Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 1107-1134, 2011

# Second order linear $q$-difference equations: nonoscillation and asymptotics

## Pavel Řehák

Pavel Řehák, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Brno, Czech Republic, e-mail: rehak@math.cas.cz

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 $q^{\mathbb{N}_0}:=\{q^k k\in\mathbb{N}_0\}$ 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 $t\to\infty$ of solutions to the $q$-difference equation $D_q^2y(t)+p(t)y(qt)=0$, where $p\colon\smash{q^{\mathbb{N}_0}}\to\mathbb{R}$. 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.

Keywords: regularly varying functions, $q$-difference equations, asymptotic behavior, oscillation

Classification (MSC 2010): 26A12, 39A12, 39A13

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