Search Advanced

AIMS Mathematics

2021, Volume 6, Issue 2: 2003-2017. doi: 10.3934/math.2021122
Previous Article Next Article
Research article

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.


    加载中


    [1] M. Ablowitz, R. G. Halburd, B. Herbst, On the extension of Painlevé property to difference equations, Nonlinearty, 13 (2000), 889–905.
    [2] T. B. Cao, R. J. Korhonen, A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables, J. Math. Anal. Appl., 444 (2016), 1114–1132.
    [3] T. B. Cao, L. Xu, Logarithmic difference lemma in several complex variables and partial difference equations, Annali di Matematica, (2019), arXiv: 1806.05423v2.
    [4] Z. X. Chen, On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations, Sci. China Math., 54 (2011), 2123–2133.
    [5] Z. X. Chen, Complex differences and difference equations, Science Press, Beijing, 2014.
    [6] Y. M. Chiang, S. J. Feng, On the Nevanlinna characteristic of f (z + η) and difference equations in the complex plane, Ramanujan J., 16 (2008), 105–129.
    [7] L. Y. Gao, Entire solutions of two types of systems of complex differential-difference equations, Acta Math. Sinica Chin. Ser., 59 (2016), 677–684.
    [8] R. G. Halburd, R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 314 (2006), 477–487.
    [9] R. G. Halburd, R. J. Korhonen, Finite-order meromorphic solutions and the discrete Painlevé equations, Proc. London Math. Soc., 94 (2007), 443–474.
    [10] R. G. Halburd, R. J. Korhonen, Nevanlinna theory for the difference operator, Annales Academiae Scientiarum Fennicae. Mathematica, 31 (2006), 463–478.
    [11] R. G. Halburd, R. J. Korhonen, Growth of meromorphic solutions of delay differential equations, Proc. Amer. Math. Soc., 145 (2017), 2513–2526.
    [12] W. K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford, 1964.
    [13] P. C. Hu, P. Li, C. C. Yang, Unicity of Meromorphic Mappings, Advances in Complex Analysis and its Applications, vol. 1. Kluwer Academic Publishers, Dordrecht, Boston, London, 2003.
    [14] R. J. Korhonen, A difference Picard theorem for meromorphic functions of several variables, Comput. Methods Funct. Theory, 12 (2012), 343–361.
    [15] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.
    [16] I. Laine, J. Rieppo, H. Silvennoinen, Remarks on complex difference equations, Comput. Methods Funct. Theory, 5 (2005), 77–88.
    [17] I. Laine, C. C. Yang, Clunie theorems for difference and q-difference polynomials, J. London. Math. Soc., 76 (2007), 556–566.
    [18] K. Liu, Meromorphic functions sharing a set with applications to difference equations, J. Math. Anal. Appl., 359 (2009), 384–393.
    [19] K. Liu, T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ., 2013 (2013), 1–10.
    [20] K. Liu, T. B. Cao, H. Z. Cao, Entire solutions of Fermat type differential-difference equations, Arch. Math., 99 (2012), 147–155.
    [21] K. Liu, I. Laine, A note on value distribution of difference polynomials, Bull. Aust. Math. Soc., 81 (2010), 353–360.
    [22] F. Lü, Z. Li, Meromorphic solutions of Fermat type partial differential equations, J. Math. Anal. Appl., 478 (2019), 864–873.
    [23] G. Pólya, On an integral function of an integral function, J. Lond. Math. Soc., 1 (1926), 12–15.
    [24] X. G. Qi, Y. Liu, L. Z. Yang, A note on solutions of some differential-difference equations, J. Contemp. Math. Anal., 52 (2017), 128–133.
    [25] L. I. Ronkin, Introduction to the Theory of Entire Functions of Several Variables, Moscow: Nauka 1971 (Russian). American Mathematical Society, Providence, 1974.
    [26] W. Stoll, Holomorphic Functions of Finite Order in Several Complex Variables, American Mathematical Society, Providence, 1974.
    [27] Z. T. Wen, Finite order solutions of meromorphic equations and difference Painlevé equations IV, Proc. Amer. Math. Soc., 144 (2016), 4247–4260.
    [28] H. Y. Xu, S. Y. Liu, Q. P. Li, Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl., 483 (2020), no. 123641.
    [29] L. Xu, T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math., 15 (2018), 1–14.
    [30] L. Xu, T. B. Cao, Correction to: Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math., 17 (2020), 1–4.
    [31] L. Yang, Value distribution theory, Springer-Verlag, Berlin, 1993.
    [32] H. X. Yi, C. C. Yang, Uniqueness theory of meromorphic functions, Kluwer Academic Publishers, Dordrecht, 2003; Chinese original: Science Press, Beijing, 1995.
    [33] J. L. Zhang, L. Z. Yang, Meromorphic solutions of Painlevé III difference equations, Acta Math. Sinica Chin. ser., 57 (2014), 181–188.
    [34] R. R. Zhang, Z. B. Huang, Entire solutions of delay differential equations of Malmquist type, Math. CV, arXiv: 1710.01505v1.
    [35] X. M. Zheng, J. Tu, Growth of meromorphic solutions of linear difference equations, J. Math. Anal. Appl., 384 (2011), 349–356.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(2194) PDF downloads(237) Cited by(4)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog