Entire solutions of Fermat type differential-difference equations

We mainly discuss entire solutions with finite order of the following Fermat type differential-difference equations $$\begin{array}{ll}(f)^{n}+f(z+c)^{m}=1;\\f^{\prime}(z)^{n}+f(z+c)^{m}=1;\\ f^{\prime}(z)^{n}+[f(z+c)-f(z)]^{m}=1,\end{array}$$ where m, n are positive integers.