×

The existence and forms of solutions for several systems of the Fermat-type difference-differential equations. (English) Zbl 1475.30083

Summary: The sets of functional equations \[\begin{cases} f^{m_{1}} + g^{n_{1}} & = R_1 , \\ f^{m_{2}} + g^{n_{2}} & = R_2 , \end{cases}\] can be regarded as the Fermat-type system, where \(m_1, m_2, n_1, n_2\) are positive integers, \( R_1, R_2\) are nonzero polynomials. The purpose of this paper is to study the existence of solutions for several Fermat-type systems of the difference-differential equations
\[\begin{cases} [ f_1^\prime ( z ) ]^2 + f_2 ( z + c )^2 & = R_1 ( z ) , \\ [ f_2^\prime ( z ) ]^2 + f_1 ( z + c )^2 & = R_2 ( z ) , \end{cases}\]
\[\begin{cases} [ f_1^\prime ( z ) ]^2 + [ f_2 ( z + c ) - f_1 ( z ) ]^2 & = R_1 ( z ) , \\ [ f_2^\prime ( z ) ]^2 + [ f_1 ( z + c ) - f_2 ( z ) ]^2 & = R_2 ( z ) , \end{cases}\]
and
\[\begin{cases} [ f_1 ( z ) ]^2 + [ f_2 ( z + c ) - f_1 ( z ) ]^2 & = R_1 ( z ) , \\ [ f_2 ( z ) ]^2 + [ f_1 ( z + c ) - f_2 ( z ) ]^2 & = R_2 ( z ). \end{cases}\]
Our results about the existence and the forms of solutions for these Fermat types systems are some improvements of the previous theorems given by Gao, Liu, Yang, Cao, Qi. Meanwhile, we give some examples to utilize the existence and forms of solutions for such systems in each case.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39A10 Additive difference equations
39B72 Systems of functional equations and inequalities
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Z.-X. Chen, “Growth and zeros of meromorphic solution of some linear difference equations”, J. Math. Anal. Appl. 373:1 (2011), 235-241. · Zbl 1208.39028
[2] M. Chen, Y. Jiang, and Z. Gao, “Growth of meromorphic solutions of certain types of \(q\)-difference differential equations”, Adv. Difference Equ. (2017), art. 37, 16 pp. · Zbl 1422.34251
[3] Y.-M. Chiang and S.-J. Feng, “On the Nevanlinna characteristic of \(f(z+\eta)\) and difference equations in the complex plane”, Ramanujan J. 16:1 (2008), 105-129. · Zbl 1152.30024
[4] L. Y. Gao, “Entire solutions of two types of systems of complex differential-difference equations”, Acta Math. Sinica (Chin. Ser.) 59:5 (2016), 677-684. · Zbl 1374.34363
[5] F. Gross, “On the equation \(f^n + g^n = 1\)”, Bull. Amer. Math. Soc. 72 (1966), 86-88. · Zbl 0131.13603
[6] R. G. Halburd and R. J. Korhonen, “Difference analogue of the lemma on the logarithmic derivative with applications to difference equations”, J. Math. Anal. Appl. 314:2 (2006), 477-487. · Zbl 1085.30026
[7] R. G. Halburd and R. J. Korhonen, “Nevanlinna theory for the difference operator”, Ann. Acad. Sci. Fenn. Math. 31:2 (2006), 463-478. · Zbl 1108.30022
[8] R. G. Halburd and R. J. Korhonen, “Finite-order meromorphic solutions and the discrete Painlevé equations”, Proc. Lond. Math. Soc. (3) 94:2 (2007), 443-474. · Zbl 1119.39014
[9] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge, “Complex difference equations of Malmquist type”, Comput. Methods Funct. Theory 1:1 (2001), 27-39. · Zbl 1013.39001
[10] I. Laine, Nevanlinna theory and complex differential equations, De Gruyter Studies in Mathematics 15, Walter de Gruyter & Co., Berlin, 1993.
[11] Z. Latreuch, “On the existence of entire solutions of certain class of nonlinear difference equations”, Mediterr. J. Math. 14 :3 (2017), art. 115, 16 pp. · Zbl 1377.30027
[12] H.-C. Li, “On existence of solutions of differential-difference equations”, Math. Methods Appl. Sci. 39:1 (2016), 144-151. · Zbl 1332.30051
[13] K. Liu, “Meromorphic functions sharing a set with applications to difference equations”, J. Math. Anal. Appl. 359:1 (2009), 384-393. · Zbl 1177.30035
[14] K. Liu, “Value distribution of differences of meromorphic functions”, Rocky Mountain J. Math. 41:5 (2011), 1567-1584. · Zbl 1236.30030
[15] K. Liu and T.-B. Cao, “Entire solutions of Fermat type \(q\)-difference differential equations”, Electron. J. Differential Equations (2013), no. 59, 10 pp. · Zbl 1287.39006
[16] K. Liu and C. J. Song, “Meromorphic solutions of complex differential-difference equations”, Results Math. 72:4 (2017), 1759-1771. · Zbl 1387.34121
[17] K. Liu and L. Yang, “On entire solutions of some differential-difference equations”, Comput. Methods Funct. Theory 13:3 (2013), 433-447. · Zbl 1314.39022
[18] K. Liu, T. Cao, and H. Cao, “Entire solutions of Fermat type differential-difference equations”, Arch. Math. (Basel) 99:2 (2012), 147-155. · Zbl 1270.34170
[19] P. Montel, Leçons sur les récurrences et leurs applications, Gauthier-Villar, Paris, 1957. · Zbl 0077.06601
[20] X. Qi and L. Yang, “Properties of meromorphic solutions to certain differential-difference equations”, Electron. J. Differential Equations (2013), no. 135, 9 pp. · Zbl 1293.39001
[21] X. Qi, Y. Liu, and L. Yang, “A note on solutions of some differential-difference equations”, Izv. Nats. Akad. Nauk Armenii Mat. 52:3 (2017), 53-60. · Zbl 1372.30023
[22] R. Taylor and A. Wiles, “Ring-theoretic properties of certain Hecke algebras”, Ann. of Math. (2) 141:3 (1995), 553-572. · Zbl 0823.11030
[23] A. Wiles, “Modular elliptic curves and Fermat’s last theorem”, Ann. of Math. (2) 141:3 (1995), 443-551. · Zbl 0823.11029
[24] H.-Y. Xu and J. Tu, “Growth of solutions to systems of \(q\)-difference differential equations”, Electron. J. Differential Equations (2016), paper no. 106, 14 pp. · Zbl 1342.39012
[25] H. Y. Xu and H. Wang, “Notes on the existence of entire solutions for several partial differential-difference equations”, Bulletin of the Iranian Mathematical Society (2020).
[26] H. Y. Xu, S. Y. Liu, and Q. P. Li, “The existence and growth of solutions for several systems of complex nonlinear difference equations”, Mediterr. J. Math. 16:1 (2019), art. 8, 30 pp. · Zbl 1416.39005
[27] 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. 483:2 (2020), art. id. 123641, 22 pp. · Zbl 1429.39012
[28] C.-C. Yang and P. Li, “On the transcendental solutions of a certain type of nonlinear differential equations”, Arch. Math. (Basel) 82:5 (2004), 442-448. · Zbl 1052.34083
[29] C.-C. Yang and H.-X. Yi, Uniqueness theory of meromorphic functions, Mathematics and its Applications 557, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 2003. · Zbl 1070.30011
[30] J. Zhang, “On some special difference equations of Malmquist type”, Bull. Korean Math. Soc. 55:1 (2018), 51-61 · Zbl 1394.30020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.