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AIMS Mathematics

2020, Volume 5, Issue 1: 685-700. doi: 10.3934/math.2020046
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Research article

The existence and forms of solutions for some Fermat-type differential-difference equations

  • 1.
    Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China
  • 2.
    School of Mathematics and Computer Science, Shangrao Normal University, Shangrao Jiangxi, 334001, China
  • 3.
    Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchan, Jiangxi, 330022, P. R. China
  • Received: 07 October 2019 Accepted: 11 December 2019 Published: 19 December 2019

  • MSC : 30D35, 39A13, 39B72

  • The main aim of this article is to investigate the existence and the forms of solutions for several complex differential-difference equations of Fermat-type. Our results about the existence and the forms of solutions for these Fermat type equations are great improvement of the previous theorems given by Liu, Yang, Cao, Zhang. Moreover, it is a very satisfactory fact that in some examples explicit solutions are given.

    Citation: Hua Wang, Hong Yan Xu, Jin Tu. The existence and forms of solutions for some Fermat-type differential-difference equations[J]. AIMS Mathematics, 2020, 5(1): 685-700. doi: 10.3934/math.2020046

    Related Papers:

  • The main aim of this article is to investigate the existence and the forms of solutions for several complex differential-difference equations of Fermat-type. Our results about the existence and the forms of solutions for these Fermat type equations are great improvement of the previous theorems given by Liu, Yang, Cao, Zhang. Moreover, it is a very satisfactory fact that in some examples explicit solutions are given.


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    [1] Z. X. Chen, Growth and zeros of meromorphic solution of some linear difference equations, J. Math. Anal. Appl., 373 (2011), 235-241.
    [2] X. K. Chang, S. Y. Liu, P. J. Zhao, et al. A generalization of linearized alternating direction method of multipliers for solving two-block separable convex programming, J. Comput. Appl. Math., 357 (2019), 251-272.
    [3] M. F. Chen, Y. Y. Jiang, Z. S. Gao, et al. Growth of meromorphic solutions of certain types of q-difference differential equations, Adv. Differ. Equ., 2017 (2017), 37.
    [4] F. Gross, On the equation fn + gn = 1, B. Am. Math. Soc., 72 (1966), 86-88.
    [5] G. G. Gundersen, J. Heittokangas, I. Laine, et al. Meromorphic solutions of generalized Schröder equations, Aequationes Math., 63 (2002), 110-135.
    [6] R. G. Halburd, R. Korhonen, Finite-order meromorphic solutions and the discrete Painlevé equations, P. Lond. Math. Soc., 94 (2007), 443-474.
    [7] R. G. Halburd, R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math., 31 (2006), 463-478.
    [8] W. K. Hayman, Meromorphic Functions, Oxford: The Clarendon Press, 1964.
    [9] J. Heittokangas, R. Korhonen, I. Laine, et al. Complex difference equations of Malmquist type, Comput. Meth. Funct. Th., 1 (2001), 27-39.
    [10] I. Laine, Nevanlinna Theory and Complex Differential Equations, Berlin: Walter de Gruyter, 1993.
    [11] Z. Latreuch, On the existence of entire solutions of certain class of nonlinear difference equations, Mediterr. J. Math., 14 (2017), 115.
    [12] H. C. Li, On existence of solutions of differential-difference equations, Math. Method. Appl. Sci., 39 (2016), 144-151.
    [13] K. Liu, T. B. Cao, X. L. Liu, The properties of differential-difference polynomials, Ukr. Math. J., 69 (2017), 85-100.
    [14] K. Liu, Meromorphic functions sharing a set with applications to difference equations, J. Math. Anal. Appl., 359 (2009), 384-393.
    [15] K. Liu, T. B. Cao, H. Z. Cao, Entire solutions of Fermat type differential-diference equations, Arch. Math., 99 (2012), 147-155.
    [16] K. Liu, L. Z. Yang, On entire solutions of some differential-difference equations, Comput. Meth. Funct. Th., 13 (2013), 433-447.
    [17] K. Liu, T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Differ. Eq., 2013 (2013), 59.
    [18] K. Liu, C. J. Song, Meromorphic solutions of complex differential-difference equations, Results Math, 72 (2017), 1759-1771.
    [19] P. Montel, Lecons sur les familles normales de fonctions analytiques et leurs applications, Paris: Gauthier-Villars, 1927,135-136.
    [20] 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.
    [21] X. G. Qi, L. Z. Yang, Properties of meromorphic solutions to certain differential-difference equations, Electron. J. Differ. Eq., 2013 (2013), 135.
    [22] J. Rieppo, On a class of complex functional equations, Ann. Acad. Sci. Fenn. Math., 32 (2007), 151-170.
    [23] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebra, Ann. Math., 141 (1995), 553-572.
    [24] A. Wiles, Modular elliptic curves and Fermats last theorem, Ann. Math., 141 (1995), 443-551.
    [25] 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), 123641.
    [26] H. Y. Xu, S. Y. Liu, Q. P. Li, The existence and growth of solutions for several systems of complex nonlinear difference equations, Mediterr. J. Math., 16 (2019), 8.
    [27] H. Y. Xu, J. Tu, Growth of solutions to systems of q-difference differential equations, Electron. J. Differ. Eq., 2016 (2016), 106.
    [28] L. Yang, Value Distribution Theory, Berlin: Springer-Verlag, 1993.
    [29] C. C. Yang, P. Li, On the transcendental solutions of a certain type of nonlinear differential equations, Arch. Math, 82 (2004), 442-448.
    [30] C. C. Yang, H. X. Yi, Uniqueness Theory of Meromorphic Functions, Dordrecht: Kluwer Academic Publishers, 2003.
    [31] J. Zhang, On some special difference equations of Malmquist type, B. Korean Math. Soc., 55 (2018), 51-61.
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