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AIMS Mathematics

2021, Volume 6, Issue 2: 2003-2017. doi: 10.3934/math.2021122
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On the entire solutions for several partial differential difference equations (systems) of Fermat type in $\mathbb{C}^2$

  • 1.
    School of Mathematics and Computer Science, Shangrao Normal University, Shangrao Jiangxi, 334001, P. R. China
  • 2.
    School of mathematics and statistics, Xidian University, Xian, Shaanxi, 710126, P. R. China
  • 3.
    Department of General Education, Beijing Union University, No.97 Bei Si Huan Dong Road Chaoyang District Beijing, 100101, P. R. China
  • 4.
    Depatment of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, P. R. China
  • Received: 21 October 2020 Accepted: 01 December 2020 Published: 07 December 2020

  • MSC : 30D35, 35M30, 32W50, 39A45

  • By utilizing the Nevanlinna theory of meromorphic functions in several complex variables, we will establish some theorems about the existence and the forms of entire solutions for several partial differential difference equations (systems) of Fermat type with two complex variables such as $ f(z)^2+\left[f(z+c)+\frac{\partial f}{\partial z_1}+\frac{\partial f}{\partial z_2}\right]^2 = 1 $ and $ \left\{ \begin{aligned} &f_1(z)^2+\left[f_2(z+c)+\frac{\partial f_1}{\partial z_1}+\frac{\partial f_1}{\partial z_2}\right]^2 = 1, \\ &f_2(z)^2+\left[f_1(z+c)+\frac{\partial f_2}{\partial z_1}+\frac{\partial f_2}{\partial z_2}\right]^2 = 1, \end{aligned}\right. $ which are some extensions and generalizations of the previous theorems given by Xu and Cao [29,30], Xu, Liu and Li [28], and Liu, Yang [18,19,20]. Moreover, we give some examples to explain that our results are precise to some extent.

    Citation: Hong Yan Xu, Zu Xing Xuan, Jun Luo, Si Min Liu. On the entire solutions for several partial differential difference equations (systems) of Fermat type in $\mathbb{C}^2$[J]. AIMS Mathematics, 2021, 6(2): 2003-2017. doi: 10.3934/math.2021122

    Related Papers:

  • By utilizing the Nevanlinna theory of meromorphic functions in several complex variables, we will establish some theorems about the existence and the forms of entire solutions for several partial differential difference equations (systems) of Fermat type with two complex variables such as $ f(z)^2+\left[f(z+c)+\frac{\partial f}{\partial z_1}+\frac{\partial f}{\partial z_2}\right]^2 = 1 $ and $ \left\{ \begin{aligned} &f_1(z)^2+\left[f_2(z+c)+\frac{\partial f_1}{\partial z_1}+\frac{\partial f_1}{\partial z_2}\right]^2 = 1, \\ &f_2(z)^2+\left[f_1(z+c)+\frac{\partial f_2}{\partial z_1}+\frac{\partial f_2}{\partial z_2}\right]^2 = 1, \end{aligned}\right. $ which are some extensions and generalizations of the previous theorems given by Xu and Cao [29,30], Xu, Liu and Li [28], and Liu, Yang [18,19,20]. Moreover, we give some examples to explain that our results are precise to some extent.


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