- Open Access
Asymptotic behavior of second-order nonlinear neutral dynamic equations
© Jing et al.; licensee Springer. 2013
- Received: 31 July 2013
- Accepted: 10 September 2013
- Published: 8 November 2013
This paper is concerned with oscillation and asymptotic behavior of a second-order neutral delay dynamic equation on an arbitrary time scale. We obtain two theorems which guarantee that every solution of the studied equation oscillates or converges to zero. These results improve and complement some known results given in the literature.
MSC:34K11, 34N05, 39A10, 39A12, 39A13, 39A21.
- asymptotic behavior
- neutral delay dynamic equation
- time scale
on an arbitrary time scale , where γ is a quotient of odd positive integers, r and p are positive rd-continuous functions on , . Also, we assume that are rd-continuous, , , , for all , and there exists a positive rd-continuous function q defined on such that .
The theory of dynamic equations on time scales, which goes back to its founder Hilger , has recently attracted attention of researchers. Several authors have expounded on various aspects of this new theory; see the survey paper written by Agarwal et al.  and the references cited therein. The books on the subject of time scales, by Bohner and Peterson [3, 4], present much of time scale calculus.
Since we are interested in oscillatory and asymptotic properties, we assume throughout this paper that the given time scale is unbounded above. We assume that , and it is convenient to assume that , and define the time scale interval of the form by . Throughout, we use the notation . By a solution of equation (1.1), we mean a non-trivial real-valued function , which has the property that z and are defined and Δ-differentiable for and satisfies equation (1.1) on . The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution x of equation (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory.
In 2007, Saker et al.  posed an open problem as follows: How to establish oscillation criteria for equation (1.1) when condition (1.5) holds? Assuming (1.5), Zhang et al. [21, 22] obtained some sufficient conditions which insure that all solutions of equation (1.1) are oscillatory.
The purpose of this paper is to present some asymptotic tests for equation (1.1) in the case where (1.5) holds. This paper is organized as follows: In the next section, we shall establish the main results. In Section 3, two examples are provided to illustrate the results obtained.
In the sequel, when we write a functional inequality without specifying its domain of validity, we assume that it holds for all sufficiently large t.
and, for sufficiently large , .
In order to prove our main results, we will use the following result; see [, Theorem 2.1].
Then every solution x of equation (1.1) is oscillatory.
then every solution x of equation (1.1) is oscillatory or .
Therefore, is strictly decreasing, and there exists a such that or for . We consider each of two cases separately.
Case 1. Assume that for . As in the proof of [, Theorem 2.1], we can obtain a contradiction to (2.1).
which yields , this is a contradiction. Hence, . By virtue of , . The proof is complete. □
Next, we establish another criterion which improves Theorem 2.2.
then every solution x of equation (1.1) is oscillatory or .
Proof Let x be a nonoscillatory solution of equation (1.1). Without loss of generality, we assume that , , and for . Then for . In view of (1.1), we get (2.3). Thus, is strictly decreasing, and there exists a such that or for . We consider each of two cases separately.
Case 1. Assume that for . Similarly to the proof of [, Theorem 2.1], we can obtain a contradiction to (2.1).
which implies that , this is a contradiction. Hence, . By , . This completes the proof. □
Remark 2.1 When , Theorems 2.2 and 2.3 improve results of Han et al. [, Theorems 2.1 and 2.2] since our results do not require condition (1.3).
In this section, we give two examples to illustrate applications of results in the previous section.
Thus, we have by Theorem 2.2 that every solution x of (3.1) is oscillatory or .
Hence, by Theorem 2.3, every solution x of (3.2) is oscillatory or .
This research is supported by the National Key Basic Research Program of P.R. China (2013CB035604) and the NNSF of P.R. China (Grant Nos. 61034007, 51277116, and 51107069).
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