- Open Access
Behavior of solutions of a third-order dynamic equation on time scales
© Şenel; licensee Springer 2013
- Received: 6 November 2012
- Accepted: 4 February 2013
- Published: 12 February 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 .
- Dynamic Equation
- Positive Function
- Oscillatory Behavior
- Product Rule
- Scale Interval
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].
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 .
In this section, we state and prove some lemmas which we will need in the proofs of our main results.
, , , , or
, , , .
This implies that is strictly increasing on . It follows from this that either on or is eventually positive and the proof is complete. □
where and is a solution of (1) that satisfies Case (ii) in Lemma 2.1. Then .
Using (3) it is possible to choose a , sufficiently large, such that for all , which is a contradiction, and this completes the proof. □
and this leads to (4) and the proof is complete. □
In this section, we establish some sufficient conditions which guarantee that every solution of (1) oscillates on or converges as .
for all sufficiently large , where , , then every solution of (1) is oscillatory or exists (finite).
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 .
then every solution of (1) is either oscillatory or exists.
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).
where , and are as in Theorem 3.1. Then every solution of (1) is either oscillatory or exists.
which is a contradiction of (12). If Case (ii) holds, then, as before, exists and the proof is complete. □
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.
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