Some generalized Jordan maps on triangular rings force additivity

In this paper, we show that a map $\delta$ over a triangular ring $\mathcal{T}$ satisfying $\delta (ab+ba)=\delta (a)b+a\tau (b)+\delta (b)a+b\tau (a)$, for all $a,b\in \mathcal{T}$ and for some maps $\tau$ over $\mathcal{T}$ satisfying $\tau (ab+ba)=\tau (a)b+a\tau (b)+\tau (b)a+b\tau (a)$, is additive. Also, it is shown that a map $T$ on $\mathcal{T}$ satisfying $T(ab)=T(a)b=aT(b)$, for all $a,b\in \mathcal{T}$, is additive. Further, we establish that if a map $D$ over $\mathcal{T}$ satisfies $(m+n)D(ab)=2mD(a)b+2naD(b)$, for all $a,b\in \mathcal{T}$ and integers $m,n\ge 1$, then $D$ is additive.