Let $f$ and $g$ be two transcendental meromorphic functions of finite order with a Borel exceptional value $\mathrm{\infty }$, let $\alpha$ $(\not\equiv 0)$ be a small function of both $f$ and $g$, let $d,k,n,m$ and $v_j(j=1,2,\cdots ,d)$ be positive integers, and let $c_j(j=1,2,\cdots ,d)$ be distinct nonzero finite values. If $n\ge max\{2k+m+\sigma +5,\sigma +2d+3\}$, where $\sigma =v_1+v_2+\cdots +v_d$, and $(f^n(z)(f^m(z)-1)\prod _{j=1}^d{f}^{v_j}(z+c_j){)}^{(k)}$ and $(g^n(z)(g^m(z)-1)\prod _{j=1}^d{g}^{v_j}(z+c_j){)}^{(k)}$ share $\alpha$ CM then $f\equiv tg$, where $t^m=t^{n+\sigma }=1.$ This result extends and improves some restlts due to ^{[1,10,14,15,19]}.

In this paper, we investigate the uniqueness of an entire function sharing a small function with its linear difference polynomial. Our results improve some results due to Li and Yi ^[11], Zhang, Chen and Huang ^[17], Zhang, Kang and Liao ^[18,19] etc.

In this paper, we study a uniqueness question of meromorphic functions concerning fixed points and mainly prove the following theorem: Let $f$ and $g$ be two nonconstant meromorphic functions, let $n,k$ be two positive integers with $n>3k+10.5-{\mathrm{\Theta }}_{min}(k+6.5)$, if ${\mathrm{\Theta }}_{min}\ge \frac{2.5}{k+6.5}$, otherwise $n>3k+8$, and let $(f^n{)}^{(k)}$ and $(g^n{)}^{(k)}$ share $z$ CM, $f$ and $g$ share $\mathrm{\infty }$ IM, then one of the following two cases holds: If $k=1$, then either $f(z)=c_1{e}^{c{z}^2}$, $g(z)=c_2{e}^{-c{z}^2}$, where $c_1,c_2$ and $c$ are three constants satisfying $4n^2(c_1{c}_2{)}^n{c}^2=-1$, or $f=tg$ for a constant $t$ such that $t^n=1$; if $k\ge 2$, then $f=tg$ for a constant $t$ such that $t^n=1$. Our results extend and improve some results due to ^[8,9,19,24].