Second order linear q-difference equations: Nonoscillation and asymptotics

A note on asymptotics and nonoscillation of linear $q$-difference equations

“…This equation is related to (1) by p(t) = (a(t) + q + 1)/(q(q − 1) 2 t 2 ) and b(t) ≡ q. In [17] we proved that, under the assumption t 2 p(t) ≤ 1/(q( √ q + 1) 2 ) we have: If the limit…”

“…It is easy to see that (16) expressed in terms of a takes the form lim t→∞ a(t) = A ∈ −∞, −2 √ q . Thus the result in [17] is a special case of Theorem 4, and recall that it can be viewed as a q-version of the sufficient condition for y ′′ + p(t)y = 0 to have regularly varying solutions, see, e.g., [14]. In both settings this condition can be easily shown to be also necessary.…”

“…Various aspects of linear q-difference equations were studied e.g. in [1,2,3,4,6,8,9,11,13,17,18]. For related topics see [10,12] and the references therein.…”

“…We conclude this section with recalling the theory of q-regular variation, see e.g. [17]. A function f : q N 0 → (0, ∞) is said to be q-regularly varying of index ϑ, ϑ ∈ R, if lim t→∞ f (qt)/f (t) = q ϑ , we write f ∈ RV q (ϑ).…”

“…Note that in contrast to the classical theory of regular variation (i.e., for functions of a real variable or of an integer variable, see e.g. [5]), the theory of q-regularly varying functions differs in several basic aspects, is simpler, and provides new types of powerful tools, because the range q N 0 is somehow natural setting for regularly varying behavior, see [17].…”