On the Oscillation of Even-Order Half-Linear Functional Difference Equations with Damping Term

Definition 1. A nontrivial sequence y_n is called a solution of (3) if it is defined for all n ≥ σ where $$, $$, and p_nQ(Δ^m−1y_n) is differenceable on $$ and satisfies (3) for all $$.

Definition 2. A nontrivial solution y_n of (3) is said to be oscillatory if the terms of the sequence y_n are not eventually positive or not eventually negative. Otherwise, the solution is called nonoscillatory. A difference equation is called oscillatory if all its solutions oscillate.

Lemma 3 (see [24].)Let y_n be defined for $$ and y(n) > 0 with Δ^my_n of constant sign for n ≥ n₀ and not identically zero. Then, there exists an integer l, 0 ≤ l ≤ m with (m + l) odd for Δ^my_n ≤ 0 and (m + l) even for Δ^my(n) ≥ 0 such that

• (i)

  l ≤ m − 1 implies (−1) ^l+iΔⁱy_n > 0 for all n ≥ n₀, l ≤ i ≤ m − 1,

• (ii)

  l ≥ 1 implies Δⁱy_n > 0 for all large n ≥ n₀, 1 ≤ i ≤ l − 1.

Lemma 4 (see [25].)Let y_n be defined for n ≥ n₀ and y_n > 0 with Δ^my_n ≤ 0 for n ≥ n₀ and not identically zero. Then, there exists a large integer n₁ ≥ n₀ such that

where l is defined as in Lemma 3. Further, if y_n is increasing, then

Lemma 5. Let y_n satisfy conditions of Lemmas 3 and 4 and Δ^m−1y_nΔ^my_n ≤ 0 for n ≥ n₁ ≥ n₀. Further, if y_n is increasing, then

where $$.

Lemma 6. Let y_n be an eventually positive solution of (3). If

then Δ^m−1y_n > 0, Δ^my_n ≤ 0, and Δy_n > 0 for all n ≥ n₁ ≥ n₀.

Proof. The fact that y_n is eventually positive solution of (3) implies y_n > 0 and $$ for all n ≥ n₁ ≥ n₀. In view of (3), we get

which leads to Hence, is decreasing and Δ^m−1y_n is eventually positive or eventually negative.

We claim that

Assume, on the contrary, that Δ^m−1y_n < 0, n ≥ n₁. Then, from (10), we obtain where $$. Therefore, from (12), we have where $$. It follows that or Consequently, we obtain Letting n → ∞ in the above inequality, one gets lim⁡_n→∞⁡Δ^m−2y_n = −∞. Hence, y_n is an eventually negative function which contradicts that y_n > 0. Therefore, inequality (11) holds.

From (3), we get

from which it follows that The above inequality implies that (Δ^m−1y_n) ^α−1 is nonincreasing. Therefore, we can write Since (Δ^m−1y_n) ^α−1 is nonincreasing and positive, then from the above inequality, we have by which we have In virtue of (21) and Lemma 3, we deduce that since m is even then l is odd. Hence Δy_n > 0 for n ≥ n₁ ≥ n₀. The proof is complete.

Theorem 7. Let condition (Λ1) hold. Further, assume that there exists a constant λ > α − 1 such that

where and M is as in Lemma 5. Then, (3) is oscillatory.

Proof. For the sake of contradiction, assume that (1) has a nonoscillatory solution y_n. Without loss of generality, we assume that y_n is eventually positive (the proof is similar when y_n is eventually negative). That is, y_n > 0,$$ and $$ for all n ≥ n₁ ≥ n₀. By Lemma 6, we have Δ^m−1y_n > 0, Δ^my_n ≤ 0, and Δy_n > 0 for n ≥ n₁. Consider the function

Taking into account that Δy_n > 0 and y_n is increasing and τ_n−k < τ_n, we deduce that Δ^my_n ≤ 0 and Δ^m−1y_n is nonincreasing. Lemmas 3 and 4, (1), and (24) yield Multiplying by (n − k) ^λ and summing up from n₁ to n − 1, we obtain or where Let Then, F has maximum value at $$. That is, Therefore, (27) can be rewritten as Hence, we have which contradicts condition (Λ2). The proof is complete.

Theorem 8. Let condition (Λ1) hold. Further, assume that there exists a function $$ such that

where M is as in Lemma 5. Then, (3) is oscillatory.

Proof. For the sake of contradiction, assume that (3) has a nonoscillatory solution y_n. Without loss of generality, we assume that y_n is eventually positive (the proof is similar when y_n is eventually negative). That is, y_n > 0, $$ and $$ for all n ≥ n₁ ≥ n₀. By Lemma 6, we have Δ^m−1y_n > 0, Δ^my_n ≤ 0, and Δy_n > 0 for n ≥ n₁. Consider the function

By utilizing the same approach as in the proof of Theorem 7, we arrive at Summing up (35) from n₁ to n − 1, we have Letting n → ∞ in the above inequality and taking the upper limit, we get a contradiction to (Λ3). The proof is complete.

Remark 9. In view of the statements of Theorems 7 and 8, one can easily deduce that condition (Λ3) is a generalization of (Λ2).

Example 10. Consider the fourth order half-linear functional difference equation with damping

where p_n = n, q_n = n, r_n = 1/n, τ_n = n − 1, m = 4, and α = 3. It is easy to see that conditions (H1)–(H3) are satisfied. It remains to check the validity of conditions Λ1 and Λ2.

For n ≥ 2, we have

It is clear that Γ1 → ∞ as n → ∞. Therfore, condition (Λ1) holds. For n ≥ 2 and λ = 3 > α − 1 = 2, we have where It is clear that Γ2 → ∞ as n → ∞. Then, condition (Λ2) holds. Thus, by the conclusion of Theorem 7, (37) is oscillatory.

Example 11. Consider the sixth order half-linear functional difference equation with damping

where p_n = n, q_n = n, r_n = n², τ_n = n − 1, m = 6, and α = 3. It is easy to see that conditions (H1)–(H3) are satisfied. In Example 10, we have seen that (Λ1) is satisfied. It remains to check the validity of condition (Λ3).

For n ≥ 2 and δ_n = n, we have

It is clear that Γ3 → ∞ as n → ∞. Then, condition (Λ3) holds. Thus, by the conclusion of Theorem 8, (41) is oscillatory.

Remark 12. It is not possible to decide the oscillatory behavior of solutions of (37) and (41) by using any of the results reported in [12, 13]. This implies that the results of our paper extend and generalize some known theorems.

Remark 13. The main results of this paper remain valid for nondelay difference equations of the form