On a class of functional differential equations having slowly varying solutions

Abstract

Abstract. Functional differential equations with deviating arguments are studied for the first time in the framework of Karamata regularly varying functions. A sharp condition is established for the existence of slowly varying solutions for a class of second order linear equations of the form x = q(t)x(g(t)), both in the retarded and in the advanced case.

On increasing solutions of half-linear delay differential equations

“…Our results are essentially new also in the linear case, i.e., when α = 2. The linear version of the existence results mentioned in the previous item can be found in [9][10][11].…”

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.…”

Regularly varying solutions of half-linear functional differential equations with retarded arguments

“…was discovered for the rst time by Kusano and Mari¢ [7], who established a sharp condition under which (1.2) has a slowly varying solution in the sense of Karamata. Theorem A (Kusano and Mari¢ [7]).…”