The exact entire solutions of certain type of nonlinear difference equations

In this paper, we consider the entire solutions of nonlinear difference equation $f^3+q(z)Δf=p_1 e^{α_1 z}+ p_2 e^{α_2 z}$ where $q$ is a polynomial, and $p_1, p_2, α_1, α_2$ are nonzero constants with $α_1\neq α_2$. It is showed that if $f$ is a non-constant entire solution of $ρ_2(f)<1$ to the above equation, then $f(z)=e_1e^{\frac{α_1 z}{3}}+e_2e^{\frac{α_2 z}{3}},$ 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].

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