Oscillation of nonlinear third order perturbed functional di erence equations

(H1) {bn}, {an} and {pn} are positive real sequences for all n ∈ N(n0); (H2) f : R → R and g : N(n0) × R × R × R → R are continuous functions, uf (u) > 0 for u = ̸ 0, and f is nondecreasing; (H3) f (uv) ≥ f (u)f (v) for uv > 0; (H4) {σ(n)} is a sequence of integers with σ(n) ≤ n and lim n→∞ σ(n) =∞; (H5) α ≥ 1 is a ratio of odd positive integers; (H6) there is a positive real sequence {qn} such that |g(n, u, v, w)| ≤ qn f (v) for all (n, u, v, w) ∈ N(n0)×R×R×R.

By a solution of equation (1.1), we mean a nontrivial real sequence {xn} which is de ned for all n ∈ N(n ) and satisfying equation (1.1). We assume that equation (1.1) possesses such solutions for all n ∈ N(n ). As usual a nontrivial solution of equation (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, and it is nonoscillatory otherwise. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
Recently in [1-6, 9-12, 14-16], the authors studied the oscillatory and asymptotic behavior of solutions of some third and higher order nonlinear delay di erence equations. In [11,15], the authors studied the oscillatory behavior of the equation In [16], the authors investigated the oscillatory behavior of the equation Our purpose in this paper is to establish oscillation results for equation (1.1) without imposing a "smallness" condition on the perturbation term. Also, we present some results on the boundedness of nonoscillatory solutions and oscillatory behavior of a particular case of equation (1.1), namely, ∆(bn ∆(an(∆xn) α )) + pn x β n = en + qn x γ n , n ∈ N(n ), (1.5) where β and γ are ratios of odd positive integers with β > γ and {en} is a real sequence. Examples are provided to illustrate the importance of the main results.

Oscillation of Equation (1.1)
In this section, we investigate the oscillatory behavior of all solutions of equation (1.1). For any integer N ∈ N(n ), we set We also assume that {τ(n)} and {η(n)} are sequences of integers satisfying and

6)
and ∆zn  Summing the last inequality from N to σ(n) − > N, we have where yn = bn ∆(an(∆xn) α ) > . Using (2.9) in (2.8) and applying (H ), the monotonicity of f we see that yn is a positive solution of the inequality It follows from Lemma 5 of [8] that the corresponding di erence equation (2.6) also has a positive solution, which is a contradiction. For Case (II), it is easy to see that So summing for j − ≥ i ≥ N, we have Hence Setting i = σ(n) and j = τ(n) in the above inequality, we obtain From (2.10) and (2.11), we see that Substituting (2.12) in (2.8), one has zn is a positive solution of the inequality ∆zn + Qn f (B(n))f (z α η(n) ) ≤ .
It follows from Lemma 5 of [8] that the corresponding di erence equation (2.7) also has a positive solution, which is a contradiction. This completes the proof of the theorem.
The next two corollaries follow immediately from known oscillation criteria for rst order delay di erence equations; for example, see [7] and [13]. Now assume that σ(n) = n − k, τ(n) = n − l and η(n) = n − m where k, l, m are positive integers such that k ≥ l ≥ m.
where zn = bn ∆(an(∆xn) α ) > . Dividing by an and then summing from σ(n) to δ(σ(n)), we obtain for all large n. Using (2.19) in (2.8) and then proceeding as in the proof of Case (II) of Theorem 2.1, we obtain a desired contradiction. This completes the proof of the theorem. Taking limit in mum on both sides of the above inequality as n → ∞, and applying conditions (3.5) -(3.7), we obtain a contradiction to {xn} being a positive solution. The proof for the case {xn} is eventually negative is similar. This completes the proof of the theorem.

Examples
In this section, we provide some examples to illustrate the above results. It is easy to see that condition (2.12) is satis ed. Hence by Corollary 2.2, the equation (4.1) is oscillatory.

(4.2)
It is easy to see that all conditions of Theorem 3.1 are satis ed with en = and so every nonoscillatory solution of (4.2) is bounded. One such bounded solution is {xn} = n .

Conclusion
In this paper, we have derived some new su cient conditions by using comparison techniques for the oscillation of all solutions of equations (1.1) and (1.5) which improve and extend that of in [16]. Finally, we provided three examples that illustrates the signi cance of the main results. It would be interesting to obtain results similar to this paper when < α < .