On Meromorphic Solutions of Non-linear Difference Equations

In this paper, using the theory of linear algebra, we investigate the non-linear difference equation of the following form in the complex plane: f(z)n+p(z)f(z+η)=β1eα1z+β2eα2z+⋯+βseαsz,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(z)^n + p(z)f(z+\eta ) = \beta _1e^{\alpha _1z}+\beta _2e^{\alpha _2z}+\cdots +\beta _se^{\alpha _sz}, \end{aligned}$$\end{document}where n, s are the positive integers, p(z)≢0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(z)\not \equiv 0$$\end{document} is a polynomial and η,β1,…,βs,α1,…,αs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta , \beta _1, \ldots , \beta _s, \alpha _1, \ldots , \alpha _s$$\end{document} are the constants with β1…βsα1…αs≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1 \ldots \beta _s\alpha _1 \ldots \alpha _s\ne 0$$\end{document}, and show that this equation just has meromorphic solutions with hyper-order at least one when n≥2+s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2+s$$\end{document}. Other cases are also obtained.