Some oscillation results for nonlinear second-order differential equations with damping

Abstract

The aim of this paper is to investigate oscillatory properties of a class of second-order nonlinear differential equations with damping. Employing the generalized Riccati transformation and a class of functions, several oscillation criteria are presented that improve the results obtained in the literature. Two examples are presented to demonstrate the main results. MSC: 34K11

On the Oscillatory Behavior of Solutions of Second-Order Non-Linear Differential Equations with Damping Term

“…An essential feature of the results in [18] is that the assumption of positivity of function U(χ(ι)) is not required. Jiang et al [15] discussed a class of equations more general than (1) and ( 2), where they employed the generalized Riccati transformation and a class of functions to establish several oscillation criteria for monotonic and non-monotonic functions to improve some results obtained in the literature. In [26], Wang and Song studied a class of second-order nonlinear differential equations with a damping term which is more general than (1) and (2).…”

“…It is notable (see [23]) that the importance of function U(χ(ι)) is closely related to the presence of a damping term in Equation (2), which makes reduction to simpler differential equations either very complicated or impossible. We note that in all of those papers, the authors assume that function U(χ(ι)) is bounded by constants as 0 < c ≤ U(χ(ι)) ≤ c 1 , even in papers [15,26], where the authors consider equations more general than (1) and (2), however, they also assumed constant bounds for the function U(χ(ι)) as well. Moreover, the authors in [22] claimed that most of oscillation criteria for Equation (2) require that function U(χ(ι)) must be bounded away from zero by positive constant c. In fact, our results in the present article show that this claim is not always necessary.…”

Oscillation criteria of third-order neutral differential equations with damping and distributed deviating arguments

“…Thus, to enrich the method of halflinear TOND differential equations with damping, here, we introduce several interesting approaches to second-order differential equations oscillation problems. In [17][18][19][20], a new operator is created. The operator is flexible in application and it is superior in estimation parameters of some second-order equations.…”

On the oscillatory behaviour of solutions of nonlinear conformable fractional differential equations

“…Very recently in [8], the authors considered linear conformable fractional differential equation and then they obtained some oscillation results. In fact, oscillation properties of solutions of the fractional(or integer) order differential(or difference) equations has been the subject of intensive investigation [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. We are strongly motivated by [8] and literature and are concerned with the oscillatory behavior of solutions of the following conformable fractional differential equation:…”