A Philos-Type Oscillation Criteria for Fourth-Order Neutral Differential Equations

Abstract

Some sufficient conditions are established for the oscillation of fourth order neutral differential equations of the form r t z ‴ t α ′ + q t x β σ t = 0 , where z t : = x t + p t x τ t . By using the technique of Riccati transformation and integral averaging method, we get conditions to ensure oscillation of solutions of this equation. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. Moreover, the importance of the obtained conditions is illustrated via some examples.

Improved Conditions for Oscillation of Functional Nonlinear Differential Equations

“…More and more scholars pay attention to the oscillatory solution of functional differential equations, see [2][3][4][5], especially for the second/third-order, see [6][7][8], or higher-order equations see [9][10][11][12][13][14][15][16][17]. With the development of the oscillation for the second-order equations, researchers began to study the oscillation for the fourth-order equations, see [18][19][20][21][22][23][24][25].…”

Behavior of Non-Oscillatory Solutions of Fourth-Order Neutral Differential Equations

“…Proof. It is well-known from [3] (Theorem 2) and [2] (Corollary 2.8) that (14) and (15) imply oscillation of (6) and (9), respectively.…”

New Aspects for Non-Existence of Kneser Solutions of Neutral Differential Equations with Odd-Order

“…There are many authors who studied the problem of oscillation of differential equations of a different order and presented many techniques in order to obtain criteria for oscillation of the studied equations, for example, [1][2][3][4][5][6][7][8][9][10][11][12].…”