Meromorphic solutions of Fermat type differential and difference equations of certain types

In this paper, we mainly consider the Fermat type differential equation $$f(z)^{n}+f’(z)^{n}=\varphi (z),$$ where $\varphi (z)=e^{h(z)}$ or $1-e^{2h(z)}$, and $h(z)$ is any entire function, and the Fermat type difference equation $$f(z)^{n}+f(z+c)^{m}=e^{P(z)},$$ where $P(z)$ is any entire function and $c$ is a non-zero constant. We also provide short proofs for some existence results without complicated computations using elliptic functions.