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

Řehák, Pavel

Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1107-1134.

Résumé

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 $q^{\mathbb {N}_0}:=\{q^k\colon k\in \mathbb {N}_0\}$ 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.

Détail

DOI : 10.1007/s10587-011-0051-9

Classification : 26A12, 39A12, 39A13, 39A21
Mots-clés : regularly varying functions; $q$-difference equations; asymptotic behavior; oscillation