On the existence of solutions of Fermat-type differential-difference equations
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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.
Abstract
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.
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