Solutions with big graph of the equation of invariant curves (Q2770186)

The main result of this paper says that under some assumptions on the given spaces \(X\), \(Y\) and the given functions \(f:X\times Y\to X\), \(g:X\times Y\to Y\) the functional equation \(\varphi(f(x,\varphi(x)))=g(x,\varphi(x))\) has a solution \(\varphi:X\to Y\) such that the set \((X\times Y)\setminus\text{Gr }\varphi\) contains no subset of \(X\times Y\) of second category having the property of Baire and contains no subset of \(X\times Y\) of positive product measure. The author generalizes results of \textit{J. Smítal} [Aequationes Math. 37, No. 2/3, 279-281 (1989; Zbl 0674.39006)] and of \textit{P. Kahlig} and \textit{J. Smítal} [Result. Math. 27, No. 3-4, 362-367 (1995; Zbl 0860.39030)].