On the solution of linear functional two-term equations with shift

Abstract

This work is dedicated to the study of linear functional equations with shift in Hölder space. Previously, for such operators, conditions for invertibility were found and the inverse operator was constructed by the authors. The operators are used in modeling systems with renewable resources. Here we propose another approach to solving functional equations with shift. With the help of an algorithm, the initial equation is reduced to the first iterated equations, then to the second iterated equation. Continuing this process, we obtain the n-th iterated equation and the limit iterated equation. We prove the theorem on the equivalence of the original and the iterated equations. Based on the analysis of the solvability of the limit equation, we find a solution to the original equation. The solution is the sum of an infinite product and a functional series. The results, and the methods for obtaining them, are transparent and not as cumbersome compared to previous works.