Oscillatory Behavior of Even-Order Half-Linear Neutral Differential Equations

Abstract

This paper discusses some sufficient conditions for oscillatory behavior of even-order half-linear neutral differential equation. An example is given to illustrate the main result.

1. Introduction

In this paper, we investigate the oscillatory behavior of even-order half-linear neutral differential equationwhere is an even integer and with is the ratio of two odd positive integers.

Throughout this paper, we assume that the following conditions are satisfied:

p, q, r and

for all , with and .

By a solution of (1), we mean a function for some which has a property and satisfies (1) on .We consider the solution of (1) on some half-line and satisfy the condition as follows:

Such solutions can be called as oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is non-oscillatory.

The oscillation problem for neutral differential (1) has been studied by many authors with various techniques. Most of the results are obtained for the case , see [1–6], , see [7, 8], and , see [2, 5, 6, 8–12]. The purpose of this paper is to extend the main results for using the Philos function method. Furthermore, the oscillation conditions are obtained in order to make it easy to apply. Also, it is our hope that the present paper will be a good contribution to study the oscillatory behavior of solutions of the even-order neutral differential equations. An example is provided to illustrate the main result.

2. Main Results

To prove our main results, we use the following notations:

We introduce a function class P by following Philos [13]. Let and . We say that the function belongs to the class P, denoted by if(i) for , and on (ii)H has a continuous and nonpositive partial derivative on with respect to the second variable

Theorem 1. Let the conditions hold. Let h, H : G be continuous functions such that H belongs to the class P and

If there exists a positive function such that for some ,for sufficiently large , wherefor some , then every solution of (1) is oscillatory.

Proof. Let be a positive solution of (1). Then, is positive for all . From equation (1), we get thatLet ; then, from (8), we get thatwhich says that is decreasing and does not change its sign which leads to two cases: (I)  (II) Now, we consider that Case (I) holds. If , then we obtainwhere . Now, we haveIntegrating the last inequality with respect to , we getAs , we have , and consequently, we see that which is a contradiction.
Now, let us take Case (II).
Since , from Lemma 1 of [5], we have . From the definition of , we see that Then, (1) becomesWe consider the Riccati transformationWe see that , and from (16), we getFrom (18), we getThe above inequality can be written asMultiplying (19) by and integrating from , we have for some and for all Integrating by parts yieldsSubstituting (5) and (21) into (20), we getNow, by applying the inequality, (22), we getSo, for every , we obtainwhich is a contradiction.
Next, oscillation result follows immediately from Theorem 1.

Corollary 1. Let the assumptions of Theorem 1 be holded except that the condition (4) is replaced by

Then, every solution of (1) is oscillatory.

Theorem 2. Let the conditions and (2) be satisfied. Let H and h be same as in Theorem 1. We suppose that

If there exist functions and such that for some  for all sufficiently large and all , where and are as in Theorem 1 and , then every solution of (1) is oscillatory.

Proof. Let be a non-oscillatory solution of (1), say and for for some . Proceeding as in the proof of Theorem 1, we again have two cases: (I)  (II) For , if Case (I) holds proceeding as in the proof of Theorem 1, we obtain a contradiction to the positivity of y.
Next, we assume case (II) holds, proceeding as in the proof of Theorem 1; we again arrive at (23), which can be written as .From (29), we haveIn view of (27), it follows from (30) thatfor all for any . Then, for all , Now, we claim thatIf not, thenBy (26), there exists a constant such thatOn the other hand, by virtue of (35) for any positive number , there exists such thatUsing integration by parts and taking (37) into account, we conclude that for all ,It follows from (36) thatand hence, there exists such thatFrom the latter inequality and (32), we see thatSince is an arbitrary positive constant, we obtainwhich contradicts (33). Thus, (34) should hold and so by (32), we havewhich contradicts (28). This completes the proof of the theorem.

Theorem 3. Let all the conditions of Theorem 2 be satisfied except the condition (21) be replaced withthen every solution of (1) is oscillatory.

Proof. The proof easily follows from the fact thatHence the proof is complete.

3. Example

In this section, we provide an example to illustrate the main result.

Example 1. We consider the fourth-order neutral delay differential equationwhere .

Hence, we have .

Simple calculation shows that by taking , that .

Letting , we see that , and

We choose , and the condition (46) becomes

Hence, all conditions of Theorem 1 are satisfied, and therefore, every solution of (46) is oscillatory.

It is noted that the conditions of Corollary 1 are not satisfied, and hence, Theorem 1 improves Corollary 1.

4. Conclusion

In this paper, we have obtained criteria for the oscillation of all solutions of (1) using the Philos type method. An example is provided to illustrate the main result. It will be of interest to study the equation under the condition [14].

It is also interesting to extend the results of this paper to stochastic type neutral differential equations and differential equations with impulses, see [15, 16].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.