Behavior of solutions of a third-order dynamic equation on time scales
Journal of Inequalities and Applications volume 2013, Article number: 47 (2013)
In this paper, we will establish some sufficient conditions which guarantee that every solution of the third-order nonlinear dynamic equation
oscillates or converges to zero on an arbitrary time scale .
The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his Ph.D. Thesis  in order to unify continuous and discrete analysis. A time scale is an arbitrary nonempty closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. A book on the subject of time scales by Bohner and Peterson  summarizes and organizes much of the time scale calculus. Many other interesting time scales exist and they give rise to plenty of applications, among them the study of population dynamic models (see ). In the last few years, there has been much research activity concerning the oscillation and nonoscillation of solutions of some dynamic equations on time scales, and we refer the reader to the papers [3–21].
Following this trend, in this paper, we will consider the third-order nonlinear dynamic equation
where , are positive, real-valued, rd-continuous functions defined on the time scale interval (throughout with ). We assume throughout that:
is a function such that , for ,
is a function such that , for ,
are rd-continuous functions such that for all .
Since we are interested in oscillatory behavior, we suppose that the time scale under consideration is not bounded above, i.e., it is a time scale interval of the form . By a solution of (1), we mean a nontrivial real-valued function satisfying equation (1) for . A solution of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory. Our attention is restricted to those solutions of (1) which exist on some half-line and satisfy for any .
2 Some lemmas
In this section, we state and prove some lemmas which we will need in the proofs of our main results.
Lemma 2.1 Suppose that x is an eventually positive solution of (1) and
Then there is a such that either
, , , , or
, , , .
Proof Let x be an eventually positive solution of (1). Then there exists such that for . From (1) we have
for . Hence, is strictly decreasing on . We claim that on . Assume not, then there is a such that , . Then we can choose a negative constant c and such that for . Dividing by and integrating from to t, we obtain
Letting , then by (2). Thus, there is a such that for , . Dividing by and integrating from to t, we obtain
which implies that as by (2), a contradiction with the fact that . Hence, we have
This implies that is strictly increasing on . It follows from this that either on or is eventually positive and the proof is complete. □
Lemma 2.2 Assume that
where and is a solution of (1) that satisfies Case (ii) in Lemma 2.1. Then .
Proof Let x be a solution of (1) satisfying Case (ii) in Lemma 2.1, that is,
Then . Assume and we now show that this leads to a contradiction. From (1) and ,
where , for . Now let
then we have
Integrating the last inequality from to t, we have
Using (3) it is possible to choose a , sufficiently large, such that for all , which is a contradiction, and this completes the proof. □
Lemma 2.3 Assume that is a solution of (1) satisfying Case (i) of Lemma 2.1. Then there exists such that
Proof From Case (i) of Lemma 2.1, we have as a solution of (1) satisfying
for . Using is a solution of (1), we get and hence is decreasing on . Hence,
and this leads to (4) and the proof is complete. □
3 Main results
In this section, we establish some sufficient conditions which guarantee that every solution of (1) oscillates on or converges as .
Theorem 3.1 Assume that (2) holds. Furthermore, assume that there exists a positive function such that is rd-continuous on and if
for all sufficiently large , where , , then every solution of (1) is oscillatory or exists (finite).
Proof Let be a nonoscillatory solution of (1). We only consider the case when is eventually positive, since the case when is eventually negative is similar. By Lemma 2.1 either Case (i) or Case (ii) in Lemma 2.1 holds. Assume satisfies Case (i) in Lemma 2.1. Define the Riccati-type function by
By the product rule,
Using (1) we have
Using (4) and , we obtain
Since is decreasing, we have
where and . From (7) we have
Integrating (8) from to t, we obtain
for all large t. This is contrary to (5) and so Case (i) is not possible. If Case (ii) in Lemma 2.1 holds, then clearly exists (finite). □
On the basis of Lemma 2.2 and Theorem 3.1, we have the following results.
Corollary 3.2 If (2) and (3) hold, then every solution of (1) is either oscillatory or .
Corollary 3.3 Assume that (2) holds. If
then every solution of (1) is either oscillatory or exists.
We consider the third-order dynamic equation
where , , . Let and . It is not difficult to verify that all conditions of Corollary 3.3 are satisfied. Hence, every solution of equation (11) is oscillatory or satisfies .
Now we establish a Kamenev-type (see ) oscillation criterion for (1).
Theorem 3.5 Assume that (2) holds. Furthermore, assume that there is a positive function such that is rd-continuous on , and for all sufficient large ,
where , and are as in Theorem 3.1. Then every solution of (1) is either oscillatory or exists.
Proof Proceeding as in Theorem 3.1, we assume that (1) has a nonoscillatory solution, say for all , where is chosen so large that Lemma 2.1 and Lemma 2.3 hold. By Lemma 2.1 there are two possible cases. First, if Case (i) holds, then by defining again by (6) as in Theorem 3.1, we have and (8), that is,
Integrating by parts of the right-hand side leads to
where . Note that
and , for . It follows from (14) that
Then from (13) we have
which is a contradiction of (12). If Case (ii) holds, then, as before, exists and the proof is complete. □
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This work was supported by the Research Fund of the Erciyes University. Project Number: FBA-11-3391. The author is grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this paper.
The author declares that they have no competing interests.
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Şenel, M.T. Behavior of solutions of a third-order dynamic equation on time scales. J Inequal Appl 2013, 47 (2013). https://doi.org/10.1186/1029-242X-2013-47
- Dynamic Equation
- Positive Function
- Oscillatory Behavior
- Product Rule
- Scale Interval