- Research Article
- Open Access
Oscillation of Second-Order Sublinear Dynamic Equations with Damping on Isolated Time Scales
© Q. Lin and B. Jia. 2010
- Received: 8 October 2010
- Accepted: 27 December 2010
- Published: 30 December 2010
This paper concerns the oscillation of solutions to the second sublinear dynamic equation with damping , on an isolated time scale which is unbounded above. In , α is the quotient of odd positive integers. As an application, we get the difference equation , where , , and is any real number, is oscillatory.
- Dynamic Equation
- Difference Equation
- Jump Operator
- Part Formula
- Closed Nonempty Subset
During the past years, there has been an increasing interest in studying the oscillation of solution of second-order damped dynamic equations on time scale which attempts to harmonize the oscillation theory for continuousness and discreteness, to include them in one comprehensive theory, and to eliminate obscurity from both. We refer the readers to the papers [1–4] and the references cited therein.
In , under the assumption of being an isolated time scale, we prove that, when is allowed to take on negative values, is sufficient for the oscillation of the dynamic equation (1.3). As an application, we get that, when is allowed to take on negative values, is sufficient for the oscillation of the dynamic equation (1.4), which improves a result of Hooker and Patula [7, Theorem 4.1] and Mingarelli .
where , , , and is any real number, is oscillatory.
Note that if , then the delta derivative is just the standard derivative, and when the delta derivative is just the forward difference operator. Hence, our results contain the discrete and continuous cases as special cases and generalize these results to arbitrary time scales (e.g., the time scale which is very important in quantum theory ).
We will need the following second mean value theorem (see [10, Theorem 5.45]).
Lemmas 2.2 and 2.4 give two lower bounds of definite integrals on time scale, respectively.
It is easy to know that, when , and, when , .
Assume that , where . Suppose that
(i)there exists a real number such that , for all ;
Then, (1.1) is oscillatory.
Since , we get , for large , which is a contradiction. Thus, (1.1) is oscillatory.
is sufficient for the oscillation of the difference equation (1.4).
then (1.4) is oscillatory.
where . It is easy to get the following.
Then, the differential equation (3.16) is oscillatory.
where , , , and is any real number.
for large . That means . It is easy to get that and is nonincreasing for large . So from Theorem 3.1, (3.18) is oscillatory.
and is nonincreasing. So from Theorem 3.1, (3.21) is oscillatory.
where , , , and is any real number.
So from Theorem 3.3, (3.23) is oscillatory.
This work was supported by the National Natural Science Foundation of China (no. 10971232).
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