<abstract><p>This article is concerned with the existence of entire solutions for the following complex second order partial differential-difference equation</p>
<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left(\frac{\partial^2 f(z_1, z_2)}{\partial z_1^2}+\frac{\partial^2 f(z_1, z_2)}{\partial z_2^2}\right)^{l}+f(z_1+c_1, z_2+c_2)^{k} = 1, $\end{document} </tex-math></disp-formula></p>
<p>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 <sup>[<xref ref-type="bibr" rid="b23">23</xref>]</sup>, Xu and Cao <sup>[<xref ref-type="bibr" rid="b35">35</xref>]</sup>, Liu, Cao and Cao <sup>[<xref ref-type="bibr" rid="b17">17</xref>]</sup>.</p></abstract>
Oscillation properties for a scalar linear difference equation of mixed type