The exact entire solutions of certain type of nonlinear difference equations

TL;DR

This paper completely characterizes the entire solutions of a specific nonlinear difference equation involving exponential terms and polynomial coefficients, confirming a conjecture and providing explicit solution forms.

Contribution

It determines the exact form of non-constant entire solutions with order less than one for a class of nonlinear difference equations, affirming a prior conjecture.

Findings

Non-constant entire solutions are exponential functions with specific exponents.

Solutions have order less than one.

Confirmed a conjecture by Zhang et al.

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 $\alpha_1\neq \alpha_2$. 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_1e^{\frac{\alpha_1 z}{3}}+e_2e^{\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].