Hyers–Ulam stability of a functional equation with several parameters

Abstract

We consider the functional equation:

$$\begin{aligned} \sum _{i=1}^{m}f(a_i x_0 + b_i x_i) + f\left( x_0 - \sum _{i=1}^{m}b_i x_i\right) = f(x_0), \end{aligned}$$

(1)

where $m \ge 2$ is an integer, $(a_i)_{i=1, \dots , m}$ and $(b_i)_{i=1, \dots , m}$ are scalars so that $\sum _{i=1}^m a_i = 0$ and $b_i \ne 0$ for every $i = 1, \dots , m$. We prove the Hyers–Ulam stability of (1) under several kinds of approximation conditions and with different methods. Unlike most of authors, when using the fixed point theorem method, we use the classical Banach contraction principle instead of the alternative fixed point theorem. We then combine (1) with some other equations and show the stability of the obtained systems. In particular, we get the stability of left (right) multipliers, ring morphisms, ring antimorphisms, Lie morphisms, Lie derivation, ring n-derivations, and so on.