Conditions for the oscillation of solutions of iterative equations
While the problem of oscillation of solutions of differential and difference equations has received a lot of attention, such problem has not been faced for many classes of functional equations in a single variable. The authors of this paper deal with such problems in the case of the linear iterative functional equation of the form: \[ \sum_{i=0}^{m+1} Q_i (t) x (g^i (t)) = 0, \quad m \geq 1 , \] where \(x\) is an unknown real-valued function and \(Q_i : I \to R \), \( i=0,\dots,m+1\) and \( g: I \to I \) are given functions (where \(I\) is an unbounded subset of \([0, \infty) \) and \( g^i \) means the ith iterate of \( g \), \( g^0 = j \), \(g^{i+1} = g \circ g^i\)). Under some natural regular conditions the existence of oscillatory solutions is proven.
scientific article; zbMATH DE number 2163672