Positive evolution families solving nonautonomous difference equations (Q1928249)

The author considers the delay Cauchy problem \(\Phi(t,f_t)=0\) (\(t\geq s\in \mathbb{R}\)), \(f_s=g\in X\), where \(X\) is a space of \(\mathbb{C}^n\)-valued functions on \([0,1]\), \(f:[s,\infty)\to \mathbb{C}^n\), and \(\Phi(\cdot,\cdot):\mathbb{R}\times W^{1,1}([0,1],\mathbb{C}^n)\to \mathbb{C}^n\) takes the form \(\Phi(t,f)=f(1)-B(t)f(0)\) for matrices \(B(t)\in M_n(\mathbb{C})\) (\(t\in \mathbb{R}\)). By transferring the problem to a nonautonomous abstract Cauchy problem and using the corresponding evolution families, the author obtains some results on the wellposedness and asymptotic behavior of periodic evolution families for the delay Cauchy problem. Moreover, an application to flows in time-dependent networks is given.