Dragan Djurčić

Double Sequences and Selections

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

Relations between Sequences and Selection Properties

We consider the set 𝕊 of sequences of positive real numbers and show that some subclasses of 𝕊 have certain nice selection and game theoretic properties.

A Theorem of Galambos-Bojanić-Seneta Type

In the theorems of Galambos-Bojanic-Senetas type, the asymptotic behavior of the functions 𝑐 [ 𝑥 ] , 𝑥 ≥ 1 , for 𝑥 → + ∞ , is investigated by the asymptotic behavior of the given sequence of positive numbers ( 𝑐 𝑛 ) , as 𝑛 → + ∞ and vice versa. The main result of this paper is one theorem of such a type for sequences of positive numbers ( 𝑐 𝑛 ) which satisfy an asymptotic condition of the Karamata type l i m 𝑛 → ∞ 𝑐 [ 𝜆 𝑛 ] / 𝑐 𝑛 > 1 , for 𝜆 > 1 .

On the class S 0 ¯ of real sequences

Abstract We prove that certain classes of sequences of positive real numbers satisfy some selection principles related to a special kind of convergence.

Summability of Sequences and Selection Properties

We prove that some classes of summable sequences of positive real numbers satisfy several selection principles related to a special kind of convergence.

On a subclass of the class of rapidly varying sequences

We define and study the class KR ∞ , s which is a proper subclass of the class R ∞ , s of rapidly varying sequences of index of variability ∞. Then, we prove a theorem of the Galambos-Bojanic-Seneta type for this class, as well as a representation theorem in the Karamata sense. We also give some important asymptotic properties and characterizations of sequences in KR ∞ , s .

Note on rapidly varying sequences

In this paper relation between rapidly varying sequence (cn) and its generated function f(x) = c[c], x ≥ 1 is considered. That relation is expressed by the concept of the Bojanić–Seneta proposition type for rapidly variability.