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# On certain asymptotic class of solutions to second order linear q-difference equations

Autoři 2012 Článek v odborném periodiku Journal of Physics A: Mathematical and Theoretical http://dx.doi.org/10.1088/1751-8113/45/5/055202 Obecná matematika q-difference equation; asymptotic behavior; regular variation; oscillation The paper deals with the linear second order $q$-difference equation $y(q^2t)+a(t)y(qt)+b(t)y(t)=0$, $b(t)\ne 0$, considered on $\{q^k:k\in\N_0\}$, $q>1$. The class of functions satisfying the relation $y(qt)/y(t)\sim\omega(t)$ as $t\to\infty$ for some function $\omega$ is introduced and studied. Sufficient and necessary conditions are established for the equation to have solutions in this class. Related results concerning estimates for solutions and (non)oscillation of all solutions are discussed. A comparison with existing results is made and some applications are given.