Smoothness of solutions for delay-difference equations
A real difference equation of the form (1) \(x(n+1) = f(n,x(n), x(n-k_ n))\), \(n \geq 0\), is considered. Here \(k_ n \in \{0, \dots, r\}\) for all \(n \geq 0\) and for some nonnegative integer \(r\). It is assumed that the partial derivatives of \(f\) with respect to the second and third variable are continuous. For \((a_ 0, \dots, a_ r) \in \mathbb{R}^{r+1}\) let \(x(n, a_ 0, \dots, a_ r)\) denote the solution of (1) satisfying the initial condition \((x(-r), \dots, x(0)) = (a_ 0, \dots, a_ r)\). It is proved that the partial derivatives of \(x\) with respect to its last \(r+1\) variables exist.
scientific article; zbMATH DE number 641438
scientific article