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 $$\begin{aligned} f^{n}(z)+q(z)\Delta _{c}f(z)=p_{1}\mathrm{e}^{\alpha _{1}z}+p_{2}\mathrm{e}^{\alpha _{2}z},\quad n\ge 2 \end{aligned}$$ and $$\begin{aligned} f^{n}(z)+q(z)\mathrm{e}^{Q(z)}f(z+c)=p_{1}\mathrm{e}^{\lambda z}+p_{2}\mathrm{e}^{-\lambda z},\quad n\ge 3 \end{aligned}$$ where q, Q are non-zero polynomials, \(c,\lambda ,p_{i},\alpha _{i}(i=1,2)\) are non-zero constants such that \(\alpha _{1}\ne \alpha _{2}\) and \(\Delta _{c}f(z)=f(z+c)-f(z)\not \equiv 0\). 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).