Asymptotic discontinuity of smooth solutions of nonlinear \(q\)-difference equations (Q2722247)

The authors study the asymptotic behavior of solutions to the functional equation NEWLINE\[NEWLINEx(qt+1)=f(x(t)),\quad t\in \mathbb R^+,\;q>1,NEWLINE\]NEWLINE where \(f\in C^1(I,I), I\) being a closed interval.NEWLINENEWLINENEWLINELet NEWLINE\[NEWLINE\Phi:=\{\varphi(t)\in C^1([0,1),I):\varphi(1-0)=f(\varphi(0)),\quad q\varphi '(1-0)=f'(\varphi(0))\varphi '(0)\}.NEWLINE\]NEWLINE For any \(\varphi(t)\in \Phi \) the initial condition \(x(t)=\varphi(t)\), \(t\in [0,1)\), determines a \(C^1\)-smooth solution NEWLINE\[NEWLINEx_{\varphi }(t)=(f^i\circ \varphi)(\alpha ^{-i}(t)),\quad t\in \alpha^i([0,1)),\;i=0,1,\ldots, NEWLINE\]NEWLINE where \(\alpha(t):=qt+1\). First the authors prove that \(x'_\varphi(t)\to 0\), \(t\to \infty \), provided that \(q>\max_{t\in I}|[f^m(t)]'|^{1/m}\) for a certain natural \(m\).NEWLINENEWLINENEWLINENext, the case is considered where the latter condition fails. Let \(V_\delta(x)\) denote the \(\delta \)-neighborhood of the point \(x\) in \(I\). The sets \(Q_f(x)=\bigcap_{\delta >0}\bigcap_{j>0} \overline{\bigcup_{i>j}f^i(V_\delta(x))}\) and \(D(f):=\{x\in I: \text{int }Q_f(x)\neq\emptyset \}\) are called the influence domain of the point \(x\) under the mapping \(f\) and the separator of the mapping \(f\) respectively. It appears that if \(f'(x)\neq 0\) \(\forall x\in \overline{D(f)}\) then the periods of \(Q_f(x)\) for all \(x\in I\) have a l.c.m. which is denoted as \(p\). Let \(\Omega(f)\) be the set of non-wandering points of \(f\) and \(\Omega_-(f)=\Omega(f)\bigcap D(f)\). Under the assumption that \(q<\min_{x\in \Omega_-(f)}|[f^m(x)]'|^{1/m}\) for some natural \(m\), the authors show that NEWLINE\[NEWLINE\lim_{i\to \infty }|x_\varphi '(\alpha^i(t))|=\infty \quad \forall t\in \{t\in [0,1]:\varphi(t)\in D(f)\}:=D(f,\varphi).NEWLINE\]NEWLINE Moreover in the case where \(D(f,\varphi)\neq\emptyset \) the solution \(x_\varphi(t)\) tends (in Hausdorff metric for graphs) to a discontinuous upper semicontinuous function. As an example, the case of a standard family of quadratic mappings \(f_{\lambda }(x)=\lambda x(1-x)\), \(\lambda \in (3,4),\) is considered.