Properties 𝐴 and 𝐵 of 𝑛th Order Linear Differential Equations with Deviating Argument

Abstract

Sufficient conditions for the nth order linear differential equation 𝑢(𝑛) (𝑡) + 𝑝(𝑡)𝑢(τ(𝑡)) = 0, 𝑛 ≥ 2, to have Property 𝐴 or Property 𝐵 are established in both the delayed and the advanced cases. These conditions essentially improve many known results not only for differential equations with deviating arguments but for ordinary differential equations as well.

Oscillation and Asymptotic Properties of Second Order Half-Linear Differential Equations with Mixed Deviating Arguments

“…Thus, our results are of high generality and, what is more, they hold for all α > 0, and our technique does not require discussing cases α ∈ (0, 1) and α > 1 separately as it is common, see [1][2][3][4][5][6][7][8][9][10].…”

“…An equation itself is said to be oscillatory if all its proper solutions are oscillatory. There are numerous papers devoted to oscillation theory of differential equations, see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12].…”

“…The first aim of this paper is to establish criteria for N 0 = ∅ and N 2 = ∅. This problem has been solved by many authors, and we mention here the pioneering works of Ladas et al [12] and Koplatadze and Chanturija [9]; in general, authors discuss the condition for N 0 = ∅ only when the deviating argument is delayed (τ(t) < t) and criteria for N 2 = ∅ only for advanced arguments (τ(t) > t), (see [1][2][3][4][5][6][7][8][9][10][11][12]).…”

An Improved Oscillation Result for a Class of Higher Order Non-canonical Delay Differential Equations

“…has been studied by Koplatadze et al [8]. They took into account the oddand even-order cases of this equation.…”

On Higher Order Functional Differential Equations with Property A

“…As early as 1893, A. Kneser [2] obtained sufficient conditions for the equation u (n) (t) + p(t)u(t) = 0 (1.4) with p ∈ L loc (R + ; R + ) to have Property A. Noteworthy results in this direction were obtained by A. Kondrat'ev [1], I. Kiguradze and T. Chanturia [3]. For a differential equation with deviating arguments, which is a special case of (1.1), similar problems were considered in [4]- [6] (see also the references therein). As to functional differential equations, they are studied well enough in [7]- [9] both in linear and nonlinear cases.…”