Existence and uniqueness conditions for solutions of linear functional equations in the classes of Lebesgue functions antiderivatives on a simple smooth curve

The article describes linear functional equations on simple smooth curves with a shift function and fixed points only at the ends of the curve. The case when the shift function has a nonzero derivative satisfying the Hölder condition is considered. The objective of the article is to find the conditions of the existence and uniqueness of such equations solution in the classes of Lebesgue functions antiderivatives with a coefficient and the right-hand part belonging to the same classes. These conditions depend on the values of the equation coefficient at the ends of the curve. It is shown that if the coefficient and the right-hand side of a functional equation belong to the class of Lebesgue functions antiderivatives, then its solution also belongs to this class. The indicators of Hölder and of classes of Lebesgue functions antiderivatives are determined for the solutions. The research method is based on F. Riesz's criterion of a function's belonging to the class of antiderivatives of Lebesgue integrable functions. The possibilities of applying linear functional equations for studying and solving singular integral equations with logarithmic singularities are shown.