Rectifiable oscillations in second-order linear differential equations

Abstract

We study the linear differential equation (P ): y (x) + f (x)y(x) = 0, on I = (0, 1), where the coefficient f (x) is strictly positive and continuous on I , and satisfies the Hartman-Wintner condition at x = 0. The four main results of the paper are: (i) a criterion for rectifiable oscillations of (P ), characterized by the integrability of 4 √ f (x) on I ; (ii) a stability result for rectifiable and unrectifiable oscillations of (P ), in terms of a perturbation on f (x); (iii) the s-dimensional fractal oscillations (for which we assume also f (x) ∼ cx −α when x → 0, α > 2, and s = max{1, 3/2 − 2/α}); and (iv) the co-existence of rectifiable and unrectifiable oscillations in the absence of the Hartman-Wintner condition on f (x). Explicit examples related to the above results are given.

Rectifiable oscillations in second-order half-linear differential equations

“…Furthermore, rectifiability of the oscillations of a linear differential equation of Euler type y + λx −α y = 0, x ∈ I , (λ > 0 for α > 2 and λ > 1/4 for α = 2), depends on the parameter α in the sense that this equation is rectifiable (resp., unrectifiable) oscillatory on I provided 2 ≤ α ≤ 4 (resp., α > 4), see [15,25]. In more general setting, one can study the rectifiable oscillations for the linear differential equation y + f (x)y = 0, x ∈ I , where the coefficient f (x) is positive and smooth in I , singular at x = 0, and satisfies the so-called Hartman-Wintner conditions near x = 0 as in the following result, see Theorem 1.4 in [11].…”

“…The purpose of this paper is to generalize our early results from [11] to the more general half-linear equation (1). It turns out that for the study of rectifiable oscillations, one does not need to use the additive property of solution-space as in the case of linear equations.…”

“…In an early paper [11], we have studied the harmonic oscillator equation (P): y + f (x)y = 0, i.e. Eq.…”

Nonrectifiable oscillatory solutions of second order linear differential equations

“…Kwong, Pašić and J. S. W. Wong [7] gave Theorem A when p(x) ≡ 1. The proof of Theorem A is based on the asymptotic formula of oscillatory solutions of (1.1), which is obtained from the Hartman-Wintner condition (1.3).…”

On one oscillatory criterion for the second order linear ordinary differential equations

“…Oscillatory analysis of second order linear ordinary differential equations is one of the important problems in the qualitative theory of differential equations and it is the subject of numerous papers (see [16] and cited works therein [1,2,4,5,[7][8][9][10][11][12][13][14][15]17]). …”