Abstract
Some new oscillation criteria are established for a third-order nonlinear mixed neutral difference equation. Our results improve and extend some known results in the literature. Several examples are given to illustrate the importance of the results.
1. Introduction
By a Riccati transformation technique, we present some new oscillation criteria for the nonlinear difference equation of the form where is a fixed integer and denotes the forward difference operator defined by and . Throughout this paper, we will assume the following hypotheses: (A1) , for , (A2) is positive, , where are constants for (A3) on for , (A4) are constants for .We set . By a solution of (1), we mean a nontrivial sequence defined on , which satisfies (1) for all . A solution of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative and nonoscillatory otherwise. Equation (1) is called oscillatory if all its solutions are oscillatory. In recent years, there has been an increasing interest in the study of the oscillatory behavior of solutions of difference equations; we refer to the books [1–5] and the papers [6–8] and the references cited therein.
In [9], via comparison with first-order oscillatory difference equations, Agarwal et al. proved that several theorems provided sufficient conditions for oscillation of all solutions of the third-order difference equation of the form depending on the condition
In [10], by a Riccati transformation technique, Schmeidel studied the oscillatory and asymptotic behavior of solutions of the third-order difference equation using the condition
In [11], by the generalized Riccati transformation technique, Saker et al. established some new oscillation criteria for a certain class of third-order nonlinear delay difference equation of the form
In [12], by a Riccati transformation technique, Selvaraj et al. established some sufficient conditions for oscillation of all solutions of the third-order nonlinear difference equation of the form using the condition
In [13], by means of a Riccati transformation technique, Thandapani et al. discussed the oscillatory behavior of the solutions of the difference equation of the form using the condition
In [14], by using condition (5) and a Riccati transformation technique, Aktas et al. obtained some new oscillation criteria of third-order nonlinear delay difference equation of the form
In [15], using condition (8), Selvaraj et al. considered nonlinear third-order difference equations of the form and they studied the oscillatory behavior of solutions of (12).
In [16], Saker investigated the third-order difference equation using the condition
In [17], under condition (5), Selvaraj et al. discussed the oscillatory behavior of neutral delay difference equation of the form
In [18], by comparison with some first difference equations whose oscillatory characters are known and by means of a Riccati transformation technique, Elabbasy et al. obtained several new sufficient conditions for the oscillation of all solutions of the nonlinear difference equation with Delay of the form
In this paper, the details of the proofs of results for nonoscillatory solutions will be carried out only for eventually positive solutions, since the arguments are similar for eventually negative solutions. The paper is organized as follows. In Section 2, we will state and prove the main oscillation theorems and in Section 3 will we provide some examples to illustrate the main results.
2. Main Results
In this section, we establish some new oscillation criteria for (1) under the following condition
In the following results, we will use the following notations: We assume that there exists a double sequence and such that(i) for ,(ii) for ,(iii) for ,(iv). We begin with these lemmas, which will be used in obtaining our main results. The proof of the following lemmas are similar to that of Lemmas 2.1, 2.2, and 2.3, respectively, in [18], and hence the details are omitted.
Lemma 1. Let be an eventually positive solution of (1) and suppose that satisfies , , and , for all .Then there exists such that
Lemma 2. Assume that (17) holds. Let be an eventually positive solution of (1). Then for sufficiently large , there are only two possible cases: (I), , or(II),.
Lemma 3. Assume that (17) holds. Let be an eventually positive solution of (1) and suppose that (II) of Lemma 2 holds. If then as .
Next, we state and prove the main theorems.
Theorem 4. Assume that (20) holds. Further, assume that and there exists a positive nondecreasing sequence , such that Then every solution of (1) is oscillatory or .
Proof. Assume that (1) has a nonoscillatory solution, say , , , , and for all . From (1), we see that and Then, is nonincreasing sequence and thus and are eventually of one sign. By Lemma 2, there exist two possible cases (I) and (II). Assume that (I) holds. From (1) and the definition of , we have Thus, It follows from that . Thus, by (24), we obtain Define a Riccati substitution Then . From (26), we have From Lemma 1,, and , we get From (27) and (28), we obtain Next, define another sequence by Then . From (30), we have From Lemma 1, and , we get Then from (30), (31), and the above inequality, we have Similarly, we define another sequence by Then . From (34), we have From Lemma 1 and , we get Then from (34), (35), and the above inequality, we have From (25), (29), (33), and (37), we obtain Using (38) and the inequality we have Summing the last inequality from to , we obtain which yields where is a finite constant. But this contradicts (21). Next we assume that (II) holds. We are then back to the proof of Lemma 3 to show that . The proof is complete.
Corollary 5. Assume that all the assumptions of Theorem 4 hold, except that condition (21) is replaced by Then every solution of (1) is oscillatory or .
Remark 6. Note that, from Theorem 4, we can obtain different conditions for oscillation of all solutions of (1) by different choices of . Let and , , and is a constant. By Theorem 4, we have the following results.
Corollary 7. Assume that all the assumptions of Theorem 4 hold, except that condition (21) is replaced by Then every solution of (1) is oscillatory or .
Corollary 8. Assume that all the assumptions of Theorem 4 hold, except that condition (21) is replaced by Then every solution of (1) is oscillatory or .
Corollary 9. Assume that all the assumptions of Theorem 4 hold, except that condition (21) is replaced by Then every solution of (1) is oscillatory or .
Theorem 10. Assume that (20) holds. Let and be a positive sequence. Furthermore, one assumes that there exists a double sequence . If then every solution of (1) is oscillatory or .
Proof. Proceeding as in the proof of Theorem 4, we assume that (1) has a nonoscillatory solution, say , , , , and for all . By Lemma 2, there are two possible cases. If (I) holds, from the proof of Theorem 4, we find that (38) holds for all . From (38), we have Therefore, we have which yields after summing by parts By completing the square, we have Then, which implies Hence, Hence, which is contrary to (47). If (II) holds, then we are back to the proof of Lemma 3 to show that . This completes the proof of Theorem 10.
Corollary 11. Assume that all the assumptions of Theorem 10 hold, except that condition (47) is replaced by Then every solution of (1) is oscillatory or .
Theorem 12. Assume that (20) holds. Further, assume that and there exists a positive nondecreasing sequence , such that Then every solution of (1) is oscillatory or .
Proof. Assume that (1) has a nonoscillatory solution, say , , , and , for all . Proceeding as in the proof of Theorem 4, we get (24). By Lemma 2, there exist two possible cases (I) and (II). Assume that (I) holds. Then, we obtain (25).
From , , we obtain
Using the Riccati transformation
Then, . From (59), we have
From Lemma 1,, and , we get
From (60) and (61), we obtain
Next, define another sequence by
Then, . From (63), we have
From Lemma 1, we get
Then from (63), (64), and the above inequality, we have
Similarly, we define another sequence by
Then, . From (67), we have
From Lemma 1 and , we get
Then from (67), (68), and the above inequality, we have
From (58), (62), (66) and (70), we obtain
Using (39) and (71), we have
Summing the last inequality from to , we obtain
which yields
where is a finite constant. But this contradicts (57). Next, we assume that (II) holds. We are then back to the proof of Lemma 3 to show that . The proof is complete.
Theorem 13. Assume that (20) holds. Let and be a positive sequence. Furthermore, assume that there exists a double sequence . If then every solution of (1) is oscillatory or .
Proof. Proceeding as in the proof of Theorem 12, we assume that (1) has a nonoscillatory solution, say , , , , and for all . By Lemma 2, there are two possible cases. If (I) holds, from the proof of Theorem 12, we find that (71) holds for all . From (71), we have The remainder of the proof is similar to that of the proof of Theorem 10 and hence the details are omitted.
3. Examples
In this section, we will show the applications of our oscillation criteria by three examples. We will see that the equations in the example are oscillatory or tend to zero based on the results in Section 2.
Example 1. Consider the following nonlinear neutral equation: where , for . Assume that (17) holds. If we take and , thus Corollary 7, asserts that every solution of (77) is oscillatory or tends to zero.
Example 2. Consider the following nonlinear neutral equation where , , and and are positive constants. We see that (17) holds. If we take , , then by Theorem 4, every solution of (78) is oscillatory or tend to zero provided that .
Example 3. Consider the following nonlinear neutral equation: where , , and are positive constants. We see that (17) holds. If we take , then by Theorem 12, every solution of (79) is oscillatory or tends to zero provided that .
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to thank the anonymous referees very much for valuable suggestions, corrections, and comments, which resulted in a great improvement of the original paper.