Modified Oscillation Results for Advanced Difference Equations of Second-Order

Abstract

In this paper, we present a new method to establish the oscillation of advanced second-order difference equations of the form using the ordinary difference equation The obtained results are new and improve the existing criteria. We provide examples to illustrate the main results.

1. Introduction

This paper is concerned with the second-order advanced difference equation where , is a nonnegative integer, and

(C₁) , , and are positive real sequences for

(C₂) is a monotone increasing sequence of integers with for

(C₃) as

By a solution of , we mean a nontrivial sequence that satisfies for all A solution of is said to be oscillatory if it is neither eventually negative nor eventually positive. Otherwise, it is said to be nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.

Oscillation phenomena take part in different models described by various differential equations, partial differential equations, and dynamic equations on time scales; see, for instance, the papers [1–6] for more details. In particular, we refer the reader to the papers [4, 6] for models from mathematical biology and physics where oscillation and/or delay actions may be formulated by means of cross-diffusion terms. In recent years, there have appeared several criteria on the oscillation of (1) for the retarded case, that is, and , using either comparison methods or/and Riccati transformation technique. On the majority, these studies use the comparison methods, which is considered to be the most powerful tool in the oscillation theory of difference equations (see, for example, [7–17] and the references cited therein). Another particular method appearing in several studies is also the summation averaging method (see, for example, [1, 18–21] and the references cited therein).

From the literature, it is well known that not many oscillation results are available by using comparison methods. In [22, 23], the authors obtained oscillation of the advanced difference equation (1) from that of the ordinary difference equation without explicitly using the information about the advanced argument Very recently in [22], the authors studied the oscillation of by assuming that for all

In this paper, we present a new method which produces the oscillation of (1) without the restriction Thus, our results generalize and complement to those reported in [14, 15, 22, 23].

2. Main Results

Without loss of generality, in studying the nonoscillatory solutions of (1), we can restrict our attention only to positive solutions.

Lemma 1. Let be a positive solution of (1). Then, and , eventually.

Proof. The proof can be found in Lemma 1, [22], and the details are omitted.

Next we present an oscillation criterion for equation (2) which will be used to prove our main results.

Lemma 2. Assume that is convergent and is a positive solution of . Then, there is an integer such that where for .

Proof. From Lemma 1, there is an such that for all Taking into account the fact that is positive and decreasing, we have

Summing up the last inequality from to and then taking , we obtain for all

Now let us define

Theorem 3. Let condition (3) hold. Then, (2) is oscillatory provided one of the following two conditions holds:
(H₁) There exists an integer such that defined by (8) satisfying (H₂) There exists an integer such that

Proof. Assume, for the sake of contradiction, that equation (2) is nonoscillatory. Let (H₁) hold, and let be a positive solution of (2) for all By Lemma 2, we have for Thus, by (11), which contradicts (9). Similarly for from (8) and (4), we have and hence, which is again a contradiction.
Next, suppose that (H₂) hold. Clearly, in view of for , we get from (4) that , which is a contradiction. The proof of the theorem is complete.

Theorem 4. Assume that eventually. Then, is oscillatory.

Proof. From (15) and (8) for we have From (8) for we get Hence, where
In general, where and . Clearly, . If converges to some positive number , then But there is no real positive solution for such an equation Thus, Then, we have Thus, from Theorem 3, it follows that (2) is oscillatory. Now using Theorem 3.5 of [23], we see that (1) is oscillatory. The proof of the theorem is complete.

Next assume that the opposite condition of (15), namely, holds.

Theorem 5. Let be a positive solution of (1) and eventually. Then, there is an integer such that for , is monotonically nondecreasing.

Proof. The proof is similar to that of Theorem 3 of [22], and hence, the details are omitted.

Next we state a comparison result, containing an advanced argument.

Theorem 6. Let (21) hold. If the difference equation is oscillatory, then (1) is oscillatory.

Proof. The proof is similar to that of Theorem 4 in [22] and hence omitted.

The above theorem ensures that any oscillation criterion established for (23) leads to an oscillation criterion for (1).

Theorem 7. Let (21) hold. Assume that there is a constant such that eventually. Then, (1) is oscillatory.

Proof. The condition (24) guarantees that (23) oscillates, which in turn implies that (1) is oscillatory. This completes the proof.

Next, we provide an example, illustrating this result.

Example 1. Consider the second-order advanced Euler type difference equation with and is an integer.

Now With and by Theorem 7, equation (25) is oscillatory provided that

For example, if , then it is required that

Note that the result in [22] cannot be applicable to (25) since

If condition (24) fails to hold , then we can derive a new oscillation criterion using the constant

Theorem 8. Let (21) hold. Assume that is a positive solution of (1) and eventually. Then, is monotonically nondecreasing.

Proof. Use [22], Theorem 2.6, to complete the proof.

Theorem 9. Let (21) and (24) hold. If the difference equation is oscillatory, then so is (1).

Theorem 10. Let (21) and (24) hold. If there exists a constant such that eventually, then (1) is oscillatory.

The proofs of Theorems 9 and 10 follow from Theorems 6 and 7, and hence, the details are omitted.

Example 2. Consider the difference equation (25). For this equation . By Theorem 10, we see that (25) is oscillatory provided that Since , Theorem 10 improves Theorem 7.

For convenience, let us use the additional condition that there is a positive constant such that

eventually. In view of (21), conditions (24) and (30) can be written in simpler form as

respectively. Repeating the above process, we have the increasing sequence defined as

Now as in [22], Theorem 2.9, one can generalize the oscillation criteria obtained in Theorems 7 and 10.

Theorem 11. Let (21) and (32) hold. If there exists a positive integer such that for and , then (1) is oscillatory.

Example 3. Consider the second-order advanced difference equation

For this equation and Through direct calculations, we get

Thus, Theorems 7 and 10 fail for equation (36). But and hence, Theorem 11 guarantees the oscillation of (36).

3. Conclusion

In this paper, we have derived a new comparison method to obtain the oscillation of second-order advanced difference equation which removed the restriction imposed on the coefficient such that as in [22]. Thus, the oscillation criteria obtained in this paper improved and complemented to the existing results. It is an interesting problem to extend the results of this paper to equation (1) when it is in noncanonical form.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The third author was supported by the Special Account for Research of ASPETE through the funding program “Strengthening research of ASPETE faculty members.”