Self-adjoint differential equations and generalized Karamata functions

Abstract

Howard and Maric have recently developed nice nonoscillation theorems for the differential equation U" + q(t)y = 0 (*) by means of regularly varying functions in the sense of Karamata. The purpose of this paper is to show that their results can be fully generalized to differential equations of the form, (p(t)y?)? + q(t)y = o (**) by using the notion of generalized Karamata functions, which is needed to comprehend how delicately the asymptotic behavior of solutions of (**) is affected by the function p(t).

Asymptotic analysis of positive solutions of generalized Emden-Fowler differential equations in the framework of regular variation

“…In fact, detailed and precise information can be acquired about existence and asymptotic behavior of regularly varying solutions of (A) provided is regularly varying (for example, [8][9][10]12]). Because of the presence of general ( ) ≡ 1 in the differential operator of equation (A) and motivated by papers [5,6] on second order linear and half-linear differential equations, we decide to choose the class RV P of generalized regularly varying functions with respect to P as the basic framework for our asymptotic analysis. Such a choice will prove to be appropriate in the sense that complete analysis can be conducted for all possible RV P -solutions of equation (A) if and are assumed to be RV P -functions (see Section 2 for the definition of ordinary and generalized regularly varying functions).…”

Asymptotic analysis of Emden-Fowler differential equations in the framework of regular variation

“…Actually, we will use a method of proof based on regular variation, which is essentially a chapter in classical real-variable theory and which has many applications as to analytic number theory, complex analysis and probability. Our selection of this method of solving the problem was motivated by the monograph of Marić [15] and the subsequent papers [4][5][6][7][9][10][11][12][13]16] which demonstrate that theory of regularly varying functions in the sense of Karamata provides a powerful tool for the asymptotic analysis of second-order linear and half-linear differential equations with or without functional arguments.…”

On a class of functional differential equations having slowly varying solutions

“…Further study of equation (A) and its generalizations in the spirit of Theorem 1.1 has been carried out by Howard and Marić [5] and Jaroš and Kusano [6], [7].…”