Symmetry and Its Role in Oscillation of Solutions of Third-Order Differential Equations

Abstract

Symmetry plays an essential role in determining the correct methods for the oscillatory properties of solutions to differential equations. This paper examines some new oscillation criteria for unbounded solutions of third-order neutral differential equations of the form (r2(ς)((r1(ς)(z′(ς))β1)′)β2)′ + ∑i=1nqi(ς)xβ3(ϕi(ς))=0. New oscillation results are established by using generalized Riccati substitution, an integral average technique in the case of unbounded neutral coefficients. Examples are given to prove the significance of new theorems.

On the Oscillatory Behavior of a Class of Even Order Nonlinear Damped Delay Differential Equations With Distributed Deviating Arguments

“…They have come up with many ways to get oscillatory criteria for fourth (or higher) order differential equations. Several studies have had very interesting results related to oscillatory properties of solutions of neutral differential equations and damped delay differential equations with/without distributed deviating arguments [4,7,8,9,10,11,12,13,15,16,17,18,19,20].…”

Nonlinear Differential Equations with Distributed Delay: Some New Oscillatory Solutions

“…Many researchers have intensively studied the topic of oscillation of fourth or higher order differential equations in depth, and many strategies for establishing oscillatory criteria for fourth or higher order differential equations have been developed. Several works, see [6][7][8][9][10][11][12][13][14][15][16][17][18], contain extremely interesting results linked to oscillatory features of solutions of neutral differential equations and damped delay differential equations with or without distributed deviating arguments.…”

Novel Oscillation Theorems and Symmetric Properties of Nonlinear Delay Differential Equations of Fourth-Order with a Middle Term

“…Several researchers have intensively investigated the oscillation of fourth-and higher-order differential equations along with numerous approaches for establishing oscillatory criteria [9][10][11][12][13]. The oscillatory features of solutions have been the subject of several investigations of neutral differential equations [14][15][16], delay differential equations [17,18], and fractional differential equations [19,20]. Dzurina et al in [21] studied the oscillation of the linear fourth-order delay differential equation with damping (r 3 (µ)(r 2 (µ)(r 1 (µ)y (µ)) ) ) + p(µ)y (µ) + q(µ)y(τ(µ)) = 0, by assuming the following:…”