Oscillatory behavior of solutions of third-order delay and advanced dynamic equations
© Adivar et al.; licensee Springer. 2014
Received: 7 October 2013
Accepted: 11 February 2014
Published: 25 February 2014
By a solution of (1.1) (or (1.2)) we mean a function , , which has the property that and satisfies (1.1) (or (1.2)) for all large . A nontrivial solution is said to be nonoscillatory if it is eventually positive or eventually negative and it is oscillatory otherwise. A dynamic equation is said to be oscillatory if all its solutions are oscillatory.
Since we are interested in the oscillatory behavior of solutions of (1.1) and (1.2) near infinity, we assume throughout this paper that our time scale is unbounded above. An excellent introduction of time scales calculus can be found in the books by Bohner and Peterson  and .
The purpose of this paper is to extend the oscillation results given in  and . Oscillation criteria for third-order dynamic equations have recently been studied in [5–8]. Other papers related with oscillation of higher-order dynamic equations can be found in [9, 10]. Well-known books concerning the oscillation theory are [11, 12].
2 Oscillation criteria for (1.1)
In this section, we investigate some oscillation criteria for solutions of the third-order delay equation (1.1).
where A and B are defined as in (1.6) and (1.7), respectively, then (1.1) is oscillatory.
- (I)Assume that and for . Then we have
- (II)Assume that , for . Then we have
Dividing the above inequality by , taking the lim sup of both sides of the above inequality as and using (2.3), we obtain a contradiction to (2.1). □
For the bounded solutions of (1.1) we have the following, which is immediate from Theorem 2.1.
Corollary 2.2 In addition to (1.3) and (1.4), assume that (2.1) and (2.3) hold. Then all bounded solutions of (1.1) are oscillatory.
Now we prove the following result.
where B is defined as in (1.7), then all bounded solutions of (1.1) are oscillatory.
By dividing the above inequality by and taking the lim sup of both sides of the resulting inequality as , we obtain a contradiction. The proof is now complete. □
where β is a ratio of two odd positive integers, and we obtain some oscillatory criteria.
where A and B are defined as in (1.6) and (1.7), respectively. Then (2.7) is oscillatory.
- (I)Assume that and for . Using (2.4) in (2.7), we have
- (II)Assume that , for . Then we have (2.6). Using (2.6) in (2.7), for , we have
Since and the right hand side of the above inequality is infinity by (2.9), we obtain a contradiction to the facts that is positive and nondecreasing. □
3 Oscillation criteria for (1.2)
In this section, we investigate some oscillation criteria for solutions of the third-order delay equation (1.2).
where A and B are defined as in (1.6) and (1.7), respectively, then (1.2) is oscillatory.
- (I)Assume that and for . Then we have
- (II)Assume that and for . Then we have
Taking the lim sup of both sides of the above inequality as , we obtain a contradiction to (2.1). The proof is complete. □
where B is defined as in (1.7), respectively, then (1.2) is oscillatory.
- (I)Assume that and for . Integrating (1.2) from t to yields
- (II)Assume that and for . Integrating
Dividing the above inequality by and taking the lim sup of both sides of the resulting inequality as , we obtain a contradiction to (3.6). The proof is complete. □
In this section we give examples to illustrate two of our main results. Recall
Theorem 4.1 ([, Theorem 1.75])
Theorem 4.2 ([, Theorem 1.79 (ii)])
Our first example illustrates Theorem 2.1.
which shows that (2.3) holds. By Theorem 2.1, (4.1) is oscillatory.
Our second example illustrates Theorem 3.2.
Thus (3.8) holds. By Theorem 3.2, (4.2) is oscillatory.
are oscillatory, then (1.1) is oscillatory.
cannot be extended since is satisfied for few time scales. While the result holds for and , this condition is not satisfied on . Being aware that time scales are not generally closed under addition, we were able to prove the following.
are oscillatory, where , then (1.1) is oscillatory.
In order to prove Theorem 5.1, it is necessary to define the function to be a nondecreasing function with respect to its second argument and with the property that for any function . It is also necessary to assume that is a bijection with for all and , and to use the following definition and theorem.
on an interval , we mean a rd-continuous function y defined on the interval , which is rd-continuously differentiable on and satisfies (5.5) for all . A solution y of (5.5) is said to be positive if for every .
This leads to a contradiction and the proof is complete. □
- (I)Assume that and for . Then (2.4) holds. Using this and (1.4) in (1.1), we obtain
- (II)Assume that and for . As in the proof of Theorem 2.1, we have (2.5). Then for , we have
where for . Similar to Case (I) above, by Theorem 5.3, there exists a positive solution y of (5.4) such that , which contradicts the fact that (5.4) is oscillatory. □
In  and , the authors prove a comparison result for (1.2) similar to the one given at the beginning of this section. That result involved . Again, since time scales are not generally closed under addition, this result cannot be extended to a general time scale .
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