On the stability of \(\alpha\)−Cauchy-Jensen type functional equation in Banach algebras

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in Banach algebras for the following \(\alpha\)-Cauchy-Jensen functional equation:$$f\left(\frac{x+y}{\alpha}+z\right)+f\left(\frac{x-y}{\alpha}+z\right)=\frac{2}{\alpha} f(x)+2 f(z),$$where \(\alpha \in \mathbb{N}_{\geq 2}\).