Continuous solutions of linear functional equations on piecewise smooth curves in mathematical models of boundary value problems with a shift

Linear functional equations on an arbitrary piecewise smooth curve are considered. Such equations are studied in connection with the theory of singular integral equations, which are a mathematical tool in the study of mathematical models of elasticity theory in which the conjugation conditions contain a boundary shift. Such equations also arise in the mathematical modeling of the transfer of charged particles and ionized radiation. The shift function is assumed to act cyclically on a set of simple curves forming a given curve, with only the ends of simple curves being periodic points. The aim of the work is to find the conditions for the existence and cardinality of a set of continuous solutions to such equations in the classes of Helder functions and primitive Lebesgue integrable functions with coefficients and the right side of the equation from the same classes. The solutions obtained have the form of convergent series and can be calculated with any degree of accuracy. The method of operation consists in reducing this equation to an equation of a special type in which all periodic points are fixed, which allows you to use the results for the case of a simple smooth curve.