Abstract
We deal with the oscillation of a generalized Emden-Fowler dynamic equation in the form . We establish some new oscillation criteria for the equation, which improve some of the main results of (H. Liu and P. Liu, 2013). Some examples are given to illustrate the new results.
1. Introduction
The theory of time scales has attracted a great deal of attention since it was first introduced by Hilger [1] in order to unify continuous and discrete analysis. For completeness, we recall the following concepts related to the notion of time scales; see [2, 3] for more details. A time scale is an arbitrary nonempty closed subset of the real numbers . In this paper, since we shall be concerned with the oscillatory behavior of solutions, we shall also assume that . We define the time scale interval by . The forward and backward jump operators are defined by where and ; here denotes the empty set. A point and is said to be left-dense if , right-dense if and , left-scattered if , and right-scattered if . The graininess function for the time scale is defined by , and for any function , the notation denotes . A function is said to be rd-continuous provided is continuous at right-dense points and at left-dense points in and left-hand limits exist and are finite. The set of all such rd-continuous functions is denoted by . We say that is differentiable at provided exists when (here by it is understood that approaches in the time scale) and when is continuous at and Note that if , then the delta derivative is just the standard derivative and when the delta derivative is just the forward difference operator. The set of functions which are differentiable and whose derivative is rd-continuous is denoted by .
In this paper, we consider the oscillatory behavior of the nontrivial solutions of the second-order Emden-Fowler dynamic equation of the form on an arbitrary time scale , with , where , and is a constant. Throughout this paper, we always assume that with ; with ;, , , and ; is a continuous function such that , for all and there exists a positive right-dense continuous function defined on such that for all and for all , where is a constant.
By a solution of (4), we mean a nontrivial real-valued function , which has the property that and satisfies (4) that holds on . The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution of (4) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is said to be nonoscillatory. The equation itself is said to be oscillatory if all its solutions are oscillatory.
Recently, there has been an increasing interest in studying the oscillation behavior of second-order dynamic equations on time scales; see for example [4–10] and the references contained therein. In [5], the authors presented some criteria for the oscillation and asymptotic behavior of (4) in the case where Also, we note further that in the proof of [5], the authors used the chain rule in the form
So the natural question which arises is can we find some new oscillation conditions for (4) which do not require (5) and (6) and, in addition, improve the main results in [5]?
The purpose of this paper is to give an affirmative answer to this question. That is, we shall establish some new criteria for the oscillation of (4) which improve the main results in [5]. We also demonstrate that our results cover certain cases which were not covered in [5]. Finally, we give two examples to illustrate the main results.
2. Main Results
For notational simplicity, define
We begin with the following lemmas.
Lemma 1. Assume that (4) has a positive solution on . Then for sufficiently large , one has
Proof. Assume that (4) has a nonoscillatory solution on . Without loss of generality, we assume that there exists a such that on . Then it follows that . From (4), we have Hence, is decreasing on . We now claim that on . If not, then there exists a such that . Therefore, that is, Integrating (11) from to , we find from that which implies that is eventually negative. This contradicts the fact that on . Thus, on . This completes the proof.
Lemma 2. Assume that (4) has a positive solution on . Then for sufficiently large ,
Proof. As in the proof of Lemma 1, there is a so that Since is decreasing on , we can choose so that , for . Then consequently, Also, we have, for hence, Therefore, by combining inequalities (16) and (18) we have from which we have This completes the proof.
Lemma 3 (see [11]). Let , where , , , and . Then
For the positive solution of (4), it follows from and Lemma 1 that, for , which implies Combining (23) , (4) one obtains where .
One may now state and prove the main results. In these, one shall consider the two cases and .
Theorem 4. Let . Assume that there exist a positive rd-continuous differentiable function and a constant such that, for some , where . Then (4) is oscillatory on .
Proof. Let be a nonoscillatory solution of (4) on . Without loss of generality, we assume that there exists a (sufficiently large) such that on , and satisfies the conclusions of Lemmas 1 and 2 on . Consider the Riccati substitution
Then . By [2, Theorem 1.20], Lemma 2, and (24), we have
By the Pötzsche chain rule [2, Theorem 1.87],
Thus,
Noting that is increasing on , we get for . Thus,
Substituting (30) into (27), we obtain
Noting that is decreasing, we have . It follows from the definition of that
Substituting (32) into (31), we obtain
Since is decreasing, there exists a constant such that for , which implies
Integrating both sides of (34) from to , we get
Hence, there exists a such that for . Then,
where . Substituting (36) into (33), we get
where
Taking , , from Lemma 3 and (37), we obtain
where . Integrating both sides of (39) from to , we have
Taking of both sides of this last inequality as , we get a contradiction to (25). This completes the proof.
Theorem 5. Let . Assume that there exist a positive rd-continuous differentiable function and a constant such that, for some , where is defined as Theorem 4. Then (4) is oscillatory on .
Proof. Assume that is a nonoscillatory solution of (4). Proceeding as in the proof of Theorem 4 we get that (33) holds, that is, Since and is increasing on , then there exist a and a positive constant such that for . Consequently, Let then , and The remainder of the proof is similar to that of Theorem 4 and is therefore omitted. This completes the proof for the case .
Remark 6. Theorems 4 and 5 remove the Conditions (5) and (6). Moreover, the authors in [5] established oscillation theorems for (4) only for the case . Our results here hold without this assumption, so our results improve the main results [5].
Remark 7. The results established here are valid for general time scales, with no additional restrictions, for example, , , and with , with , and ; see [2, 3].
3. Some Examples
In this section, we give two examples to illustrate our main results.
Example 1. Let (), , . Consider the neutral nonlinear dynamic equation where satisfies , and .
Here, It is clear that holds, and , .
Let . Noting that implies for , we get Thus, by Theorem 4, (46) is oscillatory.
Example 2. Consider the neutral dynamic equation where are constants, satisfies , and .
For (4), we let Since then holds and .
Let . Noting that implies for , we have Thus, by Theorem 5, (49) is oscillatory.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Qiaoshun Yang was supported by The Foundation of Hunan Educational Department (no. 13C753) and The Main Foundation of Jishou University (no. 2012JSUJGA23). The work of Lynn Erbe was performed during a visit to Zhongshan (Sun Yat-sen) University in Guangzhou, China. Baoguo Jia was supported by The National Natural Science Foundation of China (no. 11271380); The Guangdong Province Key Laboratory of Computational Science; and The Guangdong Province Natural Science Foundation (S2013010013050).