Asymptotic Discontinuity of Smooth Solutions of Nonlinear $q$-Difference Equations

Abstract

We investigate the asymptotic behavior of solutions of the simplest nonlinear q-difference equations having the form x(qt+ 1) = f(x(t)), q> 1, t∈ R ⁺. The study is based on a comparison of these equations with the difference equations x(t+ 1) = f(x(t)), t∈ R ⁺. It is shown that, for “not very large” q> 1, the solutions of the q-difference equation inherit the asymptotic properties of the solutions of the corresponding difference equation; in particular, we obtain an upper bound for the values of the parameter qfor which smooth bounded solutions that possess the property \(\begin{array}{*{20}c} {\max } \\ {t \in [0,T]} \\ \end{array} \left| {x'(t)} \right| \to \infty \) as T→ ∞ and tend to discontinuous upper-semicontinuous functions in the Hausdorff metric for graphs are typical of the q-difference equation.