Solutions to some generalized Fermat-type differential-difference equations

Zhiyong Xu; Junfeng Xu; Zhiyong Xu; Junfeng Xu

doi:10.3934/math.20241643

AIMS Mathematics

2024, Volume 9, Issue 12: 34488-34503. doi: 10.3934/math.20241643

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Research article

Solutions to some generalized Fermat-type differential-difference equations

Zhiyong Xu ,
Junfeng Xu ^,

School of Mathematics and Computational Sciences, Wuyi University, Jiangmen 529000, Guangdong, China

Received: 01 November 2024 Revised: 24 November 2024 Accepted: 29 November 2024 Published: 09 December 2024
MSC : 30D35, 39A13, 39B72

The main purpose of this article is to study Fermat-type complex differential-difference equations $ f^{(k)}(z)^{2}+[\alpha f(z+c)-\beta f(z)]^{2} = R(z) $. Our results improve some results due to Wang–Xu–Tu [AIMS. Mathematics, 2020], Zhang [Bull. Korean. Math. Soc, 2018], and Long–Qin [Applied Mathematics-A Journal of Chinese Universities, 2024]. Moreover, we provide some examples to show the existence of the solutions.
- meromorphic functions,
- Fermat-type equation,
- differential polynomials,
- Nevanlinna theory,
- differential-difference equation
Citation: Zhiyong Xu, Junfeng Xu. Solutions to some generalized Fermat-type differential-difference equations[J]. AIMS Mathematics, 2024, 9(12): 34488-34503. doi: 10.3934/math.20241643

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Abstract

The main purpose of this article is to study Fermat-type complex differential-difference equations $ f^{(k)}(z)^{2}+[\alpha f(z+c)-\beta f(z)]^{2} = R(z) $. Our results improve some results due to Wang–Xu–Tu [AIMS. Mathematics, 2020], Zhang [Bull. Korean. Math. Soc, 2018], and Long–Qin [Applied Mathematics-A Journal of Chinese Universities, 2024]. Moreover, we provide some examples to show the existence of the solutions.

References

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