International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 951898, 9 pages

doi:10.1155/2012/951898

Research Article

Oscillation and Asymptotic Behavior of Higher-Order Nonlinear Differential Equations

J. Džurina and B. Baculíková

Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00 Košice, Slovakia

Received 20 March 2012; Accepted 11 June 2012

Academic Editor: Feng Qi

Copyright © 2012 J. Džurina and B. Baculíková. 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 a generalization of the Philos and Staikos lemma. As a possible application of the lemma in the oscillation theory, we study the asymptotic properties and oscillation of the th order delay differential equations . The results obtained utilize also the comparison theorems.

1. Introduction

In this paper, we will study the asymptotic and oscillation behavior of the solutions of the higher-order advanced differential equations:

Throughout the paper, we will assume , and

, is the ratio of two positive odd integers, , , , .

Whenever, it is assumed

By a solution of we mean a function ,  , which has the property and 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 called to be nonoscillatory.

The problem of the oscillation of higher-order differential equations has been widely studied by many authors, who have provided many techniques for obtaining oscillatory criteria for studied equations (see, e.g., [1–19]).

Philos and Staikos lemma (see [16, 17]) essentially simplifies the examination of th-order differential equations of the form

since it provides needed relationship between and , and this fact permit us to establish just one condition for asymptotic behavior of (1.2). If we try to apply the Philos and Staikos lemma to , the strong condition appears (see, e.g., [2, 18, 20]). In this paper we offer such generalization of the Philos and Staikos lemma, where this restriction is relaxed. Moreover, the obtained lemma yields many applications in the oscillation theory. As an example of it, we offer its disposal in the comparison theory and we establish new oscillation criteria for .

2. Main Results

The following result is a well-known lemma of Kiguradze, see, for example, [6] or [13].

Lemma 2.1. Let and with , , and not identically zero on a subray of . Then there exist a and an integer , , with odd so that

on .

Now we are prepared to provide a generalization of the Philos and Staikos lemma.

Lemma 2.2. Let be as in Lemma 2.1 and numbers and assigned to by Lemma 2.1. Then for ,

for ;

for .

Proof. Let be the integer assigned to function as in Lemma 2.1. Assume that , then, for any with ,

Repeated integration in from to yields

On the other hand, if , then, for every ,

Repeated integration from to leads to

Setting (2.5) into (2.7), one gets

We have verified the first part of the lemma. Now assume that . It follows from (2.7) that

The proof is complete now.

Employing additional conditions, we are able to joint (2.5) and (2.7) to just one estimate.

Lemma 2.3. Let be as in Lemma 2.1 and . Let ,  Then for any there exists some such that

for .

Proof. Note that implies that is nonincreasing. Assume that is the integer associated with in Lemma 2.1. If , then using (2.2), we have

It is easy to see that for any there exists a such that for , which in view of (2.11) yields (2.10).

If , then proceeding similarly as above it can be shown that (2.3) implies (2.10).

If , then it follows from (2.5) that

Setting , we have

Moreover,

Therefore, for all large ,

The proof is complete now.

In the following result, we compare (2.2) and (2.3) to verify that both estimates are independent; that is, one does not result from the other, although for they are equivalent.

Remark 2.4. Let be as in Lemma 2.1, such that is increasing. At first we consider , , , and then (2.2), (2.3), and (2.10) yield

respectively.

Now we modify . Then (2.2) and (2.3) reduce to

respectively, and (2.10) is not applicable.

3. Applications

Lemma 2.2 can be applied in various techniques for investigations of the higher-order differential equations. We offer one such application in comparison theory.

Theorem 3.1. Assume that both first-order delay differential equations

are oscillatory. Moreover, for -odd assume that

Then

(i)for even, is oscillatory; (ii)for odd, every nonoscillatory solution of satisfies .

Proof. Assume that is a nonoscillatory solution of ; let us say positive. Then , and there exist a and an integer with odd such that (2.1) holds.

If , then by Lemma 2.2

Setting to , we get

Then is positive and satisfies the differential inequality:

By Theorem 1 in [15], the corresponding equation has also a positive solution. A contradiction.

If , then by Lemma 2.2

and proceeding as above, we find out that has a positive solution. A contradiction and the proof are finished for even.

Assume that ; note that it is possible only for is odd. Since , then there exists a finite . We claim that . If not, then , eventually, let us say, for . An integration of from to yields

or equivalently

Integrating times from to , we get

then

which contradicts . The proof is complete.

Employing any result (e.g., Theorem 2.1.1 in [13]) for the oscillation of and , we immediately obtain criteria for studied properties of .

Corollary 3.2. Let

Moreover, for -odd assume that holds. Then

(i)for even, is oscillatory; (ii)for odd, every nonoscillatory solution of satisfies .

If we replace by stronger condition, we can establish oscillation of even if is odd.

Theorem 3.3. Let . Assume that both first-order delay differential equations and are oscillatory. Moreover, for -odd assume that

Then is oscillatory.

Proof. Assume that is a positive solution of . Then there exists an integer assigned with by Lemma 2.1. If , then proceeding as in the proof of Theorem 3.1 we eliminate . If , then it follows from (3.7) that

which contradicts . The proof is complete.

Corollary 3.4. Let (3.9) hold. Moreover, for -odd assume that holds. Then is oscillatory.

Example 3.5. We consider the forth-order delay differential equation:

It is easy to verify that holds for ; moreover, conditions (3.9) reduce to

respectively. Thus, by Corollary 3.4, if both (3.11) and (3.12) hold, then all nonoscillatory solutions of tend to zero. For , with , one such solution is . On the other hand, condition takes the form

Therefore, by Corollary 3.4, is oscillatory, provided that all (3.11), (3.12), and (3.13) hold.

Acknowledgments

This work is the result of the project implementation: Development of the Center of Information and Communication Technologies for Knowledge Systems (ITMS project code: 26220120030) supported by the Research & Development Operational Program funded by the ERDF.

References

B. Baculíková and J. Džurina, “Oscillation of third-order neutral differential equations,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 215–226, 2010.

B. Baculíková, J. Graef, and J. Dzurina, “On the oscillation of higher order delay differential equations,” Nonlinear Oscillations, vol. 15, no. 1, pp. 13–24, 2012.

B. Baculíková and J. Džurina, “Oscillation of third-order nonlinear differential equations,” Applied Mathematics Letters, vol. 24, no. 4, pp. 466–470, 2011.

B. Baculíková, “Properties of third-order nonlinear functional differential equations with mixed arguments,” Abstract and Applied Analysis, vol. 2011, Article ID 857860, 15 pages, 2011.

B. Baculíková and J. Džurina, “Oscillation of third-order functional differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 43, pp. 1–10, 2010.

J. Džurina, “Comparison theorems for nonlinear ODEs,” Mathematica Slovaca, vol. 42, no. 3, pp. 299–315, 1992.

L. H. Erbe, Q. Kong, and B. G. Zhang, Oscillation Theory for Functional-Differential Equations, vol. 190, Marcel Dekker, New York, NY, USA, 1994.

S. R. Grace, R. P. Agarwal, R. Pavani, and E. Thandapani, “On the oscillation of certain third order nonlinear functional differential equations,” Applied Mathematics and Computation, vol. 202, no. 1, pp. 102–112, 2008.

S. R. Grace and B. S. Lalli, “Oscillation of even order differential equations with deviating arguments,” Journal of Mathematical Analysis and Applications, vol. 147, no. 2, pp. 569–579, 1990.

T. Li, Z. Han, P. Zhao, and S. Sun, “Oscillation of even-order neutral delay differential equations,” Advances in Difference Equations, vol. 2010, Article ID 184180, 9 pages, 2010.

I. T. Kiguradze and T. A. Chaturia, Asymptotic Properties of Solutions of Nonatunomous Ordinary Dierential Equations, Kluwer Academic, Dordrecht, The Netherlands, 1993.

T. Kusano and M. Naito, “Comparison theorems for functional-differential equations with deviating arguments,” Journal of the Mathematical Society of Japan, vol. 33, no. 3, pp. 509–532, 1981.

G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, vol. 110, Marcel Dekker, New York, NY, USA, 1987.

W. E. Mahfoud, “Oscillation and asymptotic behavior of solutions of th order nonlinear delay differential equations,” Journal of Differential Equations, vol. 24, no. 1, pp. 75–98, 1977.

C. G. Philos, “On 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.

C. G. Philos, “Oscillation and asymptotic behavior of linear retarded dierential equations of arbitrary order,” Tech. Rep. 57, University of Ioannina, 1981.

C. G. Philos, “On the existence of nonoscillatory solutions tending to zero at 1 for differential equations with positive delay,” Journal of the Australian Mathematical Society, vol. 36, pp. 176–186, 1984.

C. Zhang, T. Li, B. Sun, and E. Thandapani, “On the oscillation of higher-order half-linear delay differential equations,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1618–1621, 2011.

Q. Zhang, J. Yan, and L. Gao, “Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 426–430, 2010.

G. Xing, T. Li, and C. Zhang, “Oscillation of higher-order quasi-linear neutral differential equations,” Advances in Difference Equations, vol. 2011, pp. 1–10, 2011.