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.
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Derfel', G.A., Romanenko, E.Y. & Sharkovsky, A.N. Asymptotic Discontinuity of Smooth Solutions of Nonlinear q-Difference Equations. Ukrainian Mathematical Journal 52, 1841–1857 (2000). https://doi.org/10.1023/A:1010499708743
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DOI: https://doi.org/10.1023/A:1010499708743