Results on the solutions of several second order mixed type partial differential difference equations

This article is concerned with the existence of entire solutions for the following complex second order partial differential-difference equation

$${\left(\frac{{\mathrm{\partial }}^{2}f(z_1,z_2)}{\mathrm{\partial }{z}_1^2}+\frac{{\mathrm{\partial }}^{2}f(z_1,z_2)}{\mathrm{\partial }{z}_2^2}\right)}^l+f(z_1+c_1,z_2+c_2{)}^k=1,$$

where $c_1,c_2$ are constants in $\mathbb{C}$ and $k,l$ are positive integers. In addition, we also investigate the forms of finite order transcendental entire solutions for several complex second order partial differential-difference equations of Fermat type, and obtain some theorems about the existence and the forms of solutions for the above equations. Meantime, we give some examples to explain the existence of solutions for some theorems in some cases. Our results are some generalizations of the previous theorems given by Qi ^[23], Xu and Cao ^[35], Liu, Cao and Cao ^[17].