An asymptotic analysis of nonoscillatory solutions of q-difference equations via q-regular variation

We do a thorough asymptotic analysis of nonoscillatory solutions of the q-difference equation Dq(r(t)Dqy(t))+p(t)y(qt)=0 considered on the lattice {qk:k∈N0}, q>1. We classify the solutions according to various aspects that take into account their asymptotic behavior. We show relations among the asymptotic classes. For every positive solution we establish asymptotic formulae. Several discrepancies are revealed, when comparing the results with their existing differential equations or difference equations counterparts; however, it should be noted that many of our observations in the q-case have not their continuous or discrete analogies yet. Important roles in our considerations are played by the theory of q-regular variation and various transformations. The results are illustrated by examples.