On Meromorphic Solutions of Non-linear Difference Equations

Abstract

In this paper, using the theory of linear algebra, we investigate the non-linear difference equation of the following form in the complex plane:

$$\begin{aligned} f(z)^n + p(z)f(z+\eta ) = \beta _1e^{\alpha _1z}+\beta _2e^{\alpha _2z}+\cdots +\beta _se^{\alpha _sz}, \end{aligned}$$

where n, s are the positive integers, $p(z)\not \equiv 0$ is a polynomial and $\eta , \beta _1, \ldots , \beta _s, \alpha _1, \ldots , \alpha _s$ are the constants with $\beta _1 \ldots \beta _s\alpha _1 \ldots \alpha _s\ne 0$, and show that this equation just has meromorphic solutions with hyper-order at least one when $n\ge 2+s$. Other cases are also obtained.