This analysis demonstrates the stability of a bi-affine functional equation in Banach spaces, suggesting broader implications for similar equations.
In this paper, we introduce a multi-variable bi-affine functional equation of the form f∑i=1mαixi,∑j=1nβjyj=∑i=1m∑j=1nαiβjf(xi,yj), where m and n are integers and m,n≥2 and αi,βj are nonzero scalars. We investigate the Hyers–Ulam stability of this functional equation in Banach spaces using the direct method. The results obtained in this paper can be regarded as a generalization of stability results for the classical bi-Jensen functional equation and its multi-variable mean-type variants.
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Bae et al. (2026) studied this question.
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