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 -