On solvability of functional equations relating to dynamical systems with two generators (Q1400111)

The author considers the functional equation \[ F(t) - a_1(t) F(\delta_1(t)) - a_2(t) F(\delta_2(t))=h(t) \] on a finite closed interval \(I\subset \mathbb{R}\), where \(F\) is an unknown real function on \(I\), \(\delta_1\), \(\delta_2\) are given continuous maps of \(I\) into itself and \(a_1\), \(a_2\), \(h\) are given real functions on \(I\). He gives conditions on \(\delta_1\), \(\delta_2\), \(a_1\), \(a_2\) under which there exists a unique continuous solution \(F\) of the equation for an arbitrary continuous function \(h\). To obtain this result the author proves a lemma describing a property of orbits of noncommutative semigroup generated by two generators \(\delta_1\), \(\delta_2\). In the paper one can also find some applications of results concerning the considered functional equation to a Cauchy type functional equation, an integral equation and a boundary value problem for a hyperbolic partial differential equation in a bounded domain.