(𝑠)d𝑠=limπ‘‘β†’βˆžξƒ©1βˆ’1𝑀(𝑑)𝑀𝑑1ξ€Έξƒͺ=∞.(2.7)

Combining Lemmas 2.2, 2.3, and 2.4, we obtain the following result.

Lemma 2.5. Let (2.4) hold. Then trinomial (𝐸) can be represented in its binomial canonical form (𝐸𝑐).

Now we can study properties of (𝐸) with help of its canonical representation (𝐸𝑐). For our reference, let us denote for (𝐸𝑐) 𝐿0𝑦=𝑦,𝐿11𝑦=𝑣𝐿0π‘¦ξ€Έξ…ž,𝐿2𝑣𝑦=2π‘Ÿξ€·πΏ1π‘¦ξ€Έξ…ž,𝐿3𝐿𝑦=2π‘¦ξ€Έξ…ž.(2.8) Now, (𝐸𝑐) can be written as 𝐿3𝑦(𝑑)+𝑣(𝑑)π‘ž(𝑑)𝑦(𝜎(𝑑))=0.

We present a structure of the nonoscillatory solutions of (𝐸𝑐). Since (𝐸𝑐) is in a canonical form, it follows from the well-known lemma of Kiguradze (see, e.g., [7, 9, 14]) that every nonoscillatory solution 𝑦(𝑑) of (𝐸𝑐) is either of degree 0, that is,𝑦𝐿0𝑦(𝑑)>0,𝑦𝐿1𝑦(𝑑)<0,𝑦𝐿2𝑦(𝑑)>0,𝑦𝐿3𝑦(𝑑)<0,(2.9) or of degree 2, that is,𝑦𝐿0𝑦(𝑑)>0,𝑦𝐿1𝑦(𝑑)>0,𝑦𝐿2𝑦(𝑑)>0,𝑦𝐿3𝑦(𝑑)<0.(2.10)

Definition 2.6. We say that (𝐸𝑐) has property (𝐴) if its every nonoscillatory solution 𝑦(𝑑) is of degree 0; that is, it satisfies (2.9).

Now we verify that property (𝑃0) of (𝐸) and property (𝐴) of (𝐸𝑐) are equivalent in the sense that 𝑦(𝑑) satisfies (1.1) if and only if it obeys (2.9).

Theorem 2.7. Let (2.4) hold. Assume that 𝑣(𝑑) is a positive solution of (𝐸𝑣) satisfying (2.2)-(2.3). Then (𝐸𝑐) has property (𝐴) if and only if (𝐸) has property (𝑃0).

Proof. β†’ We suppose that 𝑦(𝑑) is a positive solution of (𝐸). We need to verify that 𝑦′(𝑑)<0. Since 𝑦(𝑑) is also a solution of (𝐸𝑐), then it satisfies (2.9). Therefore, 0>𝐿1𝑦(𝑑)=𝑦′(𝑑)/𝑣(𝑑).
← Assume that 𝑦(𝑑) is a positive solution of (𝐸𝑐). We will verify that (2.9) holds. Since 𝑦(𝑑) is also a solution of (𝐸), we see that π‘¦ξ…ž(𝑑)<0; that is, 𝐿1𝑦(𝑑)<0. It follows from (𝐸𝑐) that 𝐿3𝑦(𝑑)=βˆ’π‘£(𝑑)π‘ž(𝑑)𝑦(𝜎(𝑑))<0. Thus, 𝐿2𝑦(𝑑) is decreasing. If we admit 𝐿2𝑦(𝑑)<0 eventually, then L1𝑦(𝑑) is decreasing, and integrating the inequality 𝐿1𝑦(𝑑)<𝐿1𝑦(𝑑1), we get 𝑦(𝑑)<𝑦(𝑑1)+𝐿1𝑦(𝑑1)βˆ«π‘‘π‘‘1𝑣(𝑠)π‘‘π‘ β†’βˆ’βˆž as π‘‘β†’βˆž. Therefore, 𝐿2𝑦(𝑑)>0 and (2.9) holds.

The following result which can be found in [9, 14] presents the relationship between property (𝐴) of delay equation and that of equation without delay.

Theorem 2.8. Let (2.4) hold. Assume that 𝑣(𝑑) is a positive solution of (𝐸𝑣) satisfying (2.2)-(2.3). Let 𝜎(𝑑)∈𝐢1𝑑0,βˆžξ€Έξ€Έ,πœŽξ…ž(𝑑)>0.(2.11) If 𝑣2(𝑑)ξ‚΅1π‘Ÿ(𝑑)𝑦𝑣(𝑑)ξ…ž(𝑑)ξ…žξ‚Άξ…ž+π‘£ξ€·πœŽβˆ’1ξ€Έπ‘žξ€·πœŽ(𝑑)βˆ’1ξ€Έ(𝑑)πœŽξ…žξ€·πœŽβˆ’1ξ€Έ(𝑑)𝑦(𝑑)=0(𝐸2) has property (𝐴), then so does (𝐸𝑐).

Combining Theorems 2.7 and 2.8, we get a criterion that reduces property (𝑃0) of (𝐸) to the property (𝐴) of (𝐸2).

Corollary 2.9. Let (2.4) and (2.11) hold. Assume that 𝑣(𝑑) is a positive solution of (𝐸𝑣) satisfying (2.2)-(2.3). If (𝐸2) has property (𝐴) then (𝐸) has property (𝑃0).

Employing any known or future result for property (𝐴) of (𝐸2), then in view of Corollary 2.9, we immediately obtain that property (𝑃0) holds for (𝐸).

Example 2.10. We consider the third-order delay trinomial differential equation ξ‚€1π‘‘π‘¦ξ…žξ…žξ‚(𝑑)ξ…ž+𝛼(2βˆ’π›Ό)𝑑3π‘¦ξ…ž(𝑑)+π‘ž(𝑑)𝑦(𝜎(𝑑))=0,(2.12) where 0<𝛼<1 and 𝜎(𝑑) satisfies (2.11). The corresponding equation (𝐸𝑣) takes the form ξ‚€1π‘‘π‘£ξ…žξ‚(𝑑)ξ…ž+𝛼(2βˆ’π›Ό)𝑑3𝑣(𝑑)=0,(2.13) and it has the pair of the solutions 𝑣(𝑑)=𝑑𝛼 and ̂𝑣(𝑑)=𝑑2βˆ’π›Ό. Thus, 𝑣(𝑑)=𝑑𝛼 is our desirable solution, which permits to rewrite (2.12) in its canonical form. Then, by Corollary 2.9, (2.12) has property (𝑃0) if the equation 𝑑2π›Όβˆ’1ξ€·π‘‘βˆ’π›Όπ‘¦ξ…žξ€Έ(𝑑)ξ…žξ‚ξ…ž+ξ€·πœŽβˆ’1ξ€Έ(𝑑)π›Όπ‘žξ€·πœŽβˆ’1ξ€Έ(𝑑)πœŽξ…žξ€·πœŽβˆ’1(𝑑)𝑦(𝑑)=0(2.14) has property (𝐴).

Now, we enhance our results to guarantee stronger asymptotic behavior of the nonoscillatory solutions of (𝐸). We impose an additional condition on the coefficients of (𝐸) to achieve that every nonoscillatory solution of (𝐸) tends to zero as π‘‘β†’βˆž.

Corollary 2.11. Let (2.4) and (2.11) hold. Assume that 𝑣(𝑑) is a positive solution of (𝐸𝑣) satisfying (2.2)-(2.3). If (𝐸2) has property (𝐴) and ξ€œβˆžπ‘‘0𝑣𝑠3ξ€Έξ€œβˆžπ‘ 3π‘Ÿξ€·π‘ 2𝑣2𝑠2ξ€Έξ€œβˆžπ‘ 2𝑣𝑠1ξ€Έπ‘žξ€·π‘ 1ξ€Έd𝑠1d𝑠2d𝑠3=∞,(2.15) then every nonoscillatory solution 𝑦(𝑑) of (𝐸) satisfies (1.2).

Proof. Assume that 𝑦(𝑑) is a positive solution of (𝐸). Then, it follows from Corollary 2.9 that 𝑦′(𝑑)<0. Therefore, limπ‘‘β†’βˆžπ‘¦(𝑑)=β„“β‰₯0. Assume β„“>0. On the other hand, 𝑦(𝑑) is also a solution of (𝐸𝑐), and, in view of Theorem 2.7, it has to be of degree 0; that is, (2.9) is fulfilled. Then, integrating (𝐸𝑐) from 𝑑 to ∞, we get 𝐿2ξ€œπ‘¦(𝑑)β‰₯βˆžπ‘‘ξ€œπ‘£(𝑠)π‘ž(𝑠)𝑦(𝜎(𝑠))d𝑠β‰₯β„“βˆžπ‘‘π‘£(𝑠)π‘ž(𝑠)d𝑠.(2.16) Multiplying this inequality by π‘Ÿ(𝑑)/𝑣2(𝑑) and then integrating from 𝑑 to ∞, we have βˆ’πΏ1ξ€œπ‘¦(𝑑)β‰₯β„“βˆžπ‘‘π‘Ÿξ€·π‘ 2𝑣2𝑠2ξ€Έξ€œβˆžπ‘ 2𝑣𝑠1ξ€Έπ‘žξ€·π‘ 1ξ€Έd𝑠1d𝑠2.(2.17) Multiplying this by 𝑣(𝑑) and then integrating from 𝑑1 to 𝑑, we obtain 𝑦𝑑1ξ€Έξ€œβ‰₯ℓ𝑑𝑑1𝑣𝑠3ξ€Έξ€œβˆžπ‘ 3π‘Ÿξ€·π‘ 2𝑣2𝑠2ξ€Έξ€œβˆžπ‘ 2𝑣𝑠1ξ€Έπ‘žξ€·π‘ 1ξ€Έd𝑠1d𝑠2d𝑠3⟢∞asπ‘‘βŸΆβˆž.(2.18) This is a contradiction, and we deduce that β„“=0. The proof is complete.

Example 2.12. We consider once more the third-order equation (2.12). It is easy to see that (2.15) takes the form ξ€œβˆžπ‘‘0𝑠𝛼3ξ€œβˆžπ‘ 3𝑠21βˆ’2π›Όξ€œβˆžπ‘ 2𝑠𝛼1π‘žξ€·π‘ 1ξ€Έd𝑠1d𝑠2d𝑠3=∞.(2.19) Then, by Corollary 2.11, every nonoscillatory solution of (2.12) tends to zero as π‘‘β†’βˆž provided that (2.19) holds and (2.14) has property (𝐴).

In the second part of this paper, we derive criteria that enable us to deduce property (𝑃0) of (𝐸) from the oscillation of a suitable second-order differential equation. The following theorem is a modification of Tanaka's result [21].

Theorem 2.13. Let (2.4) and (2.11) hold. Assume that 𝑣(𝑑) is a positive solution of (𝐸𝑣) satisfying (2.2)-(2.3). Let ξ€œβˆžπ‘£(𝑠)π‘ž(𝑠)𝑑𝑠<∞.(2.20) If the second-order equation 𝑣2(𝑑)π‘§π‘Ÿ(𝑑)ξ…ž(𝑑)ξ…ž+𝑣(𝜎(𝑑))πœŽξ…ž(ξ€œπ‘‘)βˆžπ‘‘ξ‚Άπ‘£(𝑠)π‘ž(𝑠)𝑑𝑠𝑧(𝜎(𝑑))=0(𝐸3) is oscillatory, then (𝐸𝑐) has property (𝐴).

Proof. Assume that 𝑦(𝑑) is a positive solution of (𝐸𝑐), then 𝑦(𝑑) is either of degree 0 or of degree 2. Assume that 𝑦(𝑑) is of degree 2; that is, (2.10) holds. An integration of (𝐸𝑐) yields 𝐿2ξ€œπ‘¦(𝑑)β‰₯βˆžπ‘‘π‘£(𝑠)π‘ž(𝑠)𝑦(𝜎(𝑠))d𝑠.(2.21) On the other hand, ξ€œπ‘¦(𝑑)β‰₯𝑑𝑑1𝑣(π‘₯)𝐿1𝑦(π‘₯)dπ‘₯.(2.22) Combining the last two inequalities, we get 𝐿2ξ€œπ‘¦(𝑑)β‰₯βˆžπ‘‘ξ€œπ‘£(𝑠)π‘ž(𝑠)π‘‘πœŽ(𝑠)1𝑣(π‘₯)𝐿1β‰₯ξ€œπ‘¦(π‘₯)dπ‘₯dπ‘ βˆžπ‘‘ξ€œπ‘£(𝑠)π‘ž(𝑠)𝜎(𝑠)𝜎(𝑑)𝑣(π‘₯)𝐿1=ξ€œπ‘¦(π‘₯)dπ‘₯dπ‘ βˆžπœŽ(𝑑)𝐿1ξ€œπ‘¦(π‘₯)𝑣(π‘₯)βˆžπœŽβˆ’1(π‘₯)𝑣(𝑠)π‘ž(𝑠)d𝑠dπ‘₯.(2.23) Integrating the previous inequality from 𝑑1 to 𝑑, we see that 𝑀(𝑑)≑𝐿1𝑦(𝑑) satisfies 𝑑𝑀(𝑑)β‰₯𝑀1ξ€Έ+ξ€œπ‘‘π‘‘1π‘Ÿ(𝑠)𝑣2ξ€œ(𝑠)∞𝜎(𝑠)𝐿1ξ€œπ‘¦(π‘₯)𝑣(π‘₯)βˆžπœŽβˆ’1(π‘₯)𝑣(𝛿)π‘ž(𝛿)d𝛿dπ‘₯d𝑠.(2.24) Denoting the right-hand side of (2.24) by 𝑧(𝑑), it is easy to see that 𝑧(𝑑)>0 and 𝑣0=2(𝑑)π‘§π‘Ÿ(𝑑)ξ…ž(𝑑)ξ…ž+𝑣(𝜎(𝑑))πœŽξ…ž(ξ€œπ‘‘)βˆžπ‘‘ξ‚Άβ‰₯𝑣𝑣(𝑠)𝑔(𝑠)𝑑𝑠𝑀(𝜎(𝑑))=02(𝑑)π‘§π‘Ÿ(𝑑)ξ…žξ‚Ά(𝑑)ξ…ž+𝑣(𝜎(𝑑))πœŽξ…žξ€œ(𝑑)βˆžπ‘‘ξ‚Άπ‘£(𝑠)𝑔(𝑠)𝑑𝑠𝑧(𝜎(𝑑))=0.(2.25) By Theorem 2 in [14], the corresponding equation (𝐸3) also has a positive solution. This is a contradiction. We conclude that 𝑦(𝑑) is of degree 0; that is, (𝐸𝑐) has property (A).

If (2.20) does not hold, then we can use the following result.

Theorem 2.14. Let (2.4) and (2.11) hold. Assume that 𝑣(𝑑) is a positive solution of (𝐸𝑣) satisfying (2.2)-(2.3). If ξ€œβˆžπ‘£(𝑠)π‘ž(𝑠)d𝑠=∞,(2.26) then (𝐸𝑐) has property (A).

Proof. Assume that 𝑦(𝑑) is a positive solution of (𝐸𝑐) and 𝑦(𝑑) is of degree 2. An integration of (𝐸𝑐) yields 𝐿2𝑦𝑑1ξ€Έβ‰₯ξ€œπ‘‘π‘‘1ξ€·πœŽξ€·π‘‘π‘£(𝑠)π‘ž(𝑠)𝑦(𝜎(𝑠))d𝑠β‰₯𝑦1ξ€œξ€Έξ€Έπ‘‘π‘‘1𝑣(𝑠)π‘ž(𝑠)dπ‘ βŸΆβˆžasπ‘‘βŸΆβˆž,(2.27) which is a contradiction. Thus, 𝑦(𝑑) is of degree 0. The proof is complete now.

Taking Theorem 2.13 and Corollary 2.9 into account, we get the following criterion for property (𝑃0) of (𝐸).

Corollary 2.15. Let (2.4), (2.11), and (2.20) hold. Assume that 𝑣(𝑑) is a positive solution of (𝐸𝑣) satisfying (2.2)-(2.3). If (𝐸3) is oscillatory, then (𝐸) has property (𝑃0).

Applying any criterion for oscillation of (𝐸3), Corollary 2.15 yields a sufficient condition property (𝑃0) of (𝐸).

Corollary 2.16. Let (2.4), (2.11), and (2.20) hold. Assume that 𝑣(𝑑) is a positive solution of (𝐸𝑣) satisfying (2.2)-(2.3). If liminfπ‘‘β†’βˆžξ‚΅ξ€œπ‘‘πœŽ(𝑑)0π‘Ÿ(𝑠)𝑣2ξ€œ(𝑠)π‘‘π‘ ξ‚Άξ‚΅βˆžπ‘‘π‘£(𝜎(π‘₯))πœŽξ…žξ€œ(π‘₯)∞π‘₯ξ‚Ά>1𝑣(𝑠)𝑔(𝑠)d𝑠dπ‘₯4,(2.28) then (𝐸) has property (𝑃0).

Proof. It follows from Theorem 11 in [9] that condition (2.28) guarantees the oscillation of (𝐸3). The proof arises from Corollary 2.16.

Imposing an additional condition on the coefficients of (𝐸), we can obtain that every nonoscillatory solution of (𝐸) tends to zero as π‘‘β†’βˆž.

Corollary 2.17. Let (2.4) and (2.11) hold. Assume that 𝑣(𝑑) is a positive solution of (𝐸𝑣) satisfying (2.2)-(2.3). If (2.28) and (2.15) hold, then every nonoscillatory solution 𝑦(𝑑) of (𝐸) satisfies (1.2).

Example 2.18. We consider again (2.12). By Corollary 2.17, every nonoscillatory solution of (2.12) tends to zero as π‘‘β†’βˆž provided that (2.19) holds and liminfπ‘‘β†’βˆžπœŽ2βˆ’2π›Όξ‚΅ξ€œ(𝑑)βˆžπ‘‘πœŽπ›Ό(π‘₯)πœŽξ…žξ€œ(π‘₯)∞π‘₯𝑠𝛼>π‘ž(𝑠)d𝑠dπ‘₯2βˆ’2𝛼4.(2.29) For a special case of (2.12), namely, for ξ‚€1π‘‘π‘¦ξ…žξ…žξ‚(𝑑)ξ…ž+𝛼(2βˆ’π›Ό)𝑑3π‘¦ξ…žπ‘Ž(𝑑)+𝑑4𝑦(πœ†π‘‘))=0,(2.30) with 0<𝛼<1, 0<πœ†<1, and π‘Ž>0, we get that every nonoscillatory solution of (2.30) tends to zero as π‘‘β†’βˆž provided that π‘Žπœ†3βˆ’π›Ό(3βˆ’π›Ό)(1βˆ’π›Ό)2>1.(2.31) If we set π‘Ž=𝛽[(𝛽+1)(𝛽+3)+𝛼(2βˆ’π›Ό)]πœ†π›½, where 𝛽>0, then one such solution of (2.12) is 𝑦(𝑑)=π‘‘βˆ’π›½.
On the other hand, if for some π›Ύβˆˆ(1+𝛼,3βˆ’π›Ό) we have π‘Ž=𝛾[(π›Ύβˆ’1)(3βˆ’π›Ύ)+𝛼(π›Όβˆ’2)]πœ†βˆ’π›Ύ>0, then (2.31) is violated and (2.12) has a nonoscillatory solution 𝑦(𝑑)=𝑑𝛾 which is of degree 2.

3. Summary

In this paper, we have introduced new comparison theorems for the investigation of properties of third-order delay trinomial equations. The comparison principle established in Corollaries 2.9 and 2.11 enables us to deduce properties of the trinomial third-order equations from that of binomial third-order equations. Moreover, the comparison theorems presented in Corollaries 2.15–2.17 permit to derive properties of the trinomial third-order equations from the oscillation of suitable second-order equations. The results obtained are of high generality, are easily applicable, and are illustrated on suitable examples.

Acknowledgment

This research was supported by S.G.A. KEGA 019-025TUKE-4/2010.

References

  1. B. Baculíková, E. M. Elabbasy, S. H. Saker, and J. Džurina, β€œOscillation criteria for third-order nonlinear differential equations,” Mathematica Slovaca, vol. 58, no. 2, pp. 201–220, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  2. B. Baculíková, β€œOscillation criteria for second order nonlinear differential equations,” Archivum Mathematicum, vol. 42, no. 2, pp. 141–149, 2006. View at Zentralblatt MATH
  3. B. Baculíková, β€œOscillation of certain class of the third order neutral differential equations,” Abstract and Applied Analysis. In press.
  4. B. Baculíková and D. Lacková, β€œOscillation criteria for second order retarded differential equations,” Studies of the University of Žilina. Mathematical Series, vol. 20, no. 1, pp. 11–18, 2006.
  5. M. Bartušek, M. Cecchi, Z. Došlá, and M. Marini, β€œOscillation for third-order nonlinear differential equations with deviating argument,” Abstract and Applied Analysis, vol. 2010, Article ID 278962, 19 pages, 2010. View at Zentralblatt MATH
  6. R. Bellman, Stability Theory of Differential Equations, McGraw-Hill, New York, NY, USA, 1953.
  7. T. A. Chanturija and I. T. Kiguradze, Asymptotic Properties of Nonautonomous Ordinary Differential Equations, Nauka, Moscow, Russia, 1990.
  8. J. Džurina, β€œAsymptotic properties of the third order delay differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 26, no. 1, pp. 33–39, 1996. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  9. J. Džurina, β€œComparison theorems for nonlinear ODEs,” Mathematica Slovaca, vol. 42, no. 3, pp. 299–315, 1992.
  10. J. Džurina and R. Kotorová, β€œProperties of the third order trinomial differential equations with delay argument,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1995–2002, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. L. Erbe, β€œExistence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations,” Pacific Journal of Mathematics, vol. 64, no. 2, pp. 369–385, 1976. View at Zentralblatt MATH
  12. P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1964.
  13. G. D. Jones, β€œAn asymptotic property of solutions of y′′′+p(x)y+q(x)y=0,” Pacific Journal of Mathematics, vol. 47, pp. 135–138, 1973. View at Zentralblatt MATH
  14. 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
  15. T. Kusano, M. Naito, and K. Tanaka, β€œOscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations,” Proceedings of the Royal Society of Edinburgh A, vol. 90, no. 1-2, pp. 25–40, 1981. View at Zentralblatt MATH
  16. A. C. Lazer, β€œThe behavior of solutions of the differential equation y′′′+p(x)y+q(x)y=0,” Pacific Journal of Mathematics, vol. 17, pp. 435–466, 1966. View at Zentralblatt MATH
  17. W. E. Mahfoud, β€œComparison theorems for delay differential equations,” Pacific Journal of Mathematics, vol. 83, no. 1, pp. 187–197, 1979. View at Zentralblatt MATH
  18. N. Parhi and S. Padhi, β€œOn asymptotic behavior of delay-differential equations of third order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 3, pp. 391–403, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  19. N. Parhi and S. Padhi, β€œAsymptotic behaviour of solutions of third order delay-differential equations,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 10, pp. 1609–1620, 2002. View at Zentralblatt MATH
  20. A. Škerlík, β€œIntegral criteria of oscillation for a third order linear differential equation,” Mathematica Slovaca, vol. 45, no. 4, pp. 403–412, 1995. View at Zentralblatt MATH
  21. K. Tanaka, β€œAsymptotic analysis of odd order ordinary differential equations,” Hiroshima Mathematical Journal, vol. 10, no. 2, pp. 391–408, 1980. View at Zentralblatt MATH
  22. W. F. Trench, β€œCanonical forms and principal systems for general disconjugate equations,” Transactions of the American Mathematical Society, vol. 189, pp. 319–327, 1973.