Oscillatory Properties of Odd-Order Delay Differential Equations with Distribution Deviating Arguments

Throughout this work, new criteria for the asymptotic behavior and oscillation of a class of odd-order delay differential equations with distributed deviating arguments are established. Our method is essentially based on establishing sharper estimates for positive solutions of the studied equation, using an iterative technique. Moreover, the iterative technique allows us to test the oscillation, even when the related results fail to apply. By establishing new comparison theorems that compare the nth-order equations with one or a couple of first-order delay differential equations, we obtain new conditions for oscillation of all solutions of the studied equation. To show the importance of our results, we provide two examples.

We limit our discussion to those solutions x of Equation (1) which satisfy sup {|x (ξ)| : ξ 1 ≤ ξ 0 } > 0 for every ξ 1 ∈ [ξ x , ∞). Such a solution x is said to be non-oscillatory if x is positive or negative, ultimately; otherwise, x is said to be oscillatory. If every solutions of Equation (1) is oscillatory, then Equation (1) is called oscillatory. Oscillatory behavioral nature of solutions of various classes of neutral and delay differential equations is of great interest, and often encountered in applied problems in natural sciences, technology, and engineering, see [1,2]. Recently it has been noticed the rising interest of many researchers and papers in studying the qualitative properties of different classes of linear and non-linear differential equations, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14].
Baculikova and Dzurina [18] adopted the Riccati transformation for examining the asymptotic and oscillation behavior of the solutions of advanced differential equations of higher order where τ (ξ) > ξ and n is odd.
Based on the use of iterative technique, we first establish a more sharp estimate of the increasing/decreasing positive solutions of Equation (1). By following the comparison approach with first order delay equations, we attain oscillation of all solutions of Equation (1) under easy application conditions.
In the following, we present some useful lemmas that will be used throughout the results.

Lemma 2.
(Lemma 2 in [18]) If x is a positive solution of Equation (1), then all derivatives x (k) (ξ) , 1 ≤ k ≤ n − 1, are of constant signs, r (ξ) x (n−1) (ξ) α is non-increasing, and x satisfies either Next, we provide the following notations to help us display the results easily: moreover, we denoted the set of all positive solutions of Equation (1) with property Equation (2) or Equation (3) by X + I or X + D , respectively.

Remark 1.
All functional inequalities and properties including increasing, decreasing, positive, etc. are assumed to hold eventually, that is, they are satisfied for all ξ large enough.

Nonexistence of Increasing Positive Solutions
In this section, we obtain nonexistence criteria for increasing positive solutions of odd-order delay differential Equation (1).

Theorem 1.
Assume that η k is defined as in Lemma 3. If, for some δ k ∈ (0, 1) and some k ∈ N, the delay differential equation is oscillatory, then X + I is empty.

Corollary 1.
Assume that η k is defined as in Lemma 3. If, for some δ k ∈ (0, 1) and some k ∈ N, then X + I is empty.
Proof. Applying a well-known criterion Theorem 2 in [13] for first-order delay differential Equation (9) to be oscillatory, we obtain immediately the criterion (11).

Nonexistence of Decreasing Positive Solutions
In this section, we obtain nonexistence criteria for decreasing positive solutions of odd-order delay differential Equation (1).

Theorem 2.
Assume that µ l,k is defined as in Lemma 4. If, for some l ∈ N, then X + D is empty.
Thus, the last inequality has a positive solution. Using Theorem 1 in [10], we see that (23) also has a positive solution, which is a contradiction. The proof is complete.
Proof. Applying a well-known criterion Theorem 2 in [13] for first-order delay differential Equation (23) to be oscillatory, we obtain immediately the criterion (25).
Corollary 3. Assume that η k and µ l,k are defined as in Lemmas 3 and 4, respectively. Then, every solution of Equation (1) is oscillatory if one of the following conditions is provided: (a) Equations (11) and (21) (b) Equations (11) and (25) for some δ k ∈ (0, 1) and some k, l ∈ N.

Conclusions
Through applying an iterative approach in this work, we established sharper estimates for increasing/decreasing positive solutions of Equation (1). Using the principles of comparison, we obtained new oscillation criteria that can be used to test for oscillations, even when the previously known criteria fail to apply.
The memory effect may also appear non-locally when certain fractional derivatives with power, exponential and Mittag-Leffler laws are possibly applied. So, it would be interesting to extend the results of this paper to the fractional delay differential equations, see [28][29][30][31][32].