Meromorphic Solutions of Certain Nonlinear Difference Equations

Abstract

Let $$p_1, p_2$$ be nonzero rational functions, $$\alpha _1, \alpha _2$$ be nonzero constants, and $$P_d(z, f)$$ be a difference polynomial in f of degree d. We prove that every finite order meromorphic solution of nonlinear difference equations $$f^n+P_d(z, f)=p_1e^{\alpha _1 z}+p_2e^{\alpha _2 z}$$ has few poles provided $$n\ge 2$$ and $$d\le n-1$$ . This result shows there exists some difference between the existence of finite order meromorphic solutions of the above equations and its corresponding differential equations. We also give the forms of finite order meromorphic solutions of the above equations under some conditions on $$\alpha _1/\alpha _2$$ .