The Exact Entire Solutions of Certain Type of Nonlinear Difference Equations

In this paper, we consider the entire solutions of nonlinear difference equation $$f^{3}+q(z)\Delta f=p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}$$ , where $$q$$ is a polynomial, and $$p_{1},p_{2},\alpha_{1},$$ and $$\alpha_{2}$$ are nonzero constants with $$\alpha_{1}\neq\alpha_{2}$$ . It is showed that if $$f$$ is a nonconstant entire solution of $$\rho_{2}(f)<1$$ to the above equation, then $$f(z)=e_{1}e^{\frac{\alpha_{1}z}{3}}+e_{2}e^{\frac{\alpha_{2}z}{3}}$$ , where $$e_{1}$$ and $$e_{2}$$ are two constants. Meanwhile, we give an affirmative answer to the conjecture posed by Zhang et al in [18].