Oscillation of Fourth-Order Nonlinear Homogeneous Neutral Difference Equation

In this paper, we establish the solution of the fourth-order nonlinear homogeneous neutral functional diﬀerence equation. Moreover, we study the new oscillation criteria have been established which generalize some of the existing results of the fourth-order nonlinear homogeneous neutral functional diﬀerence equation in the literature. Likewise, a few models are given to represent the signiﬁcance of the primary outcomes.


Introduction
e investigation of the conduct of solutions of functional difference equations is a significant space of examination and is quickly developing because of the advancement of time scales and the time-scale analytics (see, e.g., [1,2]). Most papers on higher-order nonlinear neutral difference equations manage the presence of positive solutions and the asymptotic conduct of solutions. We refer the per user to a portion of the works [3][4][5][6][7][8][9] and the references referred to in that.
In [10], Migda investigated the asymptotic properties of nonoscillatory solutions of neutral difference equation of the form (1) e papers [9,10] are tantamount. Be that as it may, more accentuation might be given to [9], which manages the oscillatory, nonoscillatory, and asymptotic characters. Some oscillation criteria have been set up by applying the discrete Taylor series [11]. e inspiration of the current work has come from two bearings [9,12] and the second is expected to [13].
(2) Also, Under the condition ∞ n�0 n/r(n) < ∞, for different ranges of p(n) and q(n) ≥ 0. Also, no super linearity or sublinearity conditions are imposed on G ∈ C(R, R). It is interesting to observe that the nature of the function r(n) influences the behaviour of solutions of (2) or (3).
Parhi and Tripathy [13] studied oscillation of a class of nonlinear neutral difference equations of higher order and the behaviour of its solutions is studied separately.
In this present work, we study the oscillation behaviour of the fourth-order nonlinear homogeneous neutral functional difference equation.

Oscillation Behaviour of Neutral Difference Equation
In this section, we establish the solution of the fourth-order homogeneous neutral functional difference equation of the form where A solution � y(ϱ) of (4) is oscillatory if � y(ϱ)� y(ϱ + 1) ≠ 0 for every integer ϱ > N > 0. Otherwise, it is nonoscillatory. e NFDE (4) is oscillatory, if all its solutions are oscillatory. For the oscillation of (4), we define the operators Also, the notations Lemma 1. Let (5) hold and � u be a real valued function with L 4 � u(ϱ) ≤ 0 for large ϱ. If � u(ϱ) > 0, then one of these (8) to (9) holds and if� u(ϱ) < 0, then one of these (9) to (13) holds, where (5) and hold, then (4) is oscillatory.
Consider the potential cases 2.3 to 2.6 of Lemma 1.
is completes the proof of theorem.
is completes the proof of the theorem.

Conclusion
In this paper, we inferred new properties of the nonoscillatory solutions and using these outcomes, some new adequate are introduced for the concentrated on NFDE to have the purported property oscillatory. Our outcomes improve and supplement many known outcomes for NFDEs as well as for ordinary functional difference equations also. At last, we give two models that show the meaning of the fundamental results.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.