Some functional inequalities and their Baire category properties (Q1924289)

Motivated by problems related to the study of the iterative functional equation \(\varphi (f(x)) = g(x, \varphi (x))\), the author considers, from a topological point of view, the set of all continuous functions \(f:I \to I\) for which the unique continuous solution \(\psi: I\to [0,\infty]\) of the inequalities: \[ \psi \bigl(f(x) \bigr) \leq\beta \bigl(x,\psi (x)\bigr) \quad \text{and} \quad \alpha \bigl(x,\psi (x)\bigr) \leq\psi \bigl(f(x)\bigr) \leq\beta \bigl(x,\psi (x)\bigr) \] is the zero function.