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Research Article | Open Access
Oscillation Criteria for Nonlinear Third-Order Neutral Dynamic Equations with Damping on Time Scales
We establish several oscillation criteria for a class of third-order nonlinear dynamic equations with a damping term and a nonpositive neutral coefficient by using the Riccati transformation. Two illustrative examples are presented to show the significance of the results obtained.
In this paper, we are concerned with the oscillation of a class of third-order damped dynamic equations of neutral typeon a time scale satisfying , where , , and . Throughout, we suppose that the following conditions are satisfied: is a constant., , is positively regressive (i.e., ), and and , where ., , or , , and there exists a sequence such that and . and there exists a function such that and for .
The theory of time scales, which was firstly introduced by Hilger in [1, 2], has been enriched by researchers; see, for instance, [3, 4], monographs [5, 6], and the references cited therein. During the past decade, a great deal of interest in oscillation of solutions to different classes of dynamic equations on time scales has been shown; we refer the reader to [7–23].
Yu and Wang  studied a third-order dynamic equationAgarwal et al. [8, 10], Candan , Erbe et al. , Hassan , and Li et al.  considered a third-order retarded dynamic equationSaker et al.  studied a second-order damped dynamic equationwhereas Qiu and Wang  considered a second-order damped dynamic equationwhere , , andHan et al.  and Qiu  investigated the third-order dynamic equations with nonpositive neutral coefficientsrespectively, where .
In this paper, using the Riccati transformation, we obtain some sufficient conditions which ensure that every solution of (1) either is oscillatory or converges to a finite number asymptotically. We do not impose restrictive assumption in our results. To illustrate the significance of new results, two examples are provided in the last section. In what follows, all functional inequalities are assumed to hold for all sufficiently large . Without loss of generality, we can deal only with eventually positive solutions of (1).
Definition 1. A solution of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is termed nonoscillatory.
Definition 2. Equation (1) is said to be almost oscillatory if all its solutions either are oscillatory or converge to zero asymptotically.
2. Auxiliary Results
Lemma 4. If is an eventually positive solution of (1), then there exists a sufficiently large such that, for ,
Proof. Let be an eventually positive solution of (1). From and , there exist a and a constant such that , , , , and for . By virtue of (1) and , we conclude thatWe claim that there exists a such that for . Otherwise, assume that for . Then, by (12) and , . It follows from and that . Hence, there exists a such that for ; that is,By , there exists a sufficiently large integer such that for . For , we havewhich yields and so , which contradicts the fact that . Therefore, and hence or . The proof is complete.
Lemma 6. Let be an eventually positive solution of (1) and suppose that and are eventually positive. Assume also thatThen there exists a sufficiently large such that, for ,
Proof. Let be an eventually positive solution of (1) and assume that there exists a such that and for . DefineThen, by Lemma 4, for ,We can prove that is eventually positive. If not, then there exists a such that for . Hence, we conclude thatwhich implies that is strictly increasing on . Since , there exists a such that, for , , and soUsing , we have . Therefore, by virtue of (1) and Lemma 4, for ,Integrating (21) from to , we getIt follows from (15) thatwhich is a contradiction. Hence, is eventually positive. Then, there exists a sufficiently large such that, for ,so is strictly decreasing on . If , then . If , then . Therefore, we arrive at (16). This completes the proof.
Lemma 7. Assume that all assumptions of Lemma 6 are satisfied. For , definewhere . Then, satisfieswhere
Proof. Suppose that all assumptions of Lemma 6 hold. Differentiating (25) and using (1), we haveIf , thenAssume now that . It follows from Lemma 6 thatHence, we haveUsing Pötzsche chain rule (see [5, Theorem 1.90] for details), we obtainwhich yieldsFrom Lemma 4, we conclude thatwhich implies thatIt follows now from (31) thatThe proof is complete.
Lemma 8. Assume that is an eventually positive solution of (1) and is eventually negative. Ifthen .
Proof. Since is eventually negative, is either eventually positive or eventually negative. If is eventually negative, then there exist a constant and a such that for , which causes a contradiction as in the proof of Lemma 4. Thus, is eventually positive.
Taking into account the fact that , by Lemma 4, there exists a such that for . We prove that . Otherwise, there exists a such that for , and a similar contradiction can be obtained. Suppose that . It follows from the proof of Lemma 4 and thatIntegrating (38) from to , , we havewhich contradicts the fact that . Hence, and so when using Lemma 3. This completes the proof.
3. Main Results
Let . Definewhere is the -partial derivative of with respect to .
Theorem 9. Assume that (15) holds and there exist two functions and such that, for all sufficiently large and for some ,where and are as in Lemma 7. Then every solution of (1) is oscillatory or exists (finite).
Proof. Suppose that (1) has a nonoscillatory solution . Without loss of generality, let be eventually positive. From Lemma 5, it follows that is eventually positive or . Assume that is eventually positive. By Lemma 4, there exists a such that either or for . Let for . Define by (25). Then, by Lemma 7, (26) holds. It follows from (26) that, for some ,whereLet ,Using the inequality (a variation of the well-known Young inequality)we deduce thatTherefore, we obtainwhich implies thatThis contradicts (41). Thus, for , and so exists. By Lemma 3, exists. The proof is complete.
Proof. Proceeding as in the proof of Theorem 9, assume that for . Let be defined by (25). By virtue of Lemma 7, we arrive at (36). Let ,Using (45), we conclude thatwhich yieldsLet . It follows from (53) thatwhich contradicts (50). Therefore, for . Along the same lines as in Theorem 9, we complete the proof.
Corollary 12. Assume that (37) is satisfied and there exists a function such that, for all sufficiently large and for some ,where and are as in Lemma 7. Then conclusion of Corollary 10 remains intact.
Remark 14. If , then and we do not impose restrictive condition (15) in our theorems and corollaries.
Remark 15. Our results complement and improve those obtained by Han et al.  since we do not impose specific restrictions on .
The following examples are presented to show applications of the main results.
Example 1. Consider the third-order equationwhere , , , and . It is clear that , , , , and . Then, and . Let . Since , we haveHence, assumptions – and (15) hold. Let and . Ifwe obtainThat is, (41) is satisfied. By virtue of Theorem 9, we deduce that every solution of (59) either is oscillatory or converges to a finite number asymptotically. Furthermore, if (61) holds and , thenwhich implies that (59) is almost oscillatory by using Corollary 10.
Example 2. Consider the third-order equationwhere , , , and . It is easy to see that , , , , and . Then, and . Since , we getObviously, conditions are satisfied. Let or and . If or , then