In this paper, we study one class of nonlinear binomial differential equation

$$(f(z){)}^n+a(f^{(k)}(z){)}^n=p_1{e}^{{\lambda }_{1}z}+p_2{e}^{{\lambda }_{2}z},$$

where $n,k\in {\mathbb{N}}^{\mathbb{+}}$, $a$, $p_i$, and ${\lambda }_i$ are nonzero constants for $i=1,2$. Some results answer the previous conjecture given by Linkui Gao and Junyang Gao.

The main purpose of this article is to study Fermat-type complex differential-difference equations $f^{(k)}(z{)}^2+[\alpha f(z+c)-\beta f(z){]}^2=R(z)$. Our results improve some results due to Wang–Xu–Tu [AIMS. Mathematics, 2020], Zhang [Bull. Korean. Math. Soc, 2018], and Long–Qin [Applied Mathematics-A Journal of Chinese Universities, 2024]. Moreover, we provide some examples to show the existence of the solutions.