Asymptotic representation of intermediate solutions to a cyclic systems of second-order difference equations with regularly varying coefficients

摘要

The cyclic system of second-order difference equations ?(pi(n)|?xi(n)| αi?1?xi(n)) = qi(n)|xi+1(n + 1)| βi?1 xi+1(n + 1), for i = 1, N where xN+1 = x1, is analysed in the framework of discrete regular variation. Under the assumption that αi and βi are positive constants such that α1α2 · · · αN > β1β2 · · · βN and pi and qi are regularly varying sequences it is shown that the situation in which this system possesses regularly varying intermediate solutions can be completely characterized. Besides, precise information can be acquired about the asymptotic behavior at infinity of these solutions.