On meromorphic solutions of Fermat type delay-differential equations with two exponential terms
Xuxu Xiang, Jianren Long, Mengting Xia, Zhigao Qin
TL;DR
This work analyzes meromorphic solutions to Fermat-type delay-differential equations with two exponential terms using Nevanlinna theory. The authors derive existence results under precise growth constraints, relax prior conditions (notably replacing $n>m+2$ with $n>4$ for $m\ge2$ or $n=4$ with $m>4$), and completely resolve two Gao conjectures, aided by explicit examples. A central finding is that any meromorphic solution must have hyper-order tied to the exponential-parameter structure, often forcing exponential-type solutions $f(z)=B(z)\,e^{a z^k/n}$ with $B^n=p$ and specific relations among $a$, $m$, $n$, and the exponents $a_i$. The paper also provides a detailed classification for the differential case $f^n(z)+a(f'(z))^n=p_1e^{bz}+p_2e^{-bz}$ with $n=2,3,4$, showing no solution for $n=3$, a centered $4$-th order family with explicit coefficient constraints, and a special $2$-nd order case when $9ab^2=-4$, thus delivering a comprehensive picture of when and how such Fermat-type equations admit entire or meromorphic solutions.
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.