Multiplicative generalized Jordan $n$-derivations of unital rings with idempotents

Let $\mathfrak{A}$ be a unital ring with a nontrivial idempotent. In this paper, it is shown that under certain conditions every multiplicative generalized Jordan $n$-derivation $\mathrm{\Delta }:\mathfrak{A}\to \mathfrak{A}$ is additive. More precisely, it is proved that $\mathrm{\Delta }$ is of the form $\mathrm{\Delta }(t)=\mu t+\delta (t),$ where $\mu \in \mathcal{Z}(\mathfrak{A})$ and $\delta :\mathfrak{A}\to \mathfrak{A}$ is a Jordan $n$-derivation. The main result is then applied to some classical examples of unital rings with nontrivial idempotents such as triangular rings, matrix rings, prime rings, nest algebras, standard operator algebras, and von Neumann algebras.