More Effective Results for Testing Oscillation of Non-Canonical Neutral Delay Differential Equations

Abstract

In this work, we address an interesting problem in studying the oscillatory behavior of solutions of fourth-order neutral delay differential equations with a non-canonical operator. We obtained new criteria that improve upon previous results in the literature, concerning more than one aspect. Some examples are presented to illustrate the importance of the new results.

Noncanonical Emden–Fowler neutral differential equations: New criteria for oscillation

“…Taking lim sup t→∞ of this inequality, we arrive at a contradiction with (27). The proof is complete.…”

“…They proved that Equation () is oscillatory if $$$$ \underset{t\to \infty }{\limsup}\kern2pt {\left({\theta}_{\alpha }(t)\right)}^{\alpha}\int_{t_0}^tb\left(\varrho \right){\left(1-p\left(\beta \left(\varrho \right)\right)\frac{\theta_{\alpha}\left(\tau \left(\beta \left(\varrho \right)\right)\right)}{\theta_{\alpha}\left(\beta \left(\varrho \right)\right)}\right)}^{\alpha}\mathrm{d}\varrho >1, $$$$ or $$$$ \underset{t\to \infty }{\liminf}\int_{\beta (t)}^t{\left(\frac{1}{a(s)}\int_{t_0}^sb\left(\varrho \right)\left(1-p\left(\beta \left(\varrho \right)\right)\frac{\theta_{\alpha}\left(\tau \left(\beta \left(\varrho \right)\right)\right)}{\theta_{\alpha}\left(\beta \left(\varrho \right)\right)}\right)\mathrm{d}\varrho \right)}^{1/\alpha}\mathrm{d}s>\frac{1}{\mathrm{e}}. $$$$ Very recently, a research movement also appeared concerned with extending the development in the study of second‐order equations to even order equations; see previous studies [24–27].…”

“…Very recently, a research movement also appeared concerned with extending the development in the study of second-order equations to even order equations; see previous studies [24][25][26][27].…”

Neutral Differential Equations of Fourth-Order: New Asymptotic Properties of Solutions

“…Remark 2. By reviewing the results in [23] and by choosing η 0 = 2 and p 0 = 0.12 we have that (27) is oscillatory if h 0 > 2. 206 3.…”

New Monotonic Properties of Positive Solutions of Higher-Order Delay Differential Equations and Their Applications

“…There is an active research movement to verify the qualitative properties (oscillatory, periodicity, stability, boundedness, etc.) for solutions of these equations, see for example [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] as well as the references listed in them. Oscillation theory is one of the branches of the qualitative theory of differential equations, which deals with the issue of oscillatory and non-oscillatory behavior of solutions to differential equations, as well as discusses the issue of the zeros of the solutions and the distances between them.…”