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Solution and stability of an $$n$$-variable additive functional equation. (English) Zbl 1458.39018
Summary: In this paper, we investigate the general solution and the Hyers-Ulam stability of $$n$$-variable additive functional equation of the form $\operatorname{Im}\left(\sum_{i=1}^n(-1)^{i+1}x_i\right)=\sum_{i=1}^n(-1)^{i+1}\operatorname{Im} (x_i),$ where $$n$$ is a positive integer with $$n \ge 2$$, in Banach spaces by using the direct method.
##### MSC:
 39B52 Functional equations for functions with more general domains and/or ranges 39B72 Systems of functional equations and inequalities 39B82 Stability, separation, extension, and related topics for functional equations 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)