Second order linear q-difference equations: Nonoscillation and asymptotics

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

We study the linear second order q-difference equation y(q2t) + a(t)y(qt) + b(t)y(t) = 0 on the q-uniform lattice {qk : k ∈ N0} with q > 1, where b(t) 6= 0. We establish various conditions guaranteeing the existence of solutions satisfying certain estimates resp. (non)oscillation of all solutions resp. q-regular boundedness of solutions resp. q-regular variation of solutions. Such results may p...

Oscillation and Nonoscillation Criteria for Second-order Linear Differential Equations

Sufficient conditions for oscillation and nonoscillation of second-order linear equations are established. 1. Statement of the Problem and Formulation of Basic Results Consider the differential equation u′′ + p(t)u = 0, (1) where p : [0, +∞[→ [0, +∞[ is an integrable function. By a solution of equation (1) is understood a function u : [0,+∞[→] − ∞, +∞[ which is locally absolutely continuous tog...

q-Karamata functions and second order q-difference equations

In this paper we introduce and study q-rapidly varying functions on the lattice q0 := {qk : k ∈ N0}, q > 1, which naturally extend the recently established concept of q-regularly varying functions. These types of functions together form the class of the so-called q-Karamata functions. The theory of q-Karamata functions is then applied to half-linear q-difference equations to get information abo...

Plancherel-Rotach asymptotics of second-order difference equations with linear coefficients

Oscillation and Nonoscillation of Forced Second Order Dynamic Equations

Oscillation and nonoscillation properties of second order Sturm–Liouville dynamic equations on time scales attracted much interest. These equations include, as special cases, second order self-adjoint differential equations as well as second order Sturm–Liouville difference equations. In this paper we consider a given (homogeneous) equation and a corresponding equation with forcing term. We giv...