Bulletin of the
Korean Mathematical Society BKMS

We investigate the non-existence of finite order transcendental entire solutions of Fermat-type differential-difference equations $$\left[f(z)f'(z)\right]^{n}+P^{2}(z)f^{m}(z+\eta)=Q(z)$$ and $$\left[f(z)f'(z)\right]^{n}+P(z)[\Delta_{\eta}f(z)]^{m}=Q(z),$$ where $P(z)$ and $Q(z)$ are non-zero polynomials,~$m$ and $n$ are positive integers, and $\eta\in\mathbb{C}\setminus\{0\}$.~In addition, we discuss transcendental entire solutions of finite order of the following Fermat-type differential-difference equation $$P^{2}(z)\left[f^{(k)}(z)\right]^{2}+\left[\alpha f(z+\eta)-\beta f(z)\right]^{2}=e^{r{(z)}},$$ where $P(z)\not\equiv0$ is a polynomial, $r(z)$ is a non-constant polynomial,~$\alpha\neq0$ and $\beta$ are constants, $k$ is a positive integer, and $\eta\in\mathbb{C}\setminus\{0\}$.~Our results generalize some previous results.

Keywords: Fermat-type equation, differential-difference, entire function, Nevanlinna theory

MSC numbers: 39B32, 34M05, 30D35

Supported by: Project supported by the Natural Science Foundation of Fujian Province, China (Grants Nos. 2018J01658 and 2019J01672).