On approximate solutions of the linear functional equation of higher order

We show that, under some assumptions, every approximate solution of the linear functional equation of higher order, in single variable, generates a solution of the equation that is close to it. We also give a description of a procedure that yields such a solution, estimate the distance between those approximate and exact solutions to the equation, and discuss the problem of uniqueness. Moreover, as a consequence we obtain some results concerning the Hyers–Ulam stability of the equation.