Entire solutions for some Fermat type functional equations concerning difference and partial differential in \(\mathbb{C}^2\). (English) Zbl 1499.30292
Let \(c_j\), \(j=1, 2\) and \(a_j\), \(j=1, 2, 3, 4\) be complex constants. The authors mainly treat a functional equation of the form \[ \Big(a_1f(z+c)+a_2f(z)\Big)^2+ \Big(a_3f(z+c)+a_4f(z)\Big)^2=1\] in \(\mathbb{C}^2\). They mainly consider transcendental entire solutions \(f(z)=f(z_1,z_2)\) of finite order with \(c=(c_1,c_2)\ne(0,0)\) by the difference analogues of Nevanlinna theory, see e.g., R. G. Halburd and R. J. Korhonen [Ann. Acad. Sci. Fenn., Math. 31, No. 2, 463–478 (2006; Zbl 1108.30022)]. It is shown that \(f(z)^2+\Big(f(z+c)-f(z)\Big)^2=1\) has no transcendental entire solutions of finite order. They also obtain conditions under which the above equation admits transcendental entire solutions of finite order when \(a_4=0\) and \(a_1a_4-a_2a_3\ne0\) respectively. Some examples are given. They further discuss some variations of the above equation with a partial differential operator.
Reviewer: Katsuya Ishizaki (Chiba)
MSC:
| 30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
| 35M30 | Mixed-type systems of PDEs |
| 32W50 | Other partial differential equations of complex analysis in several variables |
| 39A45 | Difference equations in the complex domain |
Keywords:
transcendental entire solution; partial difference equation; partial differential difference equationCitations:
Zbl 1108.30022References:
| [1] | Cao, T. B.; Korhonen, R. J., A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables, J. Math. Anal. Appl., 444, 1114-1132 (2016) · Zbl 1345.32012 · doi:10.1016/j.jmaa.2016.06.050 |
| [2] | Cao, T. B.; Xu, L., Logarithmic difference lemma in several complex variables and partial difference equations, Ann. Math. Pure Appl., 199, 767-794 (2020) · Zbl 1436.39009 · doi:10.1007/s10231-019-00899-w |
| [3] | Chiang, Y. M.; Feng, S. J., On the Nevanlinna characteristic of f (z + η) and difference equations in the complex plane, Ramanujan J., 16, 105-129 (2008) · Zbl 1152.30024 · doi:10.1007/s11139-007-9101-1 |
| [4] | Courant, R.; Hilbert, D., Methods of Mathematical Physics, vol. II: Partial Differential Equations (1962), New York: Interscience, New York · Zbl 0099.29504 |
| [5] | Garabedian, P. R., Partial Differential Equations (1964), New York: Wiley, New York · Zbl 0124.30501 |
| [6] | Gross, F., On the equation f^n + g^n = 1, Bull. Amer. Math. Soc., 72, 86-88 (1966) · Zbl 0131.13603 · doi:10.1090/S0002-9904-1966-11429-5 |
| [7] | Halburd, R. G.; Korhonen, R. J., Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 314, 477-487 (2006) · Zbl 1085.30026 · doi:10.1016/j.jmaa.2005.04.010 |
| [8] | Halburd, R. G.; Korhonen, R., Finite-order meromorphic solutions and the discrete Painlevé equations, Proc. London Math. Soc., 94, 443-474 (2007) · Zbl 1119.39014 · doi:10.1112/plms/pdl012 |
| [9] | Halburd, R. G.; Korhonen, R. J., Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math., 31, 463-478 (2006) · Zbl 1108.30022 |
| [10] | Hu, P. C.; Li, B. Q., On meromorphic solutions of nonlinear partial differential equations of first order, J. Math. Anal. Appl., 377, 881-888 (2011) · Zbl 1213.35018 · doi:10.1016/j.jmaa.2010.12.004 |
| [11] | Hu, P. C.; Li, P.; Yang, C. C., Unicity of Meromorphic Mappings, Advances, Complex Analysis and its Applications, vol. 1 (2003), Dordrecht, Boston, London: Kluwer Academic Publishers, Dordrecht, Boston, London · Zbl 1074.30002 |
| [12] | Iyer, G., On certain functional equations, J. Indian. Math. Soc., 3, 312-315 (1939) · JFM 65.1326.02 |
| [13] | Khavinson, D., A note on entire solutions of the eiconal equation, Amer. Math. Monthly, 102, 159-161 (1995) · Zbl 0845.35017 · doi:10.1080/00029890.1995.11990551 |
| [14] | Korhonen, R. J., A difference Picard Theorem for meromorphic functions of several variables, Comput. Methods Funct. Theory, 12, 343-361 (2012) · Zbl 1242.32008 · doi:10.1007/BF03321831 |
| [15] | Li, B. Q., Entire solutions of \({\left( {{u_{{z_1}}}} \right)^m} + {\left( {{u_{{z_2}}}} \right)^n} = {e^g}\), Nagoya Math. J., 178, 151-162 (2005) · Zbl 1086.35021 · doi:10.1017/S0027763000009156 |
| [16] | Li, B. Q., On entire solutions of Fermat type partial differential equations, Int. J. Math., 15, 473-485 (2004) · Zbl 1053.35042 · doi:10.1142/S0129167X04002399 |
| [17] | Li, B. Q., Entire solutions of certain partial differential equations and factorization of partial derivatives, Trans. Amer. Math. Soc., 357, 3169-3177 (2004) · Zbl 1073.35057 · doi:10.1090/S0002-9947-04-03745-6 |
| [18] | Li, B. Q., Entire solutions of eiconal type equations, Arch. Math., 89, 350-357 (2007) · Zbl 1137.35337 · doi:10.1007/s00013-007-2118-2 |
| [19] | Liu, K., Meromorphic functions sharing a set with applications to difference equations, J. Math. Anal. Appl., 359, 384-393 (2009) · Zbl 1177.30035 · doi:10.1016/j.jmaa.2009.05.061 |
| [20] | K. Liu and T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ., 2013 (2013), 10 pp. · Zbl 1287.39006 |
| [21] | Liu, K.; Cao, T. B.; Cao, H. Z., Entire solutions of Fermat type differential-difference equations, Arch. Math., 99, 147-155 (2012) · Zbl 1270.34170 · doi:10.1007/s00013-012-0408-9 |
| [22] | K. Liu and X. J. Dong, Fermat type differential and difference equations, Electron. J. Diff. Equ., 2015 (2015), paper No. 159, 10 pp. · Zbl 1321.39023 |
| [23] | Liu, K.; Song, C. J., Meromorphic solutions of complex differential-difference equations, Results Math., 72, 1759-1771 (2017) · Zbl 1387.34121 · doi:10.1007/s00025-017-0736-y |
| [24] | Liu, M. L.; Gao, L. Y., Transcendental solutions of systems of complex differential-difference equations, Sci. Sin. Math., 49, 1-22 (2019) |
| [25] | F. Lu, W. R. Lu, C. P. Li and J. F. Xu, Growth and uniqueness related to complex differential and difference equations, Results Math., 4 (2019), paper No. 30, 18 pp. · Zbl 1412.30114 |
| [26] | Lu, F.; Li, Z., Meromorphic solutions of Fermat type partial differential equations, J. Math. Anal. Appl., 478, 864-873 (2019) · Zbl 1420.35027 · doi:10.1016/j.jmaa.2019.05.058 |
| [27] | Pólya, G., On an integral function of an integral function, J. London. Math. Soc., 1, 12-15 (1926) · JFM 52.0317.06 · doi:10.1112/jlms/s1-1.1.12 |
| [28] | Qi, X. G.; Liu, Y.; Yang, L. Z., A note on solutions of some differential-difference equations, J. Contemp. Math. Anal., 52, 128-133 (2017) · Zbl 1372.30023 · doi:10.3103/S1068362317030037 |
| [29] | X. G. Qi and L. Z. Yang, Entire solutions of some differential-difference equations, Bull. Iran. Math. Soc., doi:10.1007/s41980-019-00277-5. · Zbl 1439.39011 |
| [30] | Rauch, J., Partial Differential Equations (1991), New York: Springer-Verlag, New York · Zbl 0742.35001 · doi:10.1007/978-1-4612-0953-9 |
| [31] | Saleeby, E. G., Entire and meromorphic solutions of Fermat type partial differential equations, Analysis (Munich), 19, 369-376 (1999) · Zbl 0945.35021 |
| [32] | Saleeby, E. G., On entire and meromorphic solutions of \(\lambda{u^k} + \sum\nolimits_{i = 1}^n {u_{{z_i}}^m = 1} \), Complex Var, Theory Appl., 49, 101-107 (2004) · Zbl 1055.30045 |
| [33] | Ronkin, L. I., Introduction to the Theory of Entire Functions of Several Variables (1974), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0286.32004 · doi:10.1090/mmono/044 |
| [34] | Stoll, W., Holomorphic Functions of Finite Order in Several Complex Variables (1974), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0292.32003 |
| [35] | Tang, J. F.; Liao, L. W., The transcendental meromorphic solutions of a certain type of nonlinear differential equations, J. Math. Anal. Appl., 334, 517-527 (2007) · Zbl 1127.34051 · doi:10.1016/j.jmaa.2006.12.075 |
| [36] | H. Y. Xu, S. Y. Liu, and 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, 22 pp. · Zbl 1429.39012 |
| [37] | H. Y. Xu and H. Wang, Notes on the existence of entire solutions for several partial differential-difference equations, Bull. Iran. Math. Soc., 46 (2020), doi:10.1007/s41980-020-00453-y. · Zbl 1477.35013 |
| [38] | L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math., 15 (2018), paper No. 227, 14 pp. · Zbl 1403.39014 |
| [39] | L. Xu and T. B. Cao, Correction to: Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math., 17 (2020), Paper No. 8, 4 pp. · Zbl 1431.39008 |
| [40] | Yang, C. C., A generalization of a theorem of P. Montel on entire functions, Proc. Amer. Math. Soc., 26, 332-334 (1970) · Zbl 0202.36001 · doi:10.1090/S0002-9939-1970-0264080-X |
| [41] | Yang, C. C.; Li, P., On the transcendental solutions of a certain type of nonlinear differential equations, Arch. Math., 82, 442-448 (2004) · Zbl 1052.34083 · doi:10.1007/s00013-003-4796-8 |
| [42] | Zhang, J., On some special difference equations of Malmquist type, Bull. Korean Math. Soc., 55, 51-61 (2018) · Zbl 1394.30020 |
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