$k$-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.