This paper is devoted to investigate the existence and the forms of entire solutions of several Fermat type quadratic trinomial differential difference equations. Our results improve some results due to Liu and Yang [An. Stiint. Univ. Al. I. Cuza Iasi. Mat., 2016], Han and Lü [J. Contemp. Math. Anal., 2019], Luo, Xu and Hu [Open Math., 2021].

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].