On Ulam stability of generalized Hosszú functional equation

We investigated Hyers-Ulam stability (H-US) of the following generalization of the Hosszú equation (HFE): $\mathrm{{\rm Y}}(x_1+x_2-\alpha x_1{x}_2)+\mathrm{{\rm Y}}(\alpha x_1{x}_2)=\mathrm{{\rm Y}}(x_1)+\mathrm{{\rm Y}}(x_2)$, in the class of maps $\mathrm{{\rm Y}}$ from a quadratically closed field $\mathbb{K}$ into a linear space, where $\alpha \in \mathbb{K}$ is fixed. We considered this stability in cases where the linear space is equipped with either the $m$-norm or the classical norm. In this way, we extended some earlier stability outcomes obtained for maps from the set of reals $\mathbb{R}$ into a Banach space. We also proved some auxiliary stability results for the Cauchy additive equation $\psi (x+y)=\psi (x)+\psi (y)$ in $m$-Banach spaces. Finally, we discussed the symmetry issues that can be observed in these results.