TY - JOUR
T1 - Asymptotic discontinuity of smooth solutions of nonlinear q-difference equations
AU - Derfel, G. A.
AU - Romanenko, E. Yu
AU - Sharkovsky, A. N.
PY - 2000/1/1
Y1 - 2000/1/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=52849096963&partnerID=8YFLogxK
U2 - 10.1023/A:1010499708743
DO - 10.1023/A:1010499708743
M3 - Article
AN - SCOPUS:52849096963
SN - 0041-5995
VL - 52
SP - 1841
EP - 1857
JO - Ukrainian Mathematical Journal
JF - Ukrainian Mathematical Journal
IS - 12
ER -