In this talk i want to present two results regarding the problems of the problem of the unique extension of solutions of the functional equation $$\varphi (x)=h(x,elta_{xn S})\varphi\circ f_s(x))$$ in which $$h:Ximes Y^S\rightarrow Y\;\;and\;\;f_x:X\rightarrow X,\;sn S$$ whwrw $X$, $Y$ and $S$ are arbitrary sets, are given functions. nonempty set $S$.