Regularly varying sequences and second order difference equations

Second order linear q-difference equations: Nonoscillation and asymptotics

“…The definition of q-regular variation can be seen as the one which is motivated by the definition of regularly varying sequences, see e.g. [21] and also [9], [14]. But as shown next, thanks to the structure of q N0 , we are able to find a much simpler (and still equivalent) characterization which cannot exist in the classical continuous or the discrete case.…”

Positive strongly decreasing solutions of Emden-Fowler type second-order difference equations with regularly varying coefficients

“…The theory of regularly varying sequences, sometimes called Karamata sequences, was initiated in 1930 by Karamata [22] and further developed in the seventies by Galambos, Seneta and Bojanić in [5,16] and recently in [14,15]. However, until the papers of Matucci and Rehak [30,31], the relation between regularly varying sequences and difference equations has never been discussed. In these two papers, as well as in succeeding papers [32,36], the theory of regularly varying sequences has been further developed and applied in the asymptotic analysis of second-order linear and half-linear difference equations, providing necessary and sufficient conditions for the existence of regularly varying solutions of these equations.…”

On the Existence and the Asymptotic Behavior of Nonoscillatory Solutions of Second Order Quasilinear Difference Equations

“…That the class of regularly varying sequences is well suited for the study of the second order linear di¤erence equations and the half-linear di¤erence equations has been shown by Matucci and Ř ehák in [19,20,21]. They introduced the concept of normalized regularly varying sequences which play a relevant role in the study of the asymptotic behavior of nonoscillatory solutions of linear and half-linear di¤erence equations.…”

“…For that purpose, we need some basic properties of normalized regularly varying sequences listed below and proved in [3,19,20]. In view of Proposition 3.3, if for y A NRVðrÞ…”

“…There exist various necessary and su‰cient conditions for a sequence of positive numbers to be regularly varying (see [3,4,19,20]) and consequently each one of them could be used to define regularly varying sequence. The one that is the most important is the following Representation theorem (see [ and…”