Integral Comparison Criteria for Half-Linear Differential Equations Seen as a Perturbation

Abstract

In this paper, we present further developed results on Hille–Wintner-type integral comparison theorems for second-order half-linear differential equations. Compared equations are seen as perturbations of a given non-oscillatory equation, which allows studying the equations on the borderline of oscillation and non-oscillation. We bring a new comparison theorem and apply it to the so-called generalized Riemann–Weber equation (also referred to as a Euler-type equation).

Closed‐form solutions of second‐order linear difference equations close to the self‐adjoint Euler type

“…8 We also refer the reader to see related articles, [9][10][11][12][13][14][15] and also in the continuous and time-scale domain, we recommend related references. [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] Furthermore, oscillation and other properties of perturbed equations are investigated in previous studies [31][32][33][34][35][36] as well. Other recent results, where the solution of a difference equation is obtained in a closed form, are in previous studies [37][38][39][40] (see also Čermák and Jánský 41 ).…”