Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 730128, 10 pages

http://dx.doi.org/10.1155/2011/730128

## 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 , 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 Throughout the paper, we assume that and(i), , , ,(ii), , (iii) as .

By a solution of (), we mean a function ,ββ, that satisfies () on . We consider only those solutions of () which satisfy for all . We assume that () possesses such a solution. A solution of () is called oscillatory if it has arbitrarily large zeros on , and, otherwise, it is nonoscillatory. Equation () 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 [1β22]). 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 (), namely, the ordinary differential equation (without delay) has been investigated, and sufficient conditions for all its nonoscillatory solutions to satisfy or the stronger condition are presented. It is known that () 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 () satisfy (1.1) and (1.2). For our further references we define as following.

*Definition 1.1. *We say that () has property () if its every nonoscillatory solution 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 () 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 large enough.

#### 2. Main Results

It will be derived that properties of () are closely connected with the corresponding second-order differential equation as the following theorem says.

Theorem 2.1. *Let be a positive solution of (). Then () can be written as
*

*Proof. *The proof follows from the fact that

Now, in the sequel, instead of studying properties of the trinomial equation (), we will study the behavior of the binomial equation (). For our next considerations, it is desirable for () to be in a canonical form; that is, because properties of the canonical equations are nicely explored.

Now, we will study the properties of the positive solutions of () 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
**
then () possesses a positive solution .*

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

Lemma 2.3. *If is a positive solution of (), then , , and, what is more, (2.2) holds and there exists such that .*

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

Lemma 2.4. *If is a positive solution of () for which (2.3) is violated, then
**
is another positive solution of () and, for , (2.3) holds.*

*Proof. *First note that
Thus, is a positive solution of (). On the other hand, to insure that (2.3) holds for , let us denote . Then and

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

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

Now we can study properties of () with help of its canonical representation (). For our reference, let us denote for () Now, () can be written as .

We present a structure of the nonoscillatory solutions of (). Since () is in a canonical form, it follows from the well-known lemma of Kiguradze (see, e.g., [7, 9, 14]) that every nonoscillatory solution of () is either of *degree 0*, that is,
or of *degree 2*, that is,

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

Now we verify that property of () and property of () are equivalent in the sense that satisfies (1.1) if and only if it obeys (2.9).

Theorem 2.7. *Let (2.4) hold. Assume that is a positive solution of () satisfying (2.2)-(2.3). Then () has property if and only if () has property .*

*Proof. * We suppose that is a positive solution of (). We need to verify that . Since is also a solution of (), then it satisfies (2.9). Therefore, .

Assume that is a positive solution of (). We will verify that (2.9) holds. Since is also a solution of (), we see that ; that is, . It follows from () that . Thus, is decreasing. If we admit eventually, then is decreasing, and integrating the inequality , we get as . Therefore, and (2.9) holds.

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

Theorem 2.8. *Let (2.4) hold. Assume that is a positive solution of () satisfying (2.2)-(2.3). Let
**
If
**
has property , then so does ().*

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

Corollary 2.9. *Let (2.4) and (2.11) hold. Assume that is a positive solution of () satisfying (2.2)-(2.3). If () has property then () has property .*

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

*Example 2.10. *We consider the third-order delay trinomial differential equation
where and satisfies (2.11). The corresponding equation () takes the form
and it has the pair of the solutions and . Thus, is our desirable solution, which permits to rewrite (2.12) in its canonical form. Then, by Corollary 2.9, (2.12) has property if the equation
has property .

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

Corollary 2.11. *Let (2.4) and (2.11) hold. Assume that is a positive solution of () satisfying (2.2)-(2.3). If () has property and
**
then every nonoscillatory solution of () satisfies (1.2).*

*Proof. *Assume that is a positive solution of (). Then, it follows from Corollary 2.9 that . Therefore, . Assume . On the other hand, is also a solution of (), and, in view of Theorem 2.7, it has to be of * degree 0*; that is, (2.9) is fulfilled. Then, integrating () from to , we get
Multiplying this inequality by and then integrating from to , we have
Multiplying this by and then integrating from to , we obtain
This is a contradiction, and we deduce that . 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
Then, by Corollary 2.11, every nonoscillatory solution of (2.12) tends to zero as provided that (2.19) holds and (2.14) has property .

In the second part of this paper, we derive criteria that enable us to deduce property of () 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 is a positive solution of () satisfying (2.2)-(2.3). Let
**
If the second-order equation
**
is oscillatory, then () has property .*

*Proof. *Assume that is a positive solution of (), then is either of * degree 0* or of * degree 2*. Assume that is of *degree 2*; that is, (2.10) holds. An integration of () yields
On the other hand,
Combining the last two inequalities, we get
Integrating the previous inequality from to , we see that satisfies
Denoting the right-hand side of (2.24) by , it is easy to see that and
By Theorem 2 in [14], the corresponding equation () also has a positive solution. This is a contradiction. We conclude that is of *degree 0*; that is, () 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 is a positive solution of () satisfying (2.2)-(2.3). If
**
then () has property (A). *

*Proof. *Assume that is a positive solution of () and is of * degree 2*. An integration of () yields
which is a contradiction. Thus, 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 of ().

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

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

Corollary 2.16. *Let (2.4), (2.11), and (2.20) hold. Assume that is a positive solution of () satisfying (2.2)-(2.3). If
**
then () has property .*

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

Imposing an additional condition on the coefficients of (), we can obtain that every nonoscillatory solution of () tends to zero as .

Corollary 2.17. *Let (2.4) and (2.11) hold. Assume that is a positive solution of () satisfying (2.2)-(2.3). If (2.28) and (2.15) hold, then every nonoscillatory solution of () 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 provided that (2.19) holds and
For a special case of (2.12), namely, for
with , , and , we get that every nonoscillatory solution of (2.30) tends to zero as provided that
If we set , where , then one such solution of (2.12) is .

On the other hand, if for some we have , then (2.31) is violated and (2.12) has a nonoscillatory solution 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 2.15β2.17 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

- B. Baculíková, E. M. Elabbasy, S. H. Saker, and J. Džurina, βOscillation criteria for third-order nonlinear differential equations,β
*Mathematica Slovaca*, vol. 58, no. 2, pp. 201β220, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B. Baculíková, βOscillation criteria for second order nonlinear differential equations,β
*Archivum Mathematicum*, vol. 42, no. 2, pp. 141β149, 2006. View at Google Scholar Β· View at Zentralblatt MATH - B. Baculíková, βOscillation of certain class of the third order neutral differential equations,β
*Abstract and Applied Analysis*. In press. - B. Baculíková and D. Lacková, βOscillation criteria for second order retarded differential equations,β
*Studies of the University of Žilina. Mathematical Series*, vol. 20, no. 1, pp. 11β18, 2006. View at Google Scholar - M. Bartušek, M. Cecchi, Z. Došlá, and M. Marini, βOscillation for third-order nonlinear differential equations with deviating argument,β
*Abstract and Applied Analysis*, vol. 2010, Article ID 278962, 19 pages, 2010. View at Google Scholar Β· View at Zentralblatt MATH - R. Bellman,
*Stability Theory of Differential Equations*, McGraw-Hill, New York, NY, USA, 1953. - T. A. Chanturija and I. T. Kiguradze,
*Asymptotic Properties of Nonautonomous Ordinary Differential Equations*, Nauka, Moscow, Russia, 1990. - J. Džurina, βAsymptotic properties of the third order delay differential equations,β
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 26, no. 1, pp. 33β39, 1996. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Džurina, βComparison theorems for nonlinear ODEs,β
*Mathematica Slovaca*, vol. 42, no. 3, pp. 299β315, 1992. View at Google Scholar - J. Džurina and R. Kotorová, βProperties of the third order trinomial differential equations with delay argument,β
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 5-6, pp. 1995β2002, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - L. Erbe, βExistence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations,β
*Pacific Journal of Mathematics*, vol. 64, no. 2, pp. 369β385, 1976. View at Google Scholar Β· View at Zentralblatt MATH - P. Hartman,
*Ordinary Differential Equations*, John Wiley & Sons, New York, NY, USA, 1964. - G. D. Jones, βAn asymptotic property of solutions of ${y}^{\mathrm{\prime \prime \prime}}+p(x){y}^{\prime}+q(x)y=0$,β
*Pacific Journal of Mathematics*, vol. 47, pp. 135β138, 1973. View at Google Scholar Β· View at Zentralblatt MATH - T. Kusano and M. Naito, βComparison theorems for functional-differential equations with deviating arguments,β
*Journal of the Mathematical Society of Japan*, vol. 33, no. 3, pp. 509β532, 1981. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. Kusano, M. Naito, and K. Tanaka, βOscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations,β
*Proceedings of the Royal Society of Edinburgh A*, vol. 90, no. 1-2, pp. 25β40, 1981. View at Google Scholar Β· View at Zentralblatt MATH - A. C. Lazer, βThe behavior of solutions of the differential equation ${y}^{\mathrm{\prime \prime \prime}}+p(x){y}^{\prime}+q(x)y=0$,β
*Pacific Journal of Mathematics*, vol. 17, pp. 435β466, 1966. View at Google Scholar Β· View at Zentralblatt MATH - W. E. Mahfoud, βComparison theorems for delay differential equations,β
*Pacific Journal of Mathematics*, vol. 83, no. 1, pp. 187β197, 1979. View at Google Scholar Β· View at Zentralblatt MATH - N. Parhi and S. Padhi, βOn asymptotic behavior of delay-differential equations of third order,β
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 34, no. 3, pp. 391β403, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - N. Parhi and S. Padhi, βAsymptotic behaviour of solutions of third order delay-differential equations,β
*Indian Journal of Pure and Applied Mathematics*, vol. 33, no. 10, pp. 1609β1620, 2002. View at Google Scholar Β· View at Zentralblatt MATH - A. Škerlík, βIntegral criteria of oscillation for a third order linear differential equation,β
*Mathematica Slovaca*, vol. 45, no. 4, pp. 403β412, 1995. View at Google Scholar Β· View at Zentralblatt MATH - K. Tanaka, βAsymptotic analysis of odd order ordinary differential equations,β
*Hiroshima Mathematical Journal*, vol. 10, no. 2, pp. 391β408, 1980. View at Google Scholar Β· View at Zentralblatt MATH - W. F. Trench, βCanonical forms and principal systems for general disconjugate equations,β
*Transactions of the American Mathematical Society*, vol. 189, pp. 319β327, 1973. View at Google Scholar