Entire solutions of nonlinear differential-difference equations.

Cuiping Li,Feng Lü,Junfeng Xu

doi:10.1186/s40064-016-2255-9

Abstract

In this paper, we describe the properties of entire solutions of a nonlinear differential-difference equation and a Fermat type equation, and improve several previous theorems greatly. In addition, we also deduce a uniqueness result for an entire function f(z) that shares a set with its shift f(z+c), which is a generalization of a result of Liu.

Highlights

Introduction and main resultThe complex oscillation theory of meromorphic solutions of differential equations is an important topic in complex analysis
We are interested in the properties of entire solutions of difference and differential-difference equations
We spare the reader for a moment and assume some familiarity with the basics of Nevanlinna theory of meromorphic functions in C such as the first and second main theorems, and the usual notations such as the characteristic function T(r, f), the proximity function m(r, f) and the counting function N(r, f)

Summary

Introduction

Introduction and main resultThe complex oscillation theory of meromorphic solutions of differential equations is an important topic in complex analysis. Xu et al (2015) considered a general differential-difference equation to obtain the following theorem. Theorem A Consider the nonlinear differential-difference equation q(z)f n(z) + a(z)f (k)(z + 1) = p1(z)eq1(z) + p2(z)eq2(z) where p1, p2 are two nonzero polynomials, q, a are two nonzero entire functions of finite order, q1 , q2 are two nonconstant polynomials, n ≥ 2 is an integer.

Results

Conclusion