2019
Том 71
№ 5

# 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, tR +. The study is based on a comparison of these equations with the difference equations x(t+ 1) = f(x(t)), tR +. 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.

English version (Springer): Ukrainian Mathematical Journal 52 (2000), no. 12, pp 1841-1857.

Citation Example: Derfel' G. A., Romanenko Ye. Yu., Sharkovsky O. M. Asymptotic Discontinuity of Smooth Solutions of Nonlinear $q$-Difference Equations // Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1615-1629.

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