On the Iterated Method for the Solution of Functional Equations with Shift Whose Fixed Points are Located at the Ends of a Contour

In this paper, we offer an approach for solving functional equations containing a shift operator and its iterations. With the help of an algorithm, the initial equation is reduced to the first iterated equation, then, applying the same algorithm, we obtain the second iterated equation. Continuing this process, we obtain the n-th iterated equation and the limit iterated equation. We prove a theorem on the equivalence of the initial equation and the iterated equations. Based on the analysis of the solvability of the limit equation, we find a solution to the initial equation. Equations of this type appear when modeling renewable systems with elements in different states, such as being sick, healthy with immunity, and without immunity. The obtained results represent appropriate mathematical tools for the study of such systems.