Second order linear q-difference equations: Nonoscillation and asymptotics

Abstract

The paper can be understood as a completion of the q-Karamata theory along with a related discussion on the asymptotic behavior of solutions to the linear q-difference equations. The q-Karamata theory was recently introduced as the theory of regularly varying like functions on the lattice \({q^{{{\Bbb N}_0}}}: = \left\{ {{q^k}:k \in {{\Bbb N}_0}} \right\}\) with q > 1. In addition to recalling the existing concepts of q-regular variation and q-rapid variation we introduce q-regularly bounded functions and prove many related properties. The q-Karamata theory is then applied to describe (in an exhaustive way) the asymptotic behavior as t → ∞ of solutions to the q-difference equation Dq2y(t) + p(t)y(qt) = 0, where \(p:q^{\mathbb{N}_0 } \to \mathbb{R}\). We also present the existing and some new criteria of Kneser type which are related to our subject. A comparison of our results with their continuous counterparts is made. It reveals interesting differences between the continuous case and the q-case and validates the fact that q-calculus is a natural setting for the Karamata like theory and provides a powerful tool in qualitative theory of dynamic equations.