| Summary: | In this paper we introduce and study $q$-rapidly varying functions on the lattice $q^{N_0}:=\{q^k:k\in N_0\}$, $q>1$, which naturally extend the recently established concept of $q$-regularly varying functions. These types of functions together form the class of the so-called $q$-Karamata functions. The theory of $q$-Karamata functions is then applied to half-linear $q$-difference equations to get information about asymptotic behavior of nonoscillatory solutions. The obtained results can be seen as $q$-versions of the existing ones
in the linear and half-linear differential equation case. However two important aspects need to be emphasized. First, a new method of the proof is presented. This method is designed just for the $q$-calculus case and turns out to be an elegant and powerful tool also for the examination of the asymptotic behavior to many other $q$-difference equations, which then may serve to predict how their (trickily detectable) continuous counterparts look like. Second, our results show that $q^{N_0}$ is a very natural setting for the theory of $q$-rapidly and $q$-regularly varying functions and its applications, and reveal some interesting phenomena, which are not known from the continuous theory.
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