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≥2and fn(z)+q(z)eQ(z)f(z+c)=p1eλz+p2e-λz,n≥3where q, Q are non-zero polynomials, c, λ, p _i , α _i (i= 1 , 2) are non-zero constants such that α ₁ ≠ α ₂ and Δ _c f(z) = f(z+ c) - f(z) ≢ 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).