On the existence of meromorphic solutions of certain nonlinear difference equations

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

$$f\left(z\right)=A_1\left(z\right)e^{{\alpha }_1{z}^q},\text{ with }{A}_1\left(z\right)=\frac{H_{1}f}{p{\mathrm{\Delta }}_{\eta }f}\text{and }n{\alpha }_1-{\alpha }_2=0,$$

or

$$f\left(z\right)=A_2\left(z\right)e^{{\alpha }_2{z}^q},\text{ with }{A}_2\left(z\right)=\frac{H_{2}f}{p{\mathrm{\Delta }}_{\eta }f}\text{and }n{\alpha }_2-{\alpha }_1=0,$$

where $A_1$ and $A_2$ are entire function such that their characteristic functions satisfy

$$T\left(r,A_j\right)=o\left(T\left(r,f\right)\right),1\le j\le 2,$$

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).