Functional equations as mathematical models of cyclic shift coupling problems on complex curves

The paper considers linear functional equations with a shift function having a nonzero derivative satisfying the Helder condition on an arbitrary piecewise smooth curve. Such equations are studied in connection with the theory of boundary value problems for analytical functions, which are a mathematical tool in the study of mathematical models of elasticity theory in which the conjugation conditions contain a boundary shift. The shift function acts cyclically on a set of simple curves forming a given curve, and except the ends of simple curves, there are no periodic points relative to the shift function. The purpose of the study is to find conditions for the existence and uniqueness of a solution (and in the case of non-uniqueness of the cardinality of the set of solutions) of such equations in the classes of Helder and primitive Lebesgue functions with a coefficient and the right part of the same classes.