## Abstract

By Cartan’s version of Nevanlinna’s theory, we prove the following result: Let $m$ and $n$ be two positive integers satisfying $n\ge m+2$ and $m\ge 1$, let $p$ be a polynomial such that $p\not\equiv 0$, let $\eta $ be a finite complex number such that $\eta \ne 0$ and ${\mathrm{\Delta}}_{\eta}f\not\equiv 0$, let ${\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{m}$ be $m$ distinct finite nonzero complex numbers, and let ${H}_{j}$ be either an exponential polynomial of degree less than $q$ or an ordinary polynomial in $z$ such that ${H}_{j}\not\equiv 0$ for $1\le j\le m$. Suppose that $f$ is a nonconstant meromorphic solution of the difference equation

$${f}^{n}\left(z\right)+p\left(z\right){\mathrm{\Delta}}_{\eta}f\left(z\right)={H}_{1}\left(z\right){e}^{{\alpha}_{1}{z}^{q}}+{H}_{2}\left(z\right){e}^{{\alpha}_{2}{z}^{q}}+\cdots +{H}_{m}\left(z\right){e}^{{\alpha}_{m}{z}^{q}}.$$

Then $m\ge 2$ and $f$ is reduced to a transcendental entire function such that its order $\rho \left(f\right)=\infty $ or its order satisfies $\rho \left(f\right)=q$ with $m=2$, while $f$ can be either expressed as

$$f\left(z\right)={A}_{1}\left(z\right){e}^{{\alpha}_{1}{z}^{q}},\phantom{\rule{1em}{0ex}}\text{with}{A}_{1}\left(z\right)=\frac{{H}_{1}f}{p{\mathrm{\Delta}}_{\eta}f}\text{and}n{\alpha}_{1}-{\alpha}_{2}=0,$$

or

$$f\left(z\right)={A}_{2}\left(z\right){e}^{{\alpha}_{2}{z}^{q}},\phantom{\rule{1em}{0ex}}\text{with}{A}_{2}\left(z\right)=\frac{{H}_{2}f}{p{\mathrm{\Delta}}_{\eta}f}\text{and}n{\alpha}_{2}-{\alpha}_{1}=0,$$

where ${A}_{1}$ and ${A}_{2}$ are entire function such that their characteristic functions satisfy

$$T\left(r,{A}_{j}\right)=o\left(T\left(r,f\right)\right),\phantom{\rule{1em}{0ex}}1\le j\le 2,$$

as $r\notin E$ and $r\to \infty $. Here, $E\subset \left(0,+\infty \right)$ is a subset of finite linear measure.

An example is provided to show that the assumption $n\ge m+2$, in a sense, is the best possible.

This result improves and extends the corresponding results from Yang–Laine (2010), Liu–Lü–Shen–Yang (2014), and Latreuch (2017).

## Citation

Xiao-Min Li. Chen-Shuang Hao. Hong-Xun Yi. "On the existence of meromorphic solutions of certain nonlinear difference equations." Rocky Mountain J. Math. 51 (5) 1723 - 1748, October 2021. https://doi.org/10.1216/rmj.2021.51.1723

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