Oscillation Criteria for Certain Fourth-Order Nonlinear Delay Differential Equations

Abstract

In this article, we establish some new criteria for the oscillation of fourth-order nonlinear delay differential equations of the form (r2(t)(r1(t)(y (t)) α ) ) + p(t)(y (t)) α + q(t)f (y(g(t))) = 0 provided that the second-order equationis nonoscillatory or oscillatory.Mathematics Subject Classification. 34C10, 39A10.

Oscillation theorems for fourth‐order delay differential equations with a negative middle term

“…Lemma 7. Let (15) hold and v(t) ∈  0 be a positive solution of (P 2 ). Then, v(t) (t) is nondecreasing,…”

“…Theorem 3. Let (5), (13), (15) and (26) hold. Assume that the fourth-order delay differential equation…”

“…For a broader view, we refer the reader to some classical papers [2][3][4][5][6] and the references cited therein. Recently, investigation of higher-order differential equations with a middle term y (n) (t) + p(t)y (n−2) (t) + q(t)y( (t)) = 0 has received a considerable portion of interest (see, for instance Agarwal [7][8][9][10][11][12][13][14][15][16][17] ). However, a little attention has been paid to a case when a difference in the derivative order between the first and the second terms of the studied equation is greater than 2.…”

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.…”

“…for all (t, s) ∈ D 0 .Theorem 3.where P(t, s) = h(t, s) − A(s) H(t, s) and A(t), B(t) are defined in Theorem 2, and Equations(8) or (9) holds with Θ(t) as in Theorem 1. Then, every solution of Equation (1) is oscillatory.Proof.+ω(s) h(t, s) H(t, s) − H(t, s)A(s) ds ≤ H(t, t 5 )ω(t 5 ) +…”