A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay

Abstract

We consider linear differential equations with variable delay of the formIt is well-known that, for the oscillation of all solutions, it is necessary thatOur main result shows that, if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions on p and τ, condition B > 1/e implies that all solutions of the above delay differential equation are oscillatory.

A Survey on Sharp Oscillation Conditions for Delay Difference Equations

“…Pituk came up with the idea to use slowly varying functions for linear delay differential equations with a single constant delay in 2017 [32]. This was generalized in [29,33,34] for several variable delays. For difference equations, the first result can be found in [35].…”

A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays

“…where a : [t 1 , ∞) → R is a continuous function which tends to some finite limit as t → ∞, and b : [t 1 , ∞) → R is a continuously differentiable function for which lim t→∞ b (t) = 0 holds. For more information about slowly varying functions and their characterization the reader is referred to the papers [27][28][29][30] and the references cited therein. In a subsequent paper, Garab [29] studied the case of the differential equation with variable delay…”

New Explicit Oscillation Criteria for First-Order Differential Equations with Several Non-Monotone Delays

“…In the last few decades, the oscillation problem of functional differential equations has received much attention from mathematicians; see, for example, . The reader is referred to [1,2,[4][5][6]9,10,12,13,17,20,22,23,26] and [1,2,7,8,11,14,15,[17][18][19]24,27,[30][31][32][34][35][36][37] for the oscillation of Equations ( 1) and (2), respectively. The results on oscillation criteria of most of these works have iterative forms.…”

“…The results on oscillation criteria of most of these works have iterative forms. Many sharp oscillation criteria for both Equations ( 1) and ( 2) with slowly varying coefficients have been established by [18][19][20]34]. Further new oscillation conditions for Equation (2) with a non-monotone delay have been obtained by [7,8].…”