A Note on the Stability of Functional Equations via a Celebrated Direct Method

Abstract

More than ten years after Justyna Sikorska [8] attempted to solve the Heyers-Ulam stability of a single variable equation by using direct method. In this paper, we will improve the results of Justyna Sikorska by using a more efficient approach. Relations between the generalized functional equation, the dependence of their different parameters and several properties are also further explored. To achieve the problem, we try to develop some new techniques to overcome the fundamental difficulties caused by the different properties of the function and the presence of several variables in the equation. Furthermore, we continue to construct and study a couple of functional equations by making a new direct method.