Critical oscillation constant for half-linear differential equations with periodic coefficients

Abstract

We compute explicitly the oscillation constant for certain half-linear secondorder differential equations involving periodic coefficients. If these periodic functions are constants, our results reduce to the well-known oscillation constants for half-linear Euler and Riemann-Weber differential equations.

Oscillation result for half‐linear dynamic equations on timescales and its consequences

“…If our conjecture is true, only the threshold case will be unsolved (see, eg, the dashed curve in Figure ). Only for β = 0, differential equations with coefficients in this borderline case are proved to be non‐oscillatory for periodic coefficients in Hasil and Veselý (see also Došlý and Hasil and for sums of periodic coefficients in Hasil and Veselý . Nevertheless, with regard to the methods described in Veselý, we suppose that the non‐oscillation in the borderline case cannot be proved for all equations with more general coefficients than the periodic ones.…”

“…Let us considerEquation 53, where the functions r, s are rd-continuous, bounded, and -periodic such that inf t∈T r(t) > 0 and inf t∈T s(t) ≥ 0. The corollary follows from Corollary 5.1 and from the fact that the values r , s defined in(54) are constant for all t ∈ T. Hence, the positive constant can be omitted. Let us consider the timescale T with period 6 such that T ∩ [0, 6] = [0, 2] ∪ {3, 5, 6}.…”

Half-linear Euler differential equations in the critical case

“…It suggests to investigate a similar problem for the more general equation (6). (iii) In [1], motivated by the linear case treated in [7], [8], [9], we have investigated oscillatory properties of the equation…”

Conditional Oscillation of Half-Linear Differential Equations with Coefficients Having Mean Values

“…in [11] (see also [12]). Later, this result has been extended in a number of papers (e.g., [13][14][15][16][17]), where equations of the below given form (6) have been treated with coefficients , replaced by perturbations consisting of constant or periodic functions and iterated logarithms. Nevertheless, the most general result (concerning the topic of this paper) can be found in [18], where the equation…”