The exact entire solutions of certain type of nonlinear difference equations

Abstract

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 $p_1,p_2,{\alpha }_1,{\alpha }_2$ are nonzero constants with ${\alpha }_1\ne {\alpha }_2$. It is showed that if $f$ is a non-constant 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].

Entire solution, Nevanlinna theory, Difference equations, Riccati equation

References in corpus (1)

• Nevanlinna theory for the difference operator