Asymptotic behavior of a sequence defined by iteration with applications

Abstract

We consider the asymptotic behavior of some classes of sequences defined by a recurrent formula. The main result is the following: Let f: (0, ∞)2 → (0, ∞) be a continuous function such that (a) 0 < f(x, y) < px + (1 - p)y for some p ε (0, 1) and for all x, y ε (0, α), where (formula prsented) uniformly in a neighborhood of the origin, where (formula prsented). Then the sequence (xn) satisfies the following asymptotic formula: (formula prsented).