On the oscillatory behavior of solutions of third order nonlinear neutral differential equations

Abstract

The achieved results are based on the new comparison principles, which help us decrease the problem of the wavering between the third and first-order equations. Examples are given to prove the significance of new theorems.

Kamenev-Type Criteria for Testing the Asymptotic Behavior of Solutions of Third-Order Quasi-Linear Neutral Differential Equations

“…The new criteria ensure that all nonoscillatory solutions converge to zero. The results provided in this work improve and extend the well-known results in previous works; for instance, see [41][42][43][44]. It would also be of interest to use this approach to study the equation a(s) (x(s) + η(s)x(g(s))) (n−1) r ′ + q(s)x r (τ(s)) = 0, where n ≥ 3.…”

“…While using the results we obtained, we find that the solutions of ( 38) are oscillatory or tend to zero. Therefore, our results improve the results in [42,44].…”

“…Remark 2. We note that additional conditions were mentioned in [42,44], including the composition condition (τ • g = g • τ), which is a harsh condition on the delay functions, while we were able to dispense with these conditions in our results. We also note that the results we obtained are considered an expansion and extension of both [41,43], as we find that in [41], (1) was studied when 0 ≤ η(s) ≤ η < 1, and we find in [43] that Equation (1) was studied when a = 1 and η(s) ≥ 1, while in our study, Equation (1) was studied when 0 ≤ η(s) ≤ η 0 < ∞.…”

“…We find that Theorem 1 in [42] and Theorem 2 in [44] cannot be applied to this equation because τ(s) = s/2 > g(s) = s/4. While using the results we obtained, we find that the solutions of ( 38) are oscillatory or tend to zero.…”

“…They presented novel sufficient criteria for any solution of the studied equation to be either oscillating or converging to zero. Kumar and Ganesan [44] discussed the third-order nonlinear NDDE in the form…”