Solutions of complex nonlinear functional equations including second order partial differential and difference in C^2
DOI:
https://doi.org/10.58997/ejde.2023.43Keywords:
Functions of several complex variables; Fermat-type equations; entire solutions; Nevanlinna theoryAbstract
This article is devoted to exploring the existence and the form of finite order transcendental entire solutions of Fermat-type second order partial differential-difference equations $$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2 +f(z_1+c_1,z_2+c_2)^2=e^{g(z_1,z_2)} $$ and $$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2+(f(z_1+c_1,z_2+c_2) -f(z_1,z_2))^2=e^{g(z)}, $$ where \(\delta,\eta\in\mathbb{C}\) and \(g(z_1,z_2)\) is a polynomial in \(\mathbb{C}^2\). Our results improve the results of Liu and Dong [23] Liu et al. [24] and Liu and Yang [25] Several examples confirm that the form of transcendental entire solutions of finite
order in our results are precise.
For more information see https://ejde.math.txstate.edu/Volumes/2023/43/abstr.html
References
T. B. Cao; The growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc, J. Math. Anal. Appl., 352(2) (2009), 739-748.
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(2) (2016), 1114-1132.
T. B. Cao, L. Xu; Logarithmic difference lemma in several complex variables and partial difference equations, Ann. Math. Pure Appl., 199 (2020), 767-794.
Y. M. Chiang, S. J. Feng; On the Nevanlinna characteristic of f(z + ) and difference equations in the complex plane, Ramanujan J., 16 (2008), 105-129.
F. Gross; On the equation fn(z) + gn(z) = 1, Bull. Amer. Math. Soc., 72 (1966), 86-88. [6] 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.
R. G. Halburd, R. J. Korhonen; Finite-order meromorphic solutions and the discrete Painleve equations, Proc. Lond. Math. Soc. 94(2) (2007), 443-474.
G. Haldar; Solutions of Fermat-type partial differential difference equations in C2, Mediterr. J. Math., 20 (2023), 50. https://doi.org/10.1007/s00009-022-02180-6
G. Haldar, M. B. Ahamed; Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in C2, Anal. Math. Phys. 12 (2022), Article number: 113. https:10.1007/s13324-022-00722-5
Q. Han, F. Lu; On the equation fn(z)+gn(z) = e z+ , J. Contemp. Math. Anal., 54 (2019), 98-102.
W. K. Hayman; Meromorphic Functions, The Clarendon Press, Oxford, 1964.
P. C. Hu, B. Q. Li; On meromorphic solutions of nonlinear partial differential equations of firs order, J. Math. Anal. Appl., 377 (2011), 881-888.
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.
D. Khavinson; A note on entire solutions of the eiconal equation, Amer. Math. Mon., 102 (1995), 159-161.
G. Iyer; On certain functional equations, J. Indian. Math. Soc., 3 (1939), 312-315.
I. Laine; Nevanlinna Theory and Complex differential Equations, Walter de Gruyter Berlin/Newyork, 1993.
P. Lelong; Fonctionnelles Analytiques et Fonctions Enti`eres (n variables), Presses de L'Universit e de Montreal, 1968.
B. Q. Li; On entire solutions of Fermat type partial differential equations, Int. J. Math., 15 (2004), 473-485.
B. Q. Li; Entire solutions of (uz1 )m + (uz2 )n = eg, Nagoya Math. J., 178 (2005), 151-162.
B. Q. Li; Entire solutions of eiconal type equations, Arch. Math., 89 (2007), 350-357.
P. Li, C. C. Yang; On the nonexistence of entire solutions of certain type of nonlinear differential equations, J. Math. Anal. Appl., 320 (2006), 827-835.
L. W. Liao, C. C. Yang, J. J. Zhang; On meromorphic solutions of certain type of non-linear differential equations, Ann. Acad. Sci. Fenn. Math., 38 (2013), 581-593.
K. Liu, X. J. Dong; Fermat type differential and difference equations, Electron. J. differential. Equ., 2015 (159) (2015), 1-10.
K. Liu, T. B. Cao, H. Z. Cao; Entire solutions of Fermat type differential-difference equations, Arch. Math., 99 (2012), 147-155.
K. Liu, L. Z. Yang; A note on meromorphic solutions of Fermat types equations, An. Stiint. Univ. Al. I. Cuza Lasi Mat. (N. S.), 1 (2016), 317-325.
F. Lu, Z. Li; Meromorphic solutions of Fermat type partial differential equations, J. Math. Anal. Appl., 478 (2) (2019), 864-873.
P. Montel; Lecons sur les familles de nomales fonctions analytiques et leurs applications, Gauthier-Viuars Paris, (1927), 135-136.
G. P olya; On an integral function of an integral function, J. Lond. Math. Soc., 1 (1926), 12-15.
L. I. Ronkin; Introduction to the Theory of Entire Functions of Several Variables, Moscow: Nauka 1971 (Russian), American Mathematical Society, Providence, 1974.
E. G. Saleebly; Entire and meromorphic solutions of Fermat type partial differential equations, Analysis (Munich), 19 (1999), 369-376.
E. G. Saleeby; On entire and meromorphic solutions of ..., Complex Var. Theory Appl., 49 (2004), 101-107.
E. G. Saleeby; On complex analytic solutions of certain trinomial functional and partial differential equations, Aequationes Math., 85 (2013), 553-562.
W. Stoll; Holomorphic Functions of Finite Order in Several Complex Variables, American Mathematical Society, Providence, 1974.
H. Y. Xu, Z. Xuan; Some inequalities on the convergent abscissas of Laplace-Stieltjes transforms, J. Math. Inequal. 17 (1) (2023), 163-183.
H. Y. Xu, H. Li, Z. Xuan ; Some new inequalities on Laplace-Stieltjes transforms involving logarithmic growth, Fractal Fract., (2022), 6, 233.
H. Y. Xu, L. Xu; Transcendental entire solutions for several quadratic binomial and trinomial PDEs with constant coe cients, Anal. Math. Phys., 12 (2022), 64.
H. Y. Xu, X. L. Liu, Y. H. Xu; On solutions for several systems of complex nonlinear partial differential equations with two variables, Anal. Math. Phys. 10.1007/s13324-023-00811-z.
H. Y. Xu, Y. Y. Jiang; Results on entire and meromorphic solutions for several systems of quadratic trinomial functional equations with two complex variables, RACSAM (2022) 116:8, https://doi.org/10.1007/s13398-021-01154-9.
H. Y. Xu, D. W. Meng, S. Y. Liu, H. Wang; Entire solutions for several second-order partial differential-difference equations of Fermat type with two complex variables, Adv. Di er. Equ., 2021, Article number 52. https://doi.org/10.1186/s13662-020-03201-y
L. Xu and T. B. Cao; Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 1-14.
H. Y. Xu, K. Y. Zhang and X. M. Zheng; Entire and meromorphic solutions for several Fermat type partial differential difference equations in C2, Rocky Mt. J. Math. 52 (6) (2022), 2169-2187.
C. C. Yang, P. Li; On the transcendental solutions of a certain type of non-linear differential equations, Arch. Math., 82 (2004), 442-448.
X. M. Zheng, H. Y. Xu; Entire solutions of some Fermat type functional equations concerning difference and partial differential in C2, Anal. Math., 48 (2022), 199-226.
Downloads
Published
License
Copyright (c) 2023 Hong Yan Xu, Goutam Haldar
This work is licensed under a Creative Commons Attribution 4.0 International License.
