Asymptotic Behavior of Solutions to Half-Linear q-Difference Equations

Abstract

We derive necessary and sufficient conditions for (some or all) positive solutions of the half-linear $q$-difference equation $D_q\left(\mathrm{\Phi }\right(D_{q}y\left(t\right)\left)\right)+p\left(t\right)\mathrm{\Phi }\left(y\right(\text{i}tqt\left)\right)=0$, $t\in \{q^k:k\in {\mathbb{N}}_0\}$ with , $\Phi \left(u\right)={\left|u\right|}^{\alpha -1}\mathrm{sgn}u$ with , to behave like $q$-regularly varying or $q$-rapidly varying or $q$-regularly bounded functions (that is, the functions $y$, for which a special limit behavior of $y\left(\text{i}tqt\right)/y\left(t\right)$ as $t\to \mathrm{\infty }$ is prescribed). A thorough discussion on such an asymptotic behavior of solutions is provided. Related Kneser type criteria are presented.