On the Solution of Equations with Linear-Fractional Shifts

Abstract

This work represents a continuation of the studies relating to nonlinear equations, carried out by the authors. Special attention is paid to the operators with linear-fractional shifts that act on the argument of the unknown function, but also on the unknown function itself. In this work, we study homogeneous equations with such operators. The main classes of functions for which non-linear equations are considered are Hölder class real functions. Solutions of the equations have the form of infinite products or the form of infinite continued fractions; an abstract description of the solutions is also offered. The developed mathematical methods can be applied to finding the conditions of invertibility of certain operators found in modelling, as well as for the construction of their inverse operators. Subsequently, we suggest using these results for the modelling of renewable systems with elements that can be in different states: sick, healthy, immune, or vaccinated. These results can also be applied to the analysis of balance equations of the model and for finding equilibrium states of the system.