Linear functional equations in the class of antiderivatives from the Lebesgue functions on curves segments

Linear functional equations on simple smooth curves with shift function of infinite order with fixed points at the ends of the curve are considered. The purpose is to study the sets of solutions of such equations in Hölder classes of functions $H_{\mu}$, $0<{\mu}\leq 1$, and classes of antiderivatives of functions from the classes $L_p$, $p>1$, with coefficients and right-hand sides from the same classes, and to investigate the solutions behavior in a neighborhood of fixed points. The research method uses F. Riesz's criterion for a function belonging to the class of antiderivatives of functions from the $L_p$, $p>1$, classes. For solutions classes we obtain estimates of the parameters ${\mu}$ and $p$ depending on parameters of classes of coefficients and right-hand sides in the studied equations and properties of the shift function in a neighborhood of a fixed point.