Entire Solutions of Certain Type of Non-Linear Difference Equations

In this paper, we study the existence of entire solutions of finite-order of non-linear difference equations of the form fn(z)+q(z)Δcf(z)=p1eα1z+p2eα2z,n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f^{n}(z)+q(z)\Delta _{c}f(z)=p_{1}\mathrm{e}^{\alpha _{1}z}+p_{2}\mathrm{e}^{\alpha _{2}z},\quad n\ge 2 \end{aligned}$$\end{document}and fn(z)+q(z)eQ(z)f(z+c)=p1eλz+p2e-λz,n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f^{n}(z)+q(z)\mathrm{e}^{Q(z)}f(z+c)=p_{1}\mathrm{e}^{\lambda z}+p_{2}\mathrm{e}^{-\lambda z},\quad n\ge 3 \end{aligned}$$\end{document}where q, Q are non-zero polynomials, c,λ,pi,αi(i=1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c,\lambda ,p_{i},\alpha _{i}(i=1,2)$$\end{document} are non-zero constants such that α1≠α2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{1}\ne \alpha _{2}$$\end{document} and Δcf(z)=f(z+c)-f(z)≢0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{c}f(z)=f(z+c)-f(z)\not \equiv 0$$\end{document}. Our results are improvements and complements of Wen et al. (Acta Math Sin 28:1295–1306, 2012), Yang and Laine (Proc Jpn Acad Ser A Math Sci 86:10–14, 2010) and Zinelâabidine (Mediterr J Math 14:1–16, 2017).