The existence and the precise asymptotic behavior for t -> infinity of all increasing solutions of a class of second order nonlinear equations is proved. For that the use of Karamata class of regularly varying functions is essential.

An asymptotic analysis of increasing solutions x ( t ) of equations of (super-linear) Thomas–Fermi type is performed in the framework of regular variation, leading to the accurate behavior of all such x ( t ) for t → ∞ .

The existence and the asymptotics behavior for the large value of the variable of the positive solutions of generalized Thomas–Fermi equation presented in this article are proved. It is assumed that coefficient q(t) belongs to the class of regularly varying functions in the sense of Karamata. Properties of these functions and the Schauder–Tychonoff fixed point theorem are the main tools for the proofs.

An asymptotic analysis in the framework of Karamata regularly varying functions is performed for the solutions of second order linear differential and functional differential equations in the critical case i.e., when condition (1.5) as given below, holds.

Under the assumption that the second-order linear differential equation (p(t)x')' + q(t)x = 0 is nonoscillatory at infinity and has a fundamental set of solutions {f(t), g(t)}, sufficient conditions are given under which the nonlinear equation (p(t)x')' + q(t)x = F(t, x) possesses solutions x(t) and y(t) which are asymptotic to f (t) and g(t), respectively, as t -> infinity.

Regularity, in the sense of Karamata, (with nonoscillation as a consequence) and the precise asymptotic behaviour of solutions of two functional differential equations are studied.

The existence of solutions belonging to the Karamata class of functions of a class of third order nonlinear differential equations is proved via some more general results on asymptotic equivalence.

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.

The precise asymptotic behaviour at infinity of some classes of nonoscillatory solutions of the half-linear differential equations is determined.

This is the first book offering an application of regular variation to the qualitative theory of differential equations. The notion of regular variation, introduced by Karamata (1930), extended by sev

Such an approximation is obtained without hypotheses on f"(x) and without using the general linear system approach.

A simple method for proving several theorems of Tauberian and Mercerian type for(C,1)and gap(C,1)summability is presented.