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

Zuzana Pátíková

doi:10.3390/math9050502

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).

Highlights

We continue our research on Hille–Wintner-type comparison criteria for half-linear, second-order differential equations and provide an answer to one of the open problems stated in [1]
Equation (1) can be seen as a generalization of the second-order linear Sturm–Liouville linear equation, to which it reduces for p = 2, and it is well-known that many techniques for linear equations work effectively for half-linear equations too
Recall that one of the differences between half-linear and linear equations is well-visible in the notation—the attribute “half-linear” refers to the fact that the solution space of (1) has only one of the two linearity properties, where it is homogenous but not additive

Summary

Introduction

We continue our research on Hille–Wintner-type comparison criteria for half-linear, second-order differential equations and provide an answer to one of the open problems stated in [1]. Equation (1) can be seen as a generalization of the second-order linear Sturm–Liouville linear equation, to which it reduces for p = 2, and it is well-known that many techniques for linear equations work effectively for half-linear equations too. Recall that one of the differences between half-linear and linear equations is well-visible in the notation—the attribute “half-linear” refers to the fact that the solution space of (1) has only one of the two linearity properties, where it is homogenous but not additive. To refer to the most current results of the oscillation theory of (1), let us mention, for example, papers [2,3,4,5,6]

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