Solutions with big graph of an equation of the second iteration (Q5945057)

The author considers the functional equation \(\varphi(f(\varphi (x))) =g(x,\varphi(x))\), where \(f:Y\to X\), \(g:X\times Y\to Y\) and \(X,Y\) are nonempty sets. He presents the construction of a solution of this equation which has a big graph with respect to a given family \(\mathcal{R}\) of subsets of \(X \times Y\) under some set theoretical assumptions. The author also proves some topological and measure properties of functions having a big graph in case where \(\mathcal{R}\) is a subfamily of the farmily of Borel subsets of \(X\times Y\).