Oscillation Criteria for Certain nth Order Differential Equations with Deviating Arguments

Abstract

Oscillation criteria for nth order differential equations with deviating arguments of the form x n−1 t α−1 x n−1 t + F t x g t = 0 neven are established, where g ∈ C t 0 ∞ F ∈ C t 0 ∞ × , and α > 0 is a constant. 

Oscillation of Second-Order Neutral Differential Equations

“…Furthermore, sðtÞ can be a delayed argument and sðtÞ À t can even oscillate. However, to achieve such flexibility, we are forced to require, as in [4], monotonicity of t and that t s ¼ s t: The question regarding the study of oscillatory properties of equation (1) with other methods that do not require assumption ðH 3 Þ remains open at the moment.…”

Existence for nonoscillatory solutions of higher order nonlinear neutral differential equations

“…Clearly, equations (2) and (3) are special forms of the equation (1). Parhi and Rath [12], Kulenovic and Hadziomerspahic [8] proved the following results by using Banach contraction mapping principle.…”

“…We refer the reader to [1]- [15] and the references cited therein. Oscillatory and nonoscillatory behavior of solutions of the forced first order neutral functional differential equation (2) d dt [x(t) + C(t)x(t − τ )] + Q 1 (t)f 1 (x(t − σ 1 )) = g(t), t t 0 , and of the second order neutral functional differential equation with positive and negative coefficients…”

Oscillation criteria for fourth-order functional differential equations

“…Most of the work on this subject, however, has been restricted to first and second-order equations as well as equations of type (1.1) and (1.2) when α = 1 and other higher-order equations. For recent contributions, we refer to [1][2][3][4][5][6][7][8][9][10][11][12]. It appears that little is known regarding the oscillation of equation (1.1) and (1.2).…”