In the paper On asymptotic solutions of Friedmann equations (Mijajlovic et al. 2012), the theory of regularly varying functions in the sense of Karamata is applied in an asymptotic analysis of solutions of Friedmann equations. As is well known, solutions of these equations are used to represent cosmological parameters. Therefore, according to the theory of regularly varying functions all cosmological parameters depend on a function ?(t) such that limt?1 ?(t) = 0 and which appears in their integral representation. In this paper we derive a differential equation for the parameter ?(t), discuss its solutions and give some physical interpretations.

In the paper On asymptotic solutions of Friedmann equations (Mijajlovic et al. 2012), the theory of regularly varying functions in the sense of Karamata is applied in an asymptotic analysis of solutions of Friedmann equations. As is well known, solutions of these equations are used to represent cosmological parameters. Therefore, according to the theory of regularly varying functions all cosmological parameters depend on a function epsilon(t) such that lim(t ->infinity) epsilon(t) = 0 and which appears in their integral representation. In this paper we derive a differential equation for the parameter epsilon(t), discuss its solutions and give some physical interpretations.

The set of positive solutions of Thomas Fermi type differential equationx" = q(t)phi(x)is studied under the assumptions that q, phi are regularly varying functions in the sense of Karamata. It is shown that such solutions exist and their accurate asymptotic behavior at infinity is determined.

The existence of slowly and regularly varying solutions in the sense of Karamata implying nonoscillation is proved for a class of second order nonlinear retarded functional differential equations of Thomas-Fermi type. A motivation for such study is the extensively developed theory offering a number of properties of regularly and slowly varying functions ([2])-consequently of such solutions of differential equations. As an illustration, the precise asymptotic behaviour for t -> infinity of the slowly varying solutions for a subclass of considered equations is presented.

This paper surveys the known properties and presents many new results for the Legendre differential expression and the associated linear differential operators in its right-definite and left-definite Hilbert function spaces. The main tool for obtaining the best-possible form of these results is an integral operator inequality due to Chisholm, Everitt, and Littlejohn. Many of the results follow from an unpublished manuscript of Everitt and Marić and a forthcoming paper of Arvesú, Littlejohn, and Marcellán. A recent paper on left-definite differential operators by Vonhoff provides for an interesting comparison with the properties and results in this paper.

The role of the regularly varying functions in the sense of Karamata in the qualitative theory of the equations in question is presented. 1991 AMS Mathematics Subject Classication: 34E05, 26A12

In this paper we prove a theorem on the existence and asymptotic behaviour of nonoscillatory solutions of the equation x''+p(t)x=0, where p(t)=(-lambda(2)+h(t))t(-2 alpha) with lambda > 0 and 0 < alpha < 1. The coefficient p need not be onesigned. Examples show that the same asymptotic formula can hold either when p(t) is eventually negative, or when it oscillates. Moreover, the result can be applied to cases where p is negative, but is such that the classical Lioville-Green approximation formula cannot be used.

We assume p > 0 on (0, cc) and p E C”(0, cc) for an appropriate positive integer n. Usual Wronskian techniques can be used to construct and estimate linearly independent solutions. From our results we can determine the precise asymptotic behavior of the solutions in question for classes of functions p to which the usual LiouvilleeGreen approximations Cl, Chap. 4, Section 4; 2, Chap. 11, Example 9.6; 3, pp. 1 l(r122] cannot be applied. An example, for which neither the approximation

A proof of the classical Bernstein’s inequality for Legendre polynomials via the Stieltjes representation for the latter is presented.

1. In this paper we study the asymptotic behaviour of solutions of algebraic equations with real functions as coefficients, using mainly algebraic properties of the class to which the coefficients belong. To that end we introduce the notion of an m-group of functions and prove the main theorem by a procedure originated in [1]. As a corollary we obtain sufficient conditions for a class F of functions to possess the property that solutions of algebraic equations with coefficients in F are again members of F. We conclude by applying these results to the classical Hardy's logarithmico-exponential class ℋ [2]

The asymptotic behavior at infinity of solutions of the equation u′ = P(u, t)Q(u, t) is studied. P, Q are polynomials in u whose coefficients are functions of t, and belong to the Hardy class H (i.e., to the set of all real-valued functions defined by finite many ordinary algebraic, exp, and log operations.) It is proved that, for any continuously differentiable solution u(t), there exists one or the other of the asymptotic formulae u(t) ~ h(t), In u(t) ~ h(t), within the class H, i.e., h(t) ϵ H.