Oscillatory behavior of solutions of functional equations (Q5935349)

This paper is about linear iterative functional equations NEWLINE\[NEWLINEx[g^s(t)]=\sum_{k\in\{0,1,\dots ,s-1,s+1,\dots ,m+1\}}Q_k(t)x[g^k(t)]\quad (t\in I),NEWLINE\]NEWLINE where \(Q_{s-1},Q_{m+1}: I\to ]0,\infty[\); \(Q_k: I\to [0,\infty[\) for \(k\neq s\). \(I\) is a subset of \([0,\infty[,\) unbounded from above. The functions \(Q_k\) and \(g: I\to I\) are given; \(g^k\) is the \(k\)-th iterate of \(g\). The authors call a function \(x: I\to \mathbb{R},\) that satisfies the above equation, an oscillatory solution if \(\sup\{|x(s)|: s\in [t_0,\infty[\cap I\}>0\) for all \(t_0\geq 0,\) and there exists an infinite sequence \((t_n),\) tending to \(\infty\) such that \(x(t_n)x(t_{n+1})\leq 0\) \((n=1,2,\dots).\) The authors offer sufficient conditions for all solutions of the above equation to be oscillatory.