On Solvability of Functional Equations Relating to Dynamical Systems with Two Generators

In this paper, some solvability problems for functional equations of the form
$$ F(t)-a_1(t)F(\delta_1(t))-a_2(t)F(\delta_2(t))=h(t),\qquad t\in I, $$
are studied. Here $I$ is a finite closed interval in $\mathbb{R}$, $F$ is an unknown continuous function, $\delta_1$ and $\delta_2$ are given continuous maps of $I$ into itself, and $a_1(t)$, $a_2(t)$, and $h(t)$ are real-valued continuous functions on $I$. Such equations are of interest not only by themselves as an object of analysis, but they are also a necessary link in solving various problems in such diverse fields as integral and functional equations, measure theory, and boundary problems for hyperbolic differential equations. The major part of the proofs is based on the new results in the theory of dynamical systems generated by a noncommutative semigroup with two generators.