Oscillation of nonlinear third-order difference equations with mixed neutral terms

In this paper, new oscillation results for nonlinear third-order difference equations with mixed neutral terms are established. Unlike previously used techniques, which often were based on Riccati transformation and involve limsup or liminf conditions for the oscillation, the main results are obtained by means of a new approach, which is based on a comparison technique. Our new results extend, simplify, and improve existing results in the literature. Two examples with specific values of parameters are offered.


Introduction and preliminaries
Oscillation of solutions for third-order difference equations has received comparably little attention, although such equations are of importance in many fields of science such as economics, physics, mathematical biology, and other areas of mathematics [3,4,6,7,10,11,13,14,24,27,31,32,[35][36][37]. It is worth to mention that third-order difference equations may have totally different behavior from corresponding third-order differential equations; see the explicit example in [9]. On the other hand, oscillation of solutions for difference equations of first and second order have been extensively investigated in the literature; see the monographs [1,2,8] and the papers [5, 12, 15-19, 21, 23, 25, 26, 30, 33].
In this study, we consider a nonlinear third-order difference equation with mixed nonlinear neutral terms. We obtain conditions guaranteeing oscillation of solutions of this equation. The main results are proved by using a comparison technique with first-order equations. Such an approach was effectively used for other types of equations in [20,22]. To demonstrate this, we present two examples, which cannot be discussed using any of the previously established results.
The objective of this paper is to offer conditions ensuring oscillation of (1.1) whenever α 4 < 1 < α 5 or α 4 < α 5 ≤ 1 and subject to the assumption In view of the results established in the literature and to the best of our observations, there are no oscillation results for (1.1). This paper is organized as follows: In Sect. 2, we give some auxiliary results and introduce some notation. Sect. 3 features the main results of the paper. We present our investigations under two cases for (1.1). The first case is when α 4 < 1 < α 5 , and the other case is when α 4 < α 5 ≤ 1. Our approach is based on a comparison technique with first-order difference equations. In Sect. 4, two examples are provided in order to illustrate our main theorems.

Auxiliary results and notations
We start with the following fundamental result. See [22,Lemma 1], and for the proof of (I), see [ has an eventually positive solution, then so does the corresponding advanced difference equation.
Hence, there exists t 2 ≥ t 1 with y(t) < 0 for all t ≥ t 2 .
(2.15) By (2.9), (2.13), and (2.15), we have so Case PPN holds. Finally, if (2.14) does not hold, then the only other possibility is By (2.9), (2.13), and (2.16), we have so Case NPN holds. There are no other cases. See Table 1 for an illustration of the proof.
Throughout the remainder of the paper, we suppose that For convenience, we introduce the notations . Remark 2.4 1. Note that, due to the assumptions m > 2, m * > 2, and k < m -1, it is always possible to find k 0 , k 1 , k 2 , k 3 such that (2.17) holds, e.g., one may pick 2. Note that ξ 0 (t) > t holds always since m * -2k 0 > 0. Hence, equations involving ξ 0 are of advanced type. Moreover,

Main results
Now we present our first oscillation result. (3.1) Let θ 0 , θ 1 ∈ (0, 1). If the first-order advanced difference equation and the first-order delay difference equations and are oscillatory, then so is (1.1).
Proof Assume that x is a nonoscillatory solution of (1.1), say eventually. It follows from (1.1) that, eventually, Using these two inequalities, we have Since y in both Cases PPP and PPN is positive and nondecreasing, there exists C > 0 satisfying y(t) ≥ C, and so we have Next, due to (3.1), there exists κ ∈ (0, 1) such that Thus, we have Case PPP. By (3.8), we get Summing (3.9) from tk 0 to t -1, we get (3.10) Summing (3.10) again from tk 0 to t -1, we obtain In summary, y is a positive and increasing solution of Employing Lemma 2.1 (II), (3.2) also has an eventually positive solution, which is a contradiction.
Cases NNN and NPN. Throughout the remainder of the proof, we introduce Z again by (3.11). First note that, eventually, Hence, eventually, Thus, eventually, Case NNN. First note that, eventually, and therefore, eventually, and thus, eventually, -Z(t) (3.16) ≥ -Q(t)y In summary, Z is a positive and decreasing solution of Employing Lemma 2.1 (I), (3.4) also has an eventually positive solution, which is a contradiction.
Case NPN. We let First, we have, eventually, Next, we have, eventually, so, Thus, we see that -Z(t) (3.16) ≥ -Q(t)y In summary, Z is a positive and decreasing solution of Employing Lemma 2.1 (I), (3.5) also has an eventually positive solution, which is a contradiction.
We now prove the following consequence of Theorem 3.1. are oscillatory, then so is (1.1).
Proof We claim that oscillation of (3.19) implies oscillation of both (3.4) and (3.5). As all the other assumptions are the same as in Theorem 3.1, the statement then follows from Theorem 3.1. So assume that (3.19) is oscillatory. First, suppose that (3.4) is not oscillatory, say, there exists eventually positive Z satisfying From the equality in (3.20), we see that Z is eventually decreasing, and since we obtain Z(ξ 3 (t)) ≤ Z(ξ 2 (t)) eventually. Using this in (3.20), we get By Lemma 2.1 (I), (3.19) also has an eventually positive solution, a contradiction, showing that (3.4) is indeed oscillatory. Next, suppose that (3.5) is not oscillatory, say, there exists eventually positive Z satisfying By Lemma 2.1 (I), (3.19) also has an eventually positive solution, a contradiction, showing that (3.5) is indeed oscillatory as well. Thus, the proof is complete.

P(t) y(t) y(t).
Since y is positive and nondecreasing, there exists C > 0 such that y(t) ≥ C, and so we have Next, due to (3.25), there exists κ ∈ (0, 1) such that (3.7) is satisfied. This completes the proof.
As before in Theorem 3.2 and Theorem 3.4, we now obtain the following results.