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 q for which smooth bounded solutions that possess the property max t∈[0,T]|x′(t)|→∞ as T → ∞ and tend to discontinuous upper-semicontinuous functions in the Hausdorff metric for graphs are typical of the q-difference equation.
ASJC Scopus subject areas
- Mathematics (all)