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

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Authors

ŘEHÁK Pavel

Year of publication 2012
Type Article in Periodical
Magazine / Source Journal of Physics A: Mathematical and Theoretical
MU Faculty or unit

Faculty of Education

Citation
Doi http://dx.doi.org/10.1088/1751-8113/45/5/055202
Field General mathematics
Keywords q-difference equation; asymptotic behavior; regular variation; oscillation
Description 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.
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