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arXiv · 2504.15907

On meromorphic solutions of Fermat type delay-differential equations with two exponential terms

Xuxu Xiang·Jianren Long·Mengting Xia·Zhigao Qin
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Abstract

The existence of the meromorphic solutions to Fermat type delay-differential equation \begin{equation} f^n(z)+a(f^{(l)}(z+c))^m=p_1(z)e^{a_1z^k}+p_2(z)e^{a_2z^k}, \nonumber \end{equation} is derived by using Nevanlinna theory under certain conditions, where $k\ge1$, $m,$ $n$ and $l$ are integers, $p_i$ are nonzero entire functions of order less than $k$, $c$, $a$ and $a_i$ are constants, $i=1,2$. These results not only improve the previous results from Zhu et al. [J. Contemp. Math. Anal. 59(2024), 209-219], Qi et al. [Mediterr. J. Math. 21(2024), article no. 122], but also completely solve two conjectures posed by Gao et al. [Mediterr. J. Math. 20(2023), article no. 167]. Some examples are given to illustrate these results.

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Xuxu Xiang, Jianren Long, Mengting Xia, Zhigao Qin. 2025-04-22. On meromorphic solutions of Fermat type delay-differential equations with two exponential terms. https://arxiv.org/abs/2504.15907

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