International Journal of Mathematics and Mathematical Sciences

VolumeΒ 2012, Article IDΒ 951898, 9 pages

http://dx.doi.org/10.1155/2012/951898

## Oscillation and Asymptotic Behavior of Higher-Order Nonlinear Differential Equations

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: FengΒ Qi

Copyright Β© 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 2.2Setting 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.

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