Stability of the generalized logarithmic functional equations arising from fixed point theory

Abstract

Let $$\mathbb {K}$$ , $$\mathbb {F}$$ be two fields of real or complex numbers and X be a normed space over field $$\mathbb {F}$$ . In this paper, we consider the following generalized logarithmic functional equation: $$\begin{aligned} f(x^a y^b)= Af(x) + Bf(y), \end{aligned}$$ where f maps from $$\mathbb {K}$$ into X, $$a,b\in \mathbb {Z}{\setminus }\{0\}$$ and $$A, B\in \mathbb {F}$$ . By using the fixed point method, the generalized Hyers–Ulam stability results for such functional equation are proved.