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http://dx.doi.org/10.4134/BKMS.b200774

ON THE EXISTENCE OF SOLUTIONS OF FERMAT-TYPE DIFFERENTIAL-DIFFERENCE EQUATIONS  

Chen, Jun-Fan (School of Mathematics and Statistics & Fujian Key Laboratory of Mathematical Analysis and Applications Fujian Normal University)
Lin, Shu-Qing (School of Mathematics and Statistics & Fujian Key Laboratory of Mathematical Analysis and Applications Fujian Normal University)
Publication Information
Bulletin of the Korean Mathematical Society / v.58, no.4, 2021 , pp. 983-1002 More about this Journal
Abstract
We investigate the non-existence of finite order transcendental entire solutions of Fermat-type differential-difference equations [f(z)f'(z)]n + P2(z)fm(z + 𝜂) = Q(z) and [f(z)f'(z)]n + P(z)[∆𝜂f(z)]m = Q(z), where P(z) and Q(z) are non-zero polynomials, m and n are positive integers, and 𝜂 ∈ ℂ \ {0}. In addition, we discuss transcendental entire solutions of finite order of the following Fermat-type differential-difference equation P2(z) [f(k)(z)]2 + [αf(z + 𝜂) - 𝛽f(z)]2 = er(z), where $P(z){\not\equiv}0$ is a polynomial, r(z) is a non-constant polynomial, α ≠ 0 and 𝛽 are constants, k is a positive integer, and 𝜂 ∈ ℂ \ {0}. Our results generalize some previous results.
Keywords
Fermat-type equation; differential-difference; entire function; Nevanlinna theory;
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1 F. Gross, On the equation fn + gn = 1, Bull. Amer. Math. Soc. 72 (1966), 86-88. https://doi.org/10.1090/S0002-9904-1966-11429-5   DOI
2 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 (2006), no. 2, 477-487. https://doi.org/10.1016/j.jmaa.2005.04.010   DOI
3 W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
4 I. Laine, Nevanlinna Theory and Complex Differential Equations, De Gruyter Studies in Mathematics, 15, Walter de Gruyter & Co., Berlin, 1993. https://doi.org/10.1515/9783110863147
5 K. Liu and C. J. Song, Meromorphic solutions of complex differential-difference equations, Results Math. 72 (2017), no. 4, 1759-1771. https://doi.org/10.1007/s00025-017-0736-y   DOI
6 F. Lu and Q. Han, On the Fermat-type equation f3(z) + f3(z + c) = 1, Aequationes Math. 91 (2017), no. 1, 129-136. https://doi.org/10.1007/s00010-016-0443-x   DOI
7 L. Wu, C. He, W. Lu, and F. Lu, Existence of meromorphic solutions of some generalized Fermat functional equations, Aequationes Math. 94 (2020), no. 1, 59-69. https://doi.org/10.1007/s00010-019-00683-4   DOI
8 Q. Han and F. Lu, On the equation fn(z) + gn(z) = eαz+β, J. Contem. Math. Anal. 54(2019), no. 2, 98-102.   DOI
9 Z. Chen, Zeros of entire solutions to complex linear difference equations, Acta Math. Sci. Ser. B (Engl. Ed.) 32 (2012), no. 3, 1141-1148. https://doi.org/10.1016/S0252-9602(12)60086-1   DOI
10 F. Gross, On the functional equation fn + gn = hn, Amer. Math. Monthly 73 (1966), 1093-1096. https://doi.org/10.2307/2314644   DOI
11 K. Liu, T. Cao, and H. Cao, Entire solutions of Fermat type differential-difference equations, Arch. Math. (Basel) 99 (2012), no. 2, 147-155. https://doi.org/10.1007/s00013-012-0408-9   DOI
12 X. Q. Lu, L. W. Liao, and J. Wang, On meromorphic solutions of a certain type of nonlinear differential equations, Acta Math. Sin. (Engl. Ser.) 33 (2017), no. 12, 1597-1608. https://doi.org/10.1007/s10114-017-6484-9   DOI
13 C.-C. Yang, A generalization of a theorem of P. Montel on entire functions, Proc. Amer. Math. Soc. 26 (1970), 332-334. https://doi.org/10.2307/2036399   DOI
14 C.-C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 1, 10-14. http://projecteuclid.org/euclid.pja/1262271517   DOI
15 C.-C. Yang and P. Li, On the transcendental solutions of a certain type of nonlinear differential equations, Arch. Math. (Basel) 82 (2004), no. 5, 442-448. https://doi.org/10.1007/s00013-003-4796-8   DOI
16 C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, Mathematics and its Applications, 557, Kluwer Academic Publishers Group, Dordrecht, 2003.
17 Y.-M. Chiang and S.-J. Feng, On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105-129. https://doi.org/10.1007/s11139-007-9101-1   DOI
18 C. P. Zeng, B. M. Deng, and M. L. Fang, Entire solutions of systems of complex differential-difference equations, Acta Math. Sinica (Chin. Ser.) 62 (2019), no. 1, 123-136.