Abstract

We introduce the general functional equation $$\begin{aligned} \sum _{i=1}^m A_i( f(\varphi _i (\bar{x}))) + b = 0, \quad \bar{x} := (x_1, x_2, \dots , x_n) \in X^n, \end{aligned}$$ and study its Ulam–Hyers-stability and hyperstability, using a fixed point approach, where m and n are positive integers, f is a mapping from a vector space X into a Banach space $$(Y, \Vert \ \Vert )$$ , and, for every $$i \in \{1, 2, \dots , m\}$$ , $$\varphi _i$$ is a linear mapping from $$X^n$$ into X, $$A_i$$ is a continuous endomorphism of Y and $$b \in Y$$ . Our result covers most of the former ones in the literature concerning the stability and hyperstability of linear functional equations, as well as new situations.

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