Exponential Polynomials and Nonlinear Differential-Difference Equations

Junfeng Xu,Jianxun Rong

doi:10.1155/2020/6901270

Abstract

In this paper, we study finite-order entire solutions of nonlinear differential-difference equations and solve a conjecture proposed by Chen, Gao, and Zhang when the solution is an exponential polynomial. We also find that any exponential polynomial solution of a nonlinear difference equation should have special forms.

Highlights

Introduction and Main ResultExtensive application of Nevanlinna theory has prompted scholars to acquire a number of results on differential equations, difference equations, and differential-difference equations
Chen et al [6] gave an example: f(z) ez is an entire solution of finite order of the following difference equation: f2(z) + 2e− 3zf(z − log 2) e2z + e− 2z
We mainly consider the solution of two equations when the solution is an exponential polynomial

Summary

Introduction and Main Result

Extensive application of Nevanlinna theory has prompted scholars to acquire a number of results on differential equations, difference equations, and differential-difference equations. Chen et al [6] gave an example: f(z) ez is an entire solution of finite order of the following difference equation: f2(z) + 2e− 3zf(z − log 2) e2z + e− 2z. From the example, they conjectured that the conclusions of eorem 2 are still valid if n 2. If the differential-difference equation (7) has solutions f satisfying f ∈ Γ0, ρ(f) deg Q 1 and q(z) must be a constant and one of the following two relation groups holds:.

Preliminaries

Proof of Theorem 3

Proof of Theorem 5

Conclusions