On the Oscillatory Properties of Solutions of Second-Order Damped Delay Differential Equations

In the work, a new oscillation condition was created for second-order damped delay differential equations with a non-canonical operator. The new criterion is of an iterative nature which helps to apply it even when the previous relevant results fail to apply. An example is presented in order to illustrate the significance of the results.


Introduction
In this study, we focus on studying the oscillatory properties of solutions to the delay differential equation (DDE) where l ∈ I 0 := [l 0 , ∞), and under the following hypotheses: Hypothesis 1 (H1). β > 0 is a quotient of two odd integers.
DDEs, as a subclass of the functional differential equation (FDE), take into account the system's past, allowing for more accurate and efficient future prediction while also describing certain qualitative phenomena such as periodicity, oscillation, and instability. The concept of delay incorporation into systems is now proposed to play an important role in modeling when representing the time it takes to complete certain secret processes, see [1]. DDE theory has improved our understanding of the qualitative behavior of their solutions and has a wide range of applications in mathematical biology and other fields. DDE nonlinearity and sensitivity analysis has been extensively studied in recent years in a variety of fields, see [2][3][4][5][6].
The problem of determining oscillation criteria for specific FDEs has been a very active research field in the recent decades, and many references and summaries of known results can be found in the monographs by Agarwal et al. [7,8] and Gyori and Ladas [9].
Tunc and Kaymaz [23] established the oscillatory properties of DDE z (l) + h(l)z (l) + q(l)ν(g(l)) = 0, where z(l) = ν(l) + p(l)ν(τ(l)), under the condition Theorem 3. [23] Assume that σ(l) ≤ τ(l) and (8) hold. If there exists a positive function then every solution of (7) is oscillatory In an attempt to reduce the number of possible possibilities for the sign of derivatives of positive solutions, researchers study the DDEs in the canonical case, which often excludes the existence of positive decreasing solutions. On the other hand, in the noncanonical case, one of these possibilities is that the positive solutions are decreasing. The main reason for the difficulty of studying positive decreasing solutions is the probability of their convergence to zero, and this probability prevents the use of one of the most important relationships between derivatives that allows to reduce the order of the equation. It has also been noted that the conditions resulting from the exclusion of positive decreasing solutions have the largest effect on the oscillation criteria. Therefore, the main objective of this work is to study the oscillatory behavior of DDE (1) in the noncanonical case The technique used is based on obtaining criteria of an iterative nature through establishing more sharp estimates for the a 2 (l)a 1/β 0 (l)η 1+1/β (l) η β+1 (l). The iterative nature of the criteria allows us to apply them more than once, even when the other criteria fail.
Thus, from (26), ψ(l) is a positive solution of the delay differential inequality Using Theorem 1 in [24], the associated delay differential (25) has also a positive solution, which contradicts to the assumptions of the theorem. The proof is complete.
Remark 1. Table 1 shows the first value of δ m which satisfies Condition (29) for different special cases of (28). Note also that, in all cases in Table 1, δ 0 < 1/2, which does not fulfill (29).

Conclusions
We have greatly less results for DDEs with noncanonical operator than for the DDEs with canonical operator. So, in this work, new sufficient conditions for the oscillation of second-order damped DDE with noncanonical operator (1) are established. By inferring and improving some properties of positive solutions, we establish oscillation criteria of an iterative nature. For an overview of the main results, see Figure 1. It would be interesting to extend our results to neutral DDEs.