Asymptotic Discontinuity of Smooth Solutions of Nonlinear q-Difference Equations - Derfel', G. A.; Romanenko, Ye. Yu.; Sharkovsky, O. M. (Ukrains’kyi Matematychnyi Zhurnal, 2000)
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 \(array*20c t [0,T] array | x’(t) | → ∞ \) as T→ ∞ and tend to discontinuous upper-semicontinuous functions in the Hausdorff metric for graphs are typical of the q-difference equation.