Entire solutions for several Fermat type differential difference equations

Minghui Zhang; Jianbin Xiao; Mingliang Fang; Minghui Zhang; Jianbin Xiao; Mingliang Fang

doi:10.3934/math.2022646

AIMS Mathematics

2022, Volume 7, Issue 7: 11597-11613. doi: 10.3934/math.2022646

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Research article

Entire solutions for several Fermat type differential difference equations

Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, China

Received: 21 February 2022 Revised: 02 April 2022 Accepted: 08 April 2022 Published: 14 April 2022
MSC : 39A10, 30D35, 30D20, 30D05

This paper is devoted to investigate the existence and the forms of entire solutions of several Fermat type quadratic trinomial differential difference equations. Our results improve some results due to Liu and Yang [An. Stiint. Univ. Al. I. Cuza Iasi. Mat., 2016], Han and Lü [J. Contemp. Math. Anal., 2019], Luo, Xu and Hu [Open Math., 2021].
- entire solution,
- differential difference equation,
- quadratic trinomial
Citation: Minghui Zhang, Jianbin Xiao, Mingliang Fang. Entire solutions for several Fermat type differential difference equations[J]. AIMS Mathematics, 2022, 7(7): 11597-11613. doi: 10.3934/math.2022646

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Abstract

This paper is devoted to investigate the existence and the forms of entire solutions of several Fermat type quadratic trinomial differential difference equations. Our results improve some results due to Liu and Yang [An. Stiint. Univ. Al. I. Cuza Iasi. Mat., 2016], Han and Lü [J. Contemp. Math. Anal., 2019], Luo, Xu and Hu [Open Math., 2021].

References

[1]	I. N. Baker, On a class of meromorphic function, Proc. Amer. Math. Soc., 17 (1966), 819–822. https://doi.org/10.1090/S0002-9939-1966-0197732-X doi: 10.1090/S0002-9939-1966-0197732-X
[2]	L. Y. Gao, Entire solutions of two types of systems of complex differential-difference equations, Acta Math. Sin., 59 (2016), 677–684.
[3]	F. Gross, On the equation $f^n+g^n = 1$, Bull. Amer. Math. Soc., 72 (1966), 86–88. https://doi.org/10.1090/S0002-9904-1966-11429-5 doi: 10.1090/S0002-9904-1966-11429-5
[4]	F. Gross, On the equation $f^n+g^n = h^n$, Amer. Math. Monthy., 73 (1966), 1093–1096. https://doi.org/10.2307/2314644 doi: 10.2307/2314644
[5]	Q. Han, F. Lü, On the equation $f^n(z)+g^n(z) = e^{\alpha z+\beta}$, J. Contemp. Math. Anal., 54 (2019), 98–102. https://doi.org/10.3103/S1068362319020067 doi: 10.3103/S1068362319020067
[6]	W. K. Hayman, Meromorphic functions, Oxford: Clarendon Press, 1964.
[7]	I. Laine, Nevanlinna theory and complex differential equations, Berlin: De Gruyter, 1993. https://doi.org/10.1515/9783110863147
[8]	B. Q. Li, On meromorphic solutions of $f^2+g^2 = 1$, Math Z., 258 (2008), 763–771. https://doi.org/10.1007/s00209-007-0196-2 doi: 10.1007/s00209-007-0196-2
[9]	B. Q. Li, On Fermat-type functional and partial differential equations, Math Z., 258 (2008), 763–771.
[10]	B. Q. Li, Z. Ye, On meromorphic solutions of $f^3+g^3 = 1$, Arch Math., 90 (2008), 39–43. https://doi.org/10.1007/s00013-007-2371-4 doi: 10.1007/s00013-007-2371-4
[11]	K. Liu, Meromorphic functions sharing a set with applications to difference equations, J. Math. Anal. Appl., 359 (2009), 384–393. https://doi.org/10.1016/j.jmaa.2009.05.061 doi: 10.1016/j.jmaa.2009.05.061
[12]	K. Liu, T. B. Cao, H. Z. Cao, Entire solutions of Fermat type differential-difference equations, Arch. Math., 99 (2012), 147–155. https://doi.org/10.1007/s00013-012-0408-9 doi: 10.1007/s00013-012-0408-9
[13]	K. Liu, L. Z. Yang, A note on meromorphic solutions of Fermat types equations, An. Stiint. Univ. Al. I.-Mat., 62 (2016), 317–325.
[14]	J. Luo, H. Y. Xu, F. Hu, Entire solutions for several general quadratic trinomial differential difference equations, Open Math., 19 (2021), 1018–1028. https://doi.org/10.1515/math-2021-0080 doi: 10.1515/math-2021-0080
[15]	P. Montel, Leons sur les familles normales de fonctions analytiques et leurs applications, Gauthier-Villars, Parir, 1927,135–136.
[16]	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/10.1090/S0002-9939-1970-0264080-X doi: 10.1090/S0002-9939-1970-0264080-X
[17]	C. C. Yang, P. Li, On the transcendental solutions of a certain type of nonlinear differential equations, Arch. Math., 82 (2004), 442–448. https://doi.org/10.1007/s00013-003-4796-8 doi: 10.1007/s00013-003-4796-8
[18]	C. C. Yang, H. X. Yi, Uniqueness theory of meromorphic functions, Kluwer Academic Publishers Group, Dordrecht, 2003.
[19]	L. Yang, Value distribution theory, Berlin: Spring-Verlag, 1993.
[20]	H. X. Yi, L. Z. Yang, On meromorphic solutions of Fermat type functional equations, Sci. Sin. Math., 41 (2011), 907–932. https://doi.org/10.1360/012010-698 doi: 10.1360/012010-698

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