The article describes linear functional equations on simple smooth curves with a shift function having a non-zero derivative satisfying the Holder condition, and fixed points only at the ends of the curve. The objective of the article is to find the conditions of the existence and uniqueness of the solution of such equations in the Holder class functions with the coefficient and the right-hand side satisfying the Holder conditions. These conditions are obtained depending on the values of the equation coefficient at the ends of the curve. Various specifics at the ends of the curve are considered. The indicators of the Holder solutions are determined. The possibilities of applying linear functional equations to the study and solution of singular integral equations with logarithmic singularities are shown.