Let $\sigma$ be a homomorphism on Banach algebra $\cal A$. Then we introduce and study the concept of $\sigma-$cyclic amenability for $\cal A$. Moreover, for all closed two-sided $I$ of $\cal A$, we investigate the relationship between $\sigma-$cyclic amenability of $\cal A$ and ${\hat\sigma}-$cyclic amenability of ${\cal A}/I$.
