Fermat型偏微差分方程组的整函数解
Entire Solutions of Several Fermat Type Systems of Partial Differential Difference Equations
利用多变量Nevanlinna值分布理论与Nevanlinna理论的差分模拟结果,讨论了几类多变量复域Fermat型偏微差分方程组解的性质,得到了方程组有限超越整函数解的存在性条件与具体形式,推广改进了高凌云、曹廷彬、刘凯等人的结果,给出例子说明多变量与单变量方程组有限级超越整函数解之间的差异.
By making use of the Nevanlinna theory and difference Nevanlinna theory of several complex variables, we investigate several Fermat type systems of partial differential difference equations, and obtain a series of results about the existence and the forms of entire solutions of such systems, which are some improvements and generalization of the previous results given by Cao, Gao, Liu et al. We also give some examples to show that there exists significant differences in the forms of transcendental entire solutions with finite order of the systems of the equations with between several complex variables and single complex variable.
整函数 / 偏微差分方程组 / 存在性 {{custom_keyword}} /
entire function / systems of partial differential difference equations / existence {{custom_keyword}} /
[1] Biancofiore A., Stoll W., Another proof of the lemma of the logarithmic derivative in several complex variables, In:Fornaess, J. (ed.) Recent developments in several complex variables, Princeton University Press, Princeton, 1981:29-45.
[2] Cao T. B., Xu L., Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math., 2018, 15:1-14.
[3] Cao T. B., Xu L., Logarithmic difference lemma in several complex variables and partial difference equations, Ann. Mat. Pura Appl., 2020, 199(2):767-794.
[4] Cao T. B., Korhonen R. J., A new version of the second main theorem for meromorphic mappings intersecting hyper planes in several complex variables, J. Math. Anal. Appl., 2016, 444:1114-1132.
[5] Chiang Y. M., Feng S. J., On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane, Ramanujan J., 2008, 16(1):105-129.
[6] Gao L. Y., Entire solutions of two types of systems of complex differential-difference equations, Acta Math. Sinica, Chinese Series, 2016, 59:677-685.
[7] Gross F., On the equation fn+gn=1, Bull. Amer. Math. Soc., 1996, 72:86-88.
[8] Halburd R. G., Korhonen R. J., Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 2006, 314:477-487.
[9] Halburd R. G., Korhonen R. J., Finite-order meromorphic solutions and the discrete Painlevé equations, Proc. London Math. Soc., 2007, 9:443-474.
[10] Halburd R. G., Korhonen R. J., Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math., 2006, 31(2):463-478.
[11] Hu P. C., Malmquist type theorem and factorization of meromorphic solutions of partial differential equations, Complex Var., 1995, 27:269-285.
[12] Hu P. C., Yang C. C., Uniqueness of meromorphic functions on Cm, Complex Variables, 1996, 30:235-270.
[13] Hu P. C., Li P., Yang C. C., Unicity of Meromorphic Mappings, Advances in Complex Analysis and its Applications, Vol. 1. Kluwer Academic Publishers, Dordrecht, Boston, London, 2003.
[14] Korhonen R. J., A difference Picard theorem for meromorphic functions of several variables, Comput. Methods Funct. Theory, 2012, 12(1):343-361.
[15] Liu K., Meromorphic functions sharing a set with applications to difference equations, J. Math. Anal. Appl., 2009, 359:384-393.
[16] Liu K., Cao T. B., Cao H. Z., Entire solutions of Fermat type differential-difference equations, Arch. Math., 2012, 99:147-155.
[17] Liu K., Yang L. Z., On entire solutions of some differential-difference equations, Comput. Methods Funct. Theory, 2013, 13:433-447.
[18] Liu K., Cao T. B., Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ., 2013, 59:1-10.
[19] Liu M. L., Gao L. Y., Transcendental solutions of systems of complex differential-difference equations, Sci. Sinica Mathematica, 2019, 49:1-22.
[20] Montel P., Lecons sur les Familles Normales de Fonctions Analytiques et Leurs Applications, Gauthier-Villars, Paris, 1927, 135-136.
[21] Pólya G., On an integral function of an integral function, J. Lond. Math. Soc., 1926, 1:12-15.
[22] Rieppo J., On a class of complex functional equations, Ann. Acad. Sci. Fenn. Math., 2007, 32(1):151-170.
[23] Ronkin L. I., Introduction to the Theory of Entire Functions of Several Variables, Moscow:Nauka 1971(Russian); American Mathematical Society, Providence, 1974.
[24] Stoll W., Holomorphic Functions of Finite Order in Several Complex Variables, American Mathematical Society, Providence, 1974.
[25] Taylor R., Wiles A., Ring-theoretic properties of certain Hecke algebra, Ann. Math., 1995, 141:553-572.
[26] Wiles A., Modular elliptic curves and Fermats last theorem, Ann. Math., 1995, 141:443-551.
[27] Xu L., Cao T. B., Correction to:solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math., 2020, 17, Art. 8, pages, 1-4.
[28] Yang C. C., A generalization of a theorem of P. Montel on entire functions, Proc. Amer. Math. Soc., 1970, 26:332-334.
[29] Yang C. C., Li P., On the transcendental solutions of a certain type of nonlinear differential equations, Arch Math. Basel, 2004, 82:442-448.
国家自然科学基金资助项目(11561033,11371225);江西省自然科学基金(20181BAB201001)及江西省教育厅科技项目(GJJ190876,GJJ191042,GJJ190895)
/
〈 | 〉 |