Characterizations of $(m,n)$-Jordan derivations on some algebras

Let $\mathcal R$ be a ring, $\mathcal{M}$ be a $\mathcal R$-bimodule and $m,n$ be two fixed nonnegative integers with $m+n\neq0$. An additive mapping $\delta$ from $\mathcal R$ into $\mathcal{M}$ is called an \emph{$(m,n)$-Jordan derivation} if $(m+n)\delta(A^{2})=2mA\delta(A)+2n\delta(A)A$ for every $A$ in $\mathcal R$. In this paper, we prove that every $(m,n)$-Jordan derivation from a $C^{*}$-algebra into its Banach bimodule is zero. An additive mapping $\delta$ from $\mathcal R$ into $\mathcal{M}$ is called a $(m,n)$-Jordan derivable mapping at $W$ in $\mathcal R$ if $(m+n)\delta(AB+BA)=2m\delta(A)B+2m\delta(B)A+2nA\delta(B)+2nB\delta(A)$ for each $A$ and $B$ in $\mathcal R$ with $AB=BA=W$. We prove that if $\mathcal{M}$ is a unital $\mathcal A$-bimodule with a left (right) separating set generated algebraically by all idempotents in $\mathcal A$, then every $(m,n)$-Jordan derivable mapping at zero from $\mathcal A$ into $\mathcal{M}$ is identical with zero. We also show that if $\mathcal{A}$ and $\mathcal{B}$ are two unital algebras, $\mathcal{M}$ is a faithful unital $(\mathcal{A},\mathcal{B})$-bimodule and $\mathcal{U}={\left[\begin{array}{cc}\mathcal{A} &\mathcal{M} \\\mathcal{N} & \mathcal{B} \\\end{array}\right]}$ is a generalized matrix algebra, then every $(m,n)$-Jordan derivable mapping at zero from $\mathcal{U}$ into itself is equal to zero.