Properties and description of solution sets of linear functional equations on a simple smooth curve

The article investigates linear functional equations given in the field of complex numbers on simple smooth curves with a shift function of infinite order. The shift function has a nonzero derivative satisfying the Helder condition, and fixed points only at the ends of the curve. The paper gives a complete description of the sets of solutions of such equations in the classes of continuous, Helder, and primitive Lebesgue functions with a coefficient and the right side of the same classes, depending on the values of the coefficient of the equation at the ends of the curve. Sufficient conditions have been established for the solutions to belong to the specified functional classes.