Even Order Half-Linear Differential Equations with Regularly Varying Coefficients

Vojtěch Růžička

doi:10.3390/math8081236

Abstract

We establish nonoscillation criterion for the even order half-linear differential equation (−1)nfn(t)Φx(n)(n)+∑l=1n(−1)n−lβn−lfn−l(t)Φx(n−l)(n−l)=0, where β0,β1,…,βn−1 are real numbers, n∈N, Φ(s)=sp−1sgns for s∈R, p∈(1,∞) and fn−l is a regularly varying (at infinity) function of the index α−lp for l=0,1,…,n and α∈R. This equation can be understood as a generalization of the even order Euler type half-linear differential equation. We obtain this Euler type equation by rewriting the equation above as follows: the terms fn(t) and fn−l(t) are replaced by the tα and tα−lp, respectively. Unlike in other texts dealing with the Euler type equation, in this article an approach based on the theory of regularly varying functions is used. We establish a nonoscillation criterion by utilizing the variational technique.

Highlights

Consider the 2n-th order half-linear differential equation n+ ∑ (−1)n−l β n−l f n−l (t)Φ x (n−l ) = 0, (1)l =1 where β 0, β 1, . . . , β n−1 are real numbers, n ∈ N, Φ is the odd power function defined by the relationΦ(s) = |s| p−1 sgn s for s ∈ R, p ∈ (1, ∞) and for each l ∈ {0, 1, . . . , n} the function f n−l is defined, positive and continuous on [S, ∞), where S ∈ R.we assume that f n−l is a regularly varying function of the index α − l p for l = 0, 1, . . . , n and α ∈ R
We assume that f n−l is a regularly varying function of the index α − l p for l = 0, 1, . . . , n and α ∈ R
Functions belonging to SV are called slowly varying functions and the function f n−l can be equivalently described for l = 0, 1, . . . , n as follows: there exists a function Ln−l defined and continuous on [S, ∞) such that

Summary

Introduction

L =1 where β 0 , β 1 , . . . , β n−1 are real numbers, n ∈ N, Φ is the odd power function defined by the relation. We assume that f n−l is a regularly varying (at infinity) function of the index α − l p (the definition is given later) for l = 0, 1, . We write f n−l ∈ RV (α − l p), where RV (θ ) for θ ∈ R denotes the set of all regularly varying functions of the index θ. The two-term even order (Euler type and more general) half-linear differential equations are studied in [1,3,4] and in the book [5] (Section 9.4). We conclude the article with several examples and comments in the last two sections

Preliminaries

Equations with Regularly Varying Coefficients

Examples

Conclusions