Asymptotic and oscillatory characteristics of solutions of neutral differential equations

The paper investigates third-order linear neutral differential equations in the non-canonical case, aiming to simplify the complexity of such equations by transforming them into the canonical form. This transformation reduces the number of potential cases for positive solutions and their derivatives from four in the non-canonical case to two in the canonical case, significantly facilitating the derivation of results. Using an iterative method, we establish conditions that exclude the existence of positive solutions fulfilling the equation. Furthermore, by employing a comparison approach with first-order equations, we derive additional conditions that exclude the existence of Kneser-type solutions that satisfy the equation. By combining these conditions, we derive new oscillation criteria that guarantee the oscillation of all solutions satisfying the studied equation. Our findings extend and generalize existing results in the literature. We provide three illustrative examples to demonstrate our results' validity and significance.