Hyers-Ulam stability of the iterative equation with a general boundary restriction

Hyers-Ulam stability has played an important role not only in the theory of functional equations but also in a variety of branches of mathematics, such as differential equations, integral equations and linear operators. In the present paper we will discuss the Hyers-Ulam stability of the iterative equation with a general boundary restriction. By the construction of a uniformly convergent sequence of functions, we prove that for every approximate solution of such an equation, there exists an exact solution near it.