OSCILLATION CRITERIA FOR NON-CANONICAL SECOND-ORDER NONLINEAR DELAY DIFFERENCE EQUATIONS WITH A SUPERLINEAR NEUTRAL TERM

. We obtain oscillation conditions for non-canonical second-order nonlinear delay diﬀerence equations with a superlinear neutral term. To cope with non-canonical types of equations, we propose new oscillation criteria for the main equation when the neutral coeﬃcient does not satisfy any of the conditions that call it to either converge to 0 or ∞ . Our approach diﬀers from others in that we ﬁrst turn into the non-canonical equation to a canonical form and as a result, we only require one condition to weed out non-oscillatory solutions in order to induce oscillation. The conclusions made here are new and have been condensed signiﬁcantly from those found in the literature. For the sake of conﬁrmation, we provide examples that cannot be included in earlier works.

Let ν = max{τ, σ}. A solution {η(ι)} of (1.1) is a nontrivial real-valued sequence defined for all ι ≥ ι 0 −ν satisfying (1.1) for all ι ≥ ι 0 . Identically vanishing solutions in a neighborhood of infinity will not be considered in the paper. A solution of (1.1) is called oscillatory if it has arbitrarily large generalized zeros; otherwise it is called nonoscillatory. If all solutions of (1.1) are (non)oscillatory, then equation (1.1) is said to be (non)oscillatory.
Oscillation theory has expanded and developed greatly since this phenomena take part in different models from real world applications, see, e.g., the papers [9,23] dealing with biological mechanisms (for models from mathematical biology where oscillation and or delay actions may be formulated by means of cross-diffusion terms). Moreover, the study of neutral functional differential equations received significant attention because it arise in many fields such as control theory, communication, mechanical engineering, biodynamics, physics, economics and so on, see, [15,28,30] and the references therein. In view of the above observations, the researchers paid attention to the oscillation area for various classes of second-order difference, differential and dynamic equations, see [2,6,7,8,10,12,13,14,15,17,18,21,24,25,26,29,31] and the references cited therein. As far as secondorder difference equations with positive superlinear neutral terms are considered, not many results are known about the oscillation, see [3,4,11,16,19,27,32,33]. A close look at these papers reveals that the neutral coefficient {ρ(ι)} must rectify explicitly or implicitly either ρ(ι) → 0 or ρ(ι) → ∞ as ι → ∞. Further, they dealt with the non-canonical type of equations without changing its form and therefore required two conditions to eliminate all nonoscillatory solutions of these equations to get oscillatory solutions.
The purpose of the article is to study the oscillation of equation (1.1) when {ρ(ι)} fails to satisfy any of the above mentioned conditions. Our approach is different in the sense that; first we require one condition to eliminate nonoscillatory solutions of (1.1) to achieve oscillation via transforming the non-canonical equation (1.1) into canonical form. Next, we obtain oscillation of (1.1) by using comparison technique with first-order delay difference equations and Riccati transformation. Finally, we emphasize the practicality of the main results obtained via some particular examples, which cannot be discussed using any of the previously known results.
Proof. By a direct computation, we can easily show that (2.1) holds for any sequence φ(ι). Indeed, that is, the operator on the right-hand side of (2.1) is canonical. This proves the lemma.
As a result of Lemma 2.1, we see that the non-canonical equation (1.1) can be equivalently written as which is in canonical form. The next result directly follows from the above discussion.

(2.22)
Since y(ι) is increasing and y(ι)/Γ(ι) is decreasing, we have That is, For the final result of this section, the following lemma from [22] is needed.
Proof. Proceeding as in the proof of Theorem 2.6, we arrive at (2.11), that is, (2.31) Now arguing as in the proof of Theorem 2.10, we obtain Using the fact that y(ι)/Γ(ι) is decreasing and y(ι)/Γ γ (ι) is increasing, the latter inequality gives After simplification, we obtain which contradicts (2.30).

Applications
In this section, we present three examples to illustrate the emphasize of the main results.

Conclusions
By putting the equation in canonical form, we offer oscillation conditions for (1.1) in this work, which makes it easier to examine (1.1). Furthermore, the oscillation criteria developed here are novel and add to the findings previously reported in the literature. The neutral coefficient ρ(t) ∈ (0, 1) prevents the results presented in [3,4,11,16,19,27,32,33] from being applicable to our equations (3.1)-(3.3). As a result, our findings constitute a highly valuable addition to the oscillation theory of second-order neutral difference equations with superlinear neutral terms. When −1 < ρ(ι) < 0 or {ρ(ι)} is oscillatory, it is also intriguing to extend the findings of this paper.