Differential-difference equations reducible to difference and \(q\)-difference equations (Q1600405)

Asymptotic properties of solutions of the differential-difference equation \[ x'(qt+1)=h(x(t))x'(t), \quad t\geq 0,\;q\geq 1 \tag{*} \] are investigated. Let \(\varphi\in C^1([0,1];\mathbf R)\) satisfies \(\varphi'(1)=h(\varphi(0))\varphi'(0)\). A solution \(x\) of (*) satisfying \(x(t)=\varphi(t)\), \(t\in [0,1)\), is denoted by \(x_{\varphi}\). In the investigation of asymptotic properties of this solution an important role is played by the so-called attendant map \(f_{[q,\varphi]}: z\longmapsto qf(z)+\lambda [q,\varphi]\), where \(f\) is an antiderivative of \(h\) and \(\lambda[q,\varphi]=\varphi(1)-qf(\varphi(0))\). It is shown that solutions of (*) exhibit certain ``strange'' properties, comparing with solutions of ordinary differential equation of the form (*), in particular, it is shown that (*) admits unbounded solutions with the property that \(\lim_{T\to \infty}\sup_{t\in [0,T]}|x'(t)|=\infty\). A possibility to extend the results to a more general equation than (*) are discussed as well.