Some New Oscillation Criteria for Fourth-Order Nonlinear Delay Difference Equations

Abstract

In this paper, the authors studied oscillatory behavior of solutions of fourth-order delay difference equation under the conditions . New oscillation criteria have been obtained which greatly reduce the number of conditions required for the studied equation. Some examples are presented to show the strength and applicability of the main results.

1. Introduction

In this paper, we are concerned with the fourth-order delay difference equation of the formwhere

In the sequel, we will assume the following: (H1) {c₁(n)}, {c₂(n)} and {c₃(n)} are the positive real sequence for all n ≥ n₀ and satisfy (H2) {p(n)} is a nonnegative real sequence and does not vanish eventually. (H3) is nondecreasing, and for all x ≠ 0. (H4) k is a nonnegative integer.

Under a solution of (1), we mean a real-valued sequence {u(n)} defined for all n ≥ n₀ − k and satisfies equation (1) for all n ≥ n₀. We restrict our attention to only those solutions of (1) which exist for all n ≥ N ≥ n₀ and satisfy the following condition:

A solution {u(n)} of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory. The equation itself is termed oscillatory if all its solutions oscillate.

Fourth-order difference equations naturally appear in discrete-type models concerning physical, biological, and chemical phenomena (see for example [1]). In mechanical and engineering problems, questions concerning the existence of oscillatory solutions play an important role. During the last several years, there has been a constant interest in getting sufficient conditions for oscillatory behavior of different classes of fourth-order difference equations with or without deviating arguments, (see [2–19], and the references cited therein). In particular, the authors in [5, 8, 13, 15, 18, 19] established oscillation results for (1) under the following assumptions:respectively.

In [3, 4, 6, 7, 9–12, 14, 16, 17], the authors studied the oscillation properties of solutions of equation (1) under the following condition:

From the review of the literature, it seems that there is nothing known about the oscillation of equation (1) when condition (3) holds. Inspired by the ideas adopted in [20], our main aim is to fill this gap by presenting easily verifiable criteria for the oscillation of all solutions of (1). Examples illustrating the importance of the results obtained are presented.

2. Main Results

For the sake of convenience, we use the following notations throughout the paper. We denotewhere n ≥ N ≥ n₀. In what follows, we only need to consider eventually positive solutions of (1), since if {u(n)} satisfies (1), then so does {−u(n)}.

Lemma 1. Let {u(n)} be an eventually positive solution of (1). Then, there is an integer n₁ ≥ n₀ such that {u(n)} satisfies one of the following cases:(i)u(n) > 0, D₁u(n) > 0, D₂u(n) > 0, D₃u(n) > 0, and D₄u(n) ≤ 0(ii)u(n) > 0, D₁u(n) > 0, D₂u(n) > 0, D₃u(n) < 0, and D₄u(n) ≤ 0(iii)u(n) > 0, D₁u(n) > 0, D₂u(n) < 0, D₃u(n) > 0, and D₄u(n) ≤ 0(iv)u(n) > 0, D₁u(n) > 0, D₂u(n) < 0, D₃u(n) < 0, and D₄u(n) ≤ 0(v)u(n) > 0, D₁u(n) < 0, D₂u(n) > 0, D₃u(n) > 0, and D₄u(n) ≤ 0(vi)u(n) > 0, D₁u(n) < 0, D₂u(n) > 0, D₃u(n) < 0, and D₄u(n) ≤ 0(vii)u(n) > 0, D₁u(n) < 0, D₂u(n) < 0, D₃u(n) > 0, D₄u(n) ≤ 0(viii)u(n) > 0, D₁u(n) < 0, D₂u(n) < 0, D₃u(n) < 0, and D₄u(n) ≤ 0 for all n ≥ n₁.

Proof. The proof is quite obvious, and hence, we omit it.
First, we start with a lemma which ensures the nonexistence of solutions of types (i) − (iv).

Lemma 2. Let {u(n)} be a positive solution of (1). Ifthen cases (i) − (iv) of Lemma 1 are not possible.

Proof. From (H₁) and (8), one can see thatNow, let us assume that {u(n)} is an eventually positive solution of (1) satisfying one of the cases (i) − (iv) from Lemma 1 and there is an integer n₁ ≥ n₀ such that u(n − k) > 0 for n ≥ n₁. Since {u(n)} is increasing, there is a constant M > 0 and an integer n₂ ≥ n₁ such that u(n − k) ≥ M for all n ≥ n₂. Substituting this in (1), one getsSumming (10) from n₂ to n − 1, one can findIf we assume that {u(n)} belongs to either case (i) or case (iii), then from (9) and (11), we obtainwhich contradicts the fact that D₃u(n) is nonincreasing.
Next, assume that case (ii) holds. From (11), we haveorSumming (14) from n₂ to n − 1, we havewhich, in view of (9), yieldswhich obviously contradicts the fact that D₂u(n) is decreasing.
Finally, we assume that case (iv) holds. Proceeding the same way as in the above case, we get (15), that is,.Summing this inequality from n₂ to n − 1, one obtainswhich in view of (8) implies that D₁u(n₂) ≥ f(M)B(n, n₂) ⟶ ∞ as n ⟶ ∞, which contradicts the fact that D₁u(n) is decreasing. This completes the proof.
In our first main result with a simple condition, we show that any nonoscillatory solution of (1) converges to zero as n ⟶ ∞.

Theorem 1. Ifthen any solution of (1) is oscillatory or converges to zero as n ⟶ ∞.

Proof. Let {u(n)} be a nonoscillatory solution of (1). With no loss of generality, we may take n₁ ≥ n₀ such that u(n) > 0 and u(n − k) > 0 for all n ≥ n₁. From Lemma 1, one may have eight possible cases for n ≥ n₁.
From (19) and (H₁), we see that cannot be bounded, and by Lemma 2, case (i) − (iv) are impossible.
Now, let us assume that one of the cases (v) − (viii) holds. Since {u(n)} is decreasing, there is a finite nonnegative limit u(∞) = lim_{n ⟶ ∞}u(n) = M. Assume that M > 0. Then, there is a n₂ ≥ n₁ such that u(n − k) ≥ M for n ≥ n₂ and inequality (10) holds. Then, one can arrive at contradiction to (12) in cases (v) and (vii) and a contradiction to (16) in case (vi). Thus, we conclude that M = 0.
If we assume that case (viii) holds, then we get (18), that is,Summing the last inequality from n₂ to n, we obtainBut the term on the right side of the above inequality tends to ∞ as n ⟶ ∞ due to (19), which contradicts the fact that {u(n)} is decreasing. This completes the proof.
In the following, we present oscillation criteria for (1).

Theorem 2. Iffor any N ≥ n₀, wherethen (1) oscillates.

Proof. Let {u(n)} be a nonoscillatory solution of (1). With no loss of generality, we may take n₁ ≥ n₀ such that u(n) > 0 and u(n − k) > 0 for all n ≥ n₁. By Lemma 1, eight possible cases may occur for all n ≥ n₁.
First, we note that, in view of (H₁), it is necessary for the validity of (22) thatIn view of Lemma 2, the above condition implies that cases (i) − (iv) of Lemma 1 are not possible. Next, we shall consider the remaining possible cases (v) − (viii) separately.
Assume that case (v) holds. From the monotonicity of D₂u(n), we havethat is,Summing the above inequality from n to ∞, one getsUsing (27) and the increasing property of D₂u(n) in (1), there is a constant M > 0 and an integer n₂ ≥ n₁ such thatwhere M₂ = MM₁. Summing the above inequality from n₂ to n − 1, we haveFrom (H₁) and (24), one can easily see thatUsing (30) in (29), we arrive at a contradiction with the fact that {D₃u(n)} is nonincreasing.
Assume that case (vi) holds. From the monotonicity of D₃u(n), we haveTherefore,which implies that is nondecreasing. Using this property, one obtainsHence,and so is nondecreasing. Finally, we getUsing (33) in the above inequality, we getTherefore,Summing this inequality from n₁ to n − 1 and using the monotonicity of D₂u(n), we obtainFrom (31) and (38), one obtainsDividing the above inequality by −D₃u(n) and then taking the lim sup on both sides, one arrives at a contradiction with (22).
Assume that case (vii) holds. From the decreasing property of D₁u(n), one obtainsThus,which means that is nondecreasing. Summing (1) from n₁ to n − 1 and using the above property of , we obtainFurthermore, on using (H₁) and (30), it is easy to see that for any constant b > 0,This, by virtue of (42), contradicts the fact that {D₃u(n)} is nonincreasing.
Finally, assume that case (viii) holds. Summing (1) from n₁ to n − 1, we haveDividing both sides of the above inequality by c₃(n) and summing the resulting inequality again from n₁ to n − 1, one obtainsSimilarly, we obtainwhich contradicts (22). This completes the proof.
Finally, we obtain an oscillation criterion using classical Riccati-type transformation technique.

Theorem 3. If for all sufficiently large N ≥ n₀,then equation (1) oscillates.

Proof. Assume the contrary that {u(n)} is a nonoscillatory solution of (1) for all n ≥ n₀. With no loss of generality, we may take n₁ ≥ n₀ such that u(n) > 0 and u(n − k) > 0 for all n ≥ n₁. By Lemma 1, eight possible cases may occur for all n ≥ n₁. From (48), one may see thatwhich in view of (H₁) implies that B(∞, n₀) = ∞. So by Lemma 2, cases (i) − (iv) from Lemma 1 are not possible. Therefore, it is enough to consider cases (v) − (viii).
First, assume that case (v) holds. From (47), we haveThen, arguing as in the proof of Theorem 2 case (v), we obtain a contradiction.
Next, assume case (vi) holds. Define the functionIn view of (31) and (36), one can obtainand hence,Also, arguing as in the proof of Theorem 2 case (vi), we obtain from (31) and (33) thatBy (1), (54), and the monotonicity of u(n), we conclude thatMultiplying the above inequality by (n + 1) and summing the resulting inequality from n₁ to n − 1, we obtainNow, applying the summation by parts formula and then rearranging, we obtainTherefore, in view of (53) thatwhich contradicts (47).
Assume now that case (vii) holds. Note thatis necessary for (47). Then, for any constant M > 0, we haveProceeding as in the proof of Theorem 2 case (vii), we obtain a contradiction.
Finally, assume that case (viii) holds. DefineFrom (45), we haveOn the other hand, from the monotonicity of D₁u(n), we haveorThen, using (62), we haveNow, multiplying the last inequality by (n + 1) and then summing from n₁ to n − 1, one obtainsHence, in view of (64),which contradicts (48). The proof is now complete.

3. Examples

In this section, we provide some examples to illustrate the applicability and strength of the results obtained in the previous section.

Example 1. Let us consider the following fourth-order delay difference equation:where q₀ > 0 and k is a positive integer. It is easy to verify that condition (19) is satisfied and by Theorem 1, one can conclude that any nonoscillatory solution of (68) converges to zero as n ⟶ ∞.

Example 2. Consider the following fourth-order delay difference equation:where q₀ > 0 and k is a positive integer. By simple calculation, we see that and . Hence, by Theorem 2, equation (69) is oscillatory if q₀ > 6. The same conclusion follows from Theorem 3 if .

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.