  Let $n\in \Bbb N,$ and let $A,B$ be two rings. An additive map $h: A\to B$ is
called n-Jordan homomorphism if $h(a^n)=(h(a))^n$ for all $a \in {A}$. Every
Jordan homomorphism is an n-Jordan homomorphism, for all $n\geq 2,$ but the
converse is false, in general. In this paper we investigate the n-Jordan
homomorphisms on Banach algebras. Indeed some results related to continuity are
given as well.
