We investigate the class R ∞,s of rapidly varying sequences from different points of view. It is shown:(1) a sequence (c n ) n∈N of positive real numbers is rapidly varying if and only if the corresponding function x → c [x] , x 1, is rapidly varying; (2) the class R ∞,s satisfies some selection properties, as well as gametheoretical and Ramsey-theoretical conditions.

In this paper we consider the class of functions K c introduced by W. Matuszewska (1964, Studia Math. 24, 271-279) and W. Matuszewska and W. Orlicz (1965, Studia Math. 26, 11-24). As a main result we describe, in terms of the class K c , when two strictly increasing functions, as well as their inverse functions, are asymptoticaly equivalent in the strong sense. This result also gives a proper characterization of the class K c .

We present some results about selection properties in the class of double sequences of real numbers.

In this paper we prove that in the class of all measurable positive functions w . Ž . defined on the interval a, qϱ a ) 0 , the class of functions which preserve the Ä w . q Ž . strong asymptotic equivalence on the set of functions x: a, qϱ ¬ ‫ޒ‬ , x t ª 4 qϱ, t ª qϱ , is a class of O O-regularly varying functions with continuous index function. We also prove a representation theorem for functions from this class.

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In this paper we investigate the connection between the asymptotic relations of subordination and the negligence with the generalized inverse function in the class of all nondecreasing and unbounded functions, which are defined on a half-axis [a, +∞) (a > 0). In the main theorems we prove a characterization of all nondecreasing, unbounded slowly varying functions.

We continue investigation of certain classes of sequences of positive real numbers which are important in asymptotic analysis of divergent processes. Among others, we define the notion of quotient speed, give representation results for some kinds of sequences and improve several our earlier results concerning selection principles and games.

We discuss the relationship between the strong asymptotic equivalence relation and the generalized inverse in the class A of all nondecreasing and unbounded functions, defined and positive on a half-axis [a, +∞) (a > 0). In the main theorem, we prove a proper characterization of the function class IRV ∩ A , where IRV is the class of all O-regularly varying functions (in the sense of Karamata) having continuous index function.