In this paper, we investigate the meromorphic solutions of the Fermat-type differential equations f’(z)n + f(z+c)m = eAz+B (c ≠ 0) over the complex plane C for positive integers m, n, and A, B, c are constants. Our results improve and extend some earlier results given by Liu et al. Moreover, some examples are presented to show the preciseness of our results.
Laine, I. (1993) Nevanlinna Theory and Complex Differential Equations. Walter de Gruyte, Berlin. https://doi.org/10.1515/9783110863147
[5]
Montel, P. (1927) Lecons sur les familles normales de fonctions analytiques et leurs applications. Gauthier Villars, Paris, 135-136.
[6]
Gross, F. (1966) On the Equation fn + gn = 1. I. Bulletin of the American Mathematical Society, 72, 86-88. https://doi.org/10.1090/S0002-9904-1966-11429-5
[7]
Gross, F. (1968) On the Equation fn + gn = 1. II. Bulletin of the American Mathematical Society, 74, 647-648. https://doi.org/10.1090/S0002-9904-1968-11975-5
[8]
Baker, I.N. (1966) On a Class of Meromorphic Functions. Proceedings of the American Mathematical Society, 17, 819-822. https://doi.org/10.1090/S0002-9939-1966-0197732-X
[9]
Yang, C.C. (1970) A Generalization of a Theorem of P. Montel on Entire Functions. Proceedings of the American Mathematical Society, 26, 332-334. https://doi.org/10.1090/S0002-9939-1970-0264080-X
[10]
Hu, P.C., Li, P. and Yang, C.C. (2003) Unicity of Meromorphic Mappings. Springer, Berlin. https://doi.org/10.1007/978-1-4757-3775-2
[11]
Halburd, R.G. and Korhonen, R.J. (2006) Difference Analogue of the Lemma on the Logarithmic Derivative with Applications to Difference Equations. Journal of Mathematical Analysis and Applications, 314, 477-487. https://doi.org/10.1016/j.jmaa.2005.04.010
[12]
Chiang, Y.M. and Feng, S.J. (2008) On the Nevanlinna Characteristic of f(z + η) and Difference Equations in the Complex Plane. The Ramanujan Journal, 16, 105-129. https://doi.org/10.1007/s11139-007-9101-1
[13]
Liu, K., Cao, T.B. and Cao, H.Z. (2012) Entire Solutions of Fermat Type Differential-Difference Equations. Archiv der Mathematik, 99, 147-155. https://doi.org/10.1007/s00013-012-0408-9
[14]
Liu, K., Yang, L.Z. and Liu, X.L. (2011) Existence of Entire Solutions of Nonlinear Difference Equations. Czechoslovak Mathematical Journal, 61, 565-576. https://doi.org/10.1007/s10587-011-0075-1
[15]
Liu, K., Yang, L.Z. and Liu, X.L. (2013) On Entire Solutions of Some Differential-Difference Equations. Computational Methods and Function Theory, 13, 433-447. https://doi.org/10.1007/s40315-013-0030-2
[16]
Lü, F. and Han, Q. (2017) On the Fermat-Type Equation f3(z) + f3(z + c) = 1. Aequationes Mathematicae, 91, 129-136. https://doi.org/10.1007/s00010-016-0443-x
[17]
Lü, F. and Han, Q. (2019) On the Equation fn(z) + gn(z) = eAz+B. Journal of Contemporary Mathematical Analysis, 54, 98-102. https://doi.org/10.3103/S1068362319020067
[18]
Xu, H.Y., Liu, S. and Wang, H. (2021) Entire Solutions for Several Second-Order Partial Differential-Difference Equations of Fermat Type with Two Complex Variables. Advances in Difference Equations, 1, 1-24. https://doi.org/10.1186/s13662-020-03201-y
[19]
Steinmetz, N. (1978) Zur wertverteilung von exponentialpolynomen. Manuscripta Mathematica, 26, 155-167. https://doi.org/10.1007/BF01167971
[20]
Yang, L. (1993) Value Distribution Theory. Springer-Verlag, Berlin.
[21]
Goldberg, A.A. and Ostrovskii, I.V. (2008) Value Distribution of Meromorphic Functions. American Mathematical Society, Providence.