On continuity of solutions to some equations of iteration theory (Q1774141)

The authors consider the functional equation \[ F(s+t,x)=h(s,x,g(t,f(s,x))), \] where \(F:(0,\infty) \times X_1 \rightarrow Y\), \(f:(0,\infty) \times X_1 \rightarrow X_2\), \(g:(0,\infty) \times X_2 \rightarrow Z\), \(h:(0,\infty) \times X_1 \times Z \rightarrow Y\) and \(X_1\), \(X_2\) are locally compact separable spaces, \(Y\) is a metric space, \(Z\) is a separable metric space. Under the assumption that for every \(t\in (0,\infty)\) the functions \(F(t, \cdot)\), \(f(t, \cdot)\), \(g(t, \cdot)\) \(h(t,\cdot, \cdot)\) are continuous and for every \(x\in X_2\) the function \(g(\cdot, x)\mid _{M}\) is Lebesgue measurable for a set \(M\subset (0, \infty)\) of positive Lebesgue measure the authors prove using the \textit{K.-G. Grosse-Erdmann} theorem [ibid. 37, No.~2/3, 233--251 (1989; Zbl 0676.39007)] that for every \(t_0\in (0,\infty)\) such that \(M \cap (0,t_0)\) is of positive Lebesgue measure the function \(F\mid _{(t_0,\infty) \times X_1}\) is continuous. A similar result is obtained for the Baire measurability. As a corollary the authors get an extension to the non-compact case of a result by \textit{M. C. Zdun} [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 29, No.~1, 113--116 (1989; Zbl 0707.39006)] which says that Carathéodory solutions of the translation equation \[ F(s+t,x)=F(t,F(s,x)) \] are continuous.