International Journal of Differential Equations

Volume 2014, Article ID 791631, 6 pages

http://dx.doi.org/10.1155/2014/791631

## On the Oscillation of Even-Order Half-Linear Functional Difference Equations with Damping Term

^{1}Department of Mathematics, Faculty of Science and Literatures, Kastamonu University, 037100 Kastamonu, Turkey^{2}Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia

Received 6 February 2014; Revised 2 May 2014; Accepted 6 May 2014; Published 19 May 2014

Academic Editor: S. R. Grace

Copyright © 2014 Yaşar Bolat and Jehad Alzabut. 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

We investigate the oscillatory behavior of solutions of the th order half-linear functional difference equations with damping term of the form , , where is even and , is a fixed real number. Our main results are obtained via employing the generalized Riccati transformation. We provide two examples to illustrate the effectiveness of the proposed results.

#### 1. Introduction

Consider the second order half-linear difference equation: where is the forward difference operator and , are sequences of nonnegative real numbers with . The study of (1) has been initiated by Rehák in [1]. It is well known that there is a close similarity between (1) and the linear second order difference equation. Indeed, if is a solution of (1), then so is for any constant . Thus, (1) has one half of linearity properties [2].

In the presence of damping, (1) has been extended further to the second order half-linear difference equation with damping term of the form where is a sequence of nonnegative real numbers. It is to be noted that neither (1) nor (2) has involved a delaying term. There are numerous numbers of oscillation criteria established in the literature for the solutions of (1) and (2). Most of these results were obtained by using certain efficient tools among them we name the Riccati transformation, variational principle, and some inequality techniques; see, for instance, the monograph [3] in which many contributions have been cited therein and to the recent papers [4–9].

Let be defined by ; is a fixed real number and . Consider the th order half-linear functional difference equation with damping term of the form where is even number, and (H1) with for all ;(H2) and with and ;(H3) with and .For close results regarding the continuous counterparts of (1), (2), and (3), the reader is suggested to consult [10–14].

A primary purpose of this paper is to establish sufficient conditions that guarantee the oscillation of solutions of (3). Our main results are obtained via employing the generalized Riccati transformation. In view of (3), one can easily figure out that it is formulated in more general form so that it includes some particular cases which have been studied in the literature; see [15–23] for more details. To the best of authors’ observation, however, no published result has been concerned with the investigation of oscillatory behavior of solutions of (3) or its continuous counterpart. Therefore, our paper is new and presents a new approach.

#### 2. Main Results

We start by recalling the following standard definitions.

*Definition 1. *A nontrivial sequence is called a solution of (3) if it is defined for all where , , and is differenceable on and satisfies (3) for all .

*Definition 2. *A nontrivial solution of (3) is said to be oscillatory if the terms of the sequence are not eventually positive or not eventually negative. Otherwise, the solution is called nonoscillatory. A difference equation is called oscillatory if all its solutions oscillate.

To obtain our main results, we need the following essential lemmas. The first of these is the discrete analogue of the well-known Kiguradze’s lemma.

Lemma 3 (see [24]). *Let be defined for and with of constant sign for and not identically zero. Then, there exists an integer , with odd for and even for such that*(i)* implies for all , ,*(ii)* implies for all large , .*

Lemma 4 (see [25]). *Let be defined for and with for and not identically zero. Then, there exists a large integer such that
**
where is defined as in Lemma 3. Further, if is increasing, then
*

Lemma 5. *Let satisfy conditions of Lemmas 3 and 4 and for . Further, if is increasing, then
**
where .*

The proof of Lemma 5 is straightforward and it can be achieved by using the last inequality of Lemma 4.

Lemma 6. *Let be an eventually positive solution of (3). If
**
then , , and for all .*

*Proof. *The fact that is eventually positive solution of (3) implies and for all . In view of (3), we get
which leads to
Hence,
is decreasing and is eventually positive or eventually negative.

We claim that
Assume, on the contrary, that , . Then, from (10), we obtain
where . Therefore, from (12), we have
where . It follows that
or
Consequently, we obtain
Letting in the above inequality, one gets . Hence, is an eventually negative function which contradicts that . Therefore, inequality (11) holds.

From (3), we get
from which it follows that
The above inequality implies that is nonincreasing. Therefore, we can write
Since is nonincreasing and positive, then from the above inequality, we have
by which we have
In virtue of (21) and Lemma 3, we deduce that since is even then is odd. Hence for . The proof is complete.

Theorem 7. *Let condition hold. Further, assume that there exists a constant such that
**
where
**
and is as in Lemma 5. Then, (3) is oscillatory.*

*Proof. *For the sake of contradiction, assume that (1) has a nonoscillatory solution . Without loss of generality, we assume that is eventually positive (the proof is similar when is eventually negative). That is, , and for all . By Lemma 6, we have , , and for . Consider the function
Taking into account that and is increasing and , we deduce that and is nonincreasing. Lemmas 3 and 4, (1), and (24) yield
Multiplying by and summing up from to , we obtain
or
where
Let
Then, has maximum value at . That is,
Therefore, (27) can be rewritten as
Hence, we have
which contradicts condition . The proof is complete.

Theorem 8. *Let condition hold. Further, assume that there exists a function such that
**
where is as in Lemma 5. Then, (3) is oscillatory.*

*Proof. *For the sake of contradiction, assume that (3) has a nonoscillatory solution . Without loss of generality, we assume that is eventually positive (the proof is similar when is eventually negative). That is, , and for all . By Lemma 6, we have , , and for . Consider the function
By utilizing the same approach as in the proof of Theorem 7, we arrive at
Summing up (35) from to , we have
Letting in the above inequality and taking the upper limit, we get a contradiction to . The proof is complete.

*Remark 9. *In view of the statements of Theorems 7 and 8, one can easily deduce that condition is a generalization of .

*Example 10. *Consider the fourth order half-linear functional difference equation with damping
where , , , , , and . It is easy to see that conditions (H1)*–*(H3) are satisfied. It remains to check the validity of conditions and .

For , we have
It is clear that as . Therfore, condition holds. For and , we have
where
It is clear that as . Then, condition holds. Thus, by the conclusion of Theorem 7, (37) is oscillatory.

*Example 11. *Consider the sixth order half-linear functional difference equation with damping
where , , , , , and . It is easy to see that conditions (H1)*–*(H3) are satisfied. In Example 10, we have seen that is satisfied. It remains to check the validity of condition .

For and , we have
It is clear that as . Then, condition holds. Thus, by the conclusion of Theorem 8, (41) is oscillatory.

*Remark 12. *It is not possible to decide the oscillatory behavior of solutions of (37) and (41) by using any of the results reported in [12, 13]. This implies that the results of our paper extend and generalize some known theorems.

*Remark 13. *The main results of this paper remain valid for nondelay difference equations of the form

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The authors would like to express thier sincere thanks to the referee for pointing out several suggestions and corrections that helped making the contents of this paper more accurate.

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