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# Asymptotic behavior of solutions to half-linear q-difference equations

Autoři 2011 Článek v odborném periodiku Abstract and Applied Analysis Obecná matematika regular variation; q-difference equation; asymptotic behavior; oscillation We derive necessary and sufficient conditions for (some or all) positive solutions of the half-linear $q$-difference equation $D_q(\Phi(D_q y(t)))+p(t)\Phi(y(qt))=0$, $t\in\{q^k:k\in\N_0\}$ with $q>1$, $\Phi(u)=|u|^{\alpha-1}\sgn u$ with $\alpha>1$, to behave like $q$-regularly varying or $q$-rapidly varying or $q$-regularly bounded functions (i.e., the functions $y$, for which a special limit behavior of $y(qt)/y(t)$ as $t\to\infty$ is prescribed). A thorough discussion on such an asymptotic behavior of solutions is provided. Related Kneser type criteria are presented.