$k$-power centralizing and $k$-power skew-centralizing maps on‎ ‎triangular rings

‎Let $mathcal A$ and $mathcal B$ be unital rings‎, ‎and $mathcal M$‎ ‎be an $(mathcal A‎, ‎mathcal B)$-bimodule‎, ‎which is faithful as a‎ ‎left $mathcal A$-module and also as a right $mathcal B$-module‎. ‎Let ${mathcal U}=mbox{rm Tri}(mathcal A‎, ‎mathcal M‎, ‎mathcal‎ ‎B)$ be the triangular ring and ${mathcal Z}({mathcal U})$ its‎ ‎center‎. ‎Assume that $f:{mathcal U}rightarrow{mathcal U}$ is a map‎ ‎satisfying $f(x+y)-f(x)-f(y)in{mathcal Z}({mathcal U})$ for all‎ ‎$x, yin{mathcal U}$ and $k$ is a positive integer‎. ‎It is shown‎ ‎that‎, ‎under some mild conditions‎, ‎the following statements are‎ ‎equivalent‎: ‎(1) $[f(x),x^k]in{mathcal Z}({mathcal U})$ for all‎ ‎$xin{mathcal U}$; (2) $[f(x),x^k]=0$ for all $xin{mathcal U}$;‎ ‎(3) $[f(x),x]=0$ for all $xin{mathcal U}$; (4) there exist a‎ ‎central element $zin{mathcal Z}({mathcal U})$ and an additive‎ ‎modulo ${mathcal Z}({mathcal U})$ map $h:{mathcal‎ ‎U}rightarrow{mathcal Z}({mathcal U})$ such that $f(x)=zx+h(x)$‎ ‎for all $xin{mathcal U}$‎. ‎It is also shown that there is no‎ ‎nonzero additive $k$-skew-centralizing maps on triangular rings.