Characterizations of Jordan mappings on some rings and algebras through zero products

Let $\mathcal{U}=\left[ \begin{array}{cc} \mathcal{A} & \mathcal{M} \mathcal{N}& \mathcal{B} \end{array} \right]$ be a generalized matrix ring, where $\mathcal{A}$ and $\mathcal{B}$ are 2-torsion free. We prove that if $\phi :\mathcal{U}\rightarrow \mathcal{U}$ is an additive mapping such that $\phi(U)\circ V+U\circ \phi(V)=0$ whenever $UV=VU=0,$ then $\phi=\delta+\eta$, where $\delta$ is a Jordan derivation and $\eta$ is a multiplier. As its applications, we prove that the similar conclusion remains valid on full matrix algebras, unital prime rings with a nontrivial idempotent, unital standard operator algebras, CDCSL algebras and von Neumann algebras.