Entire Solutions of a Certain Type of Nonlinear Differential-Difference Equations

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), and the partial differential difference equations f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z12=eg(z),f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z2+∂f(z)∂z1+∂f(z)∂z22=eg(z). We establish some theorems about the forms of the finite order transcendental entire solutions of these functional equations. We also list a series of examples to explain the existence of the finite order transcendental entire solutions of such equations. Meantime, some examples show that there exists a very significant difference with the previous literature on the growth order of the finite order transcendental entire solutions. Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order. These results make a few improvements of the previous theorems given by Xu and Cao, Liu and Yang.

Download Full-text

Polynomial difference equations which have entire solutions of finite order

Proceedings of the Japan Academy Series A Mathematical Sciences ◽

10.3792/pjaa.61.144 ◽

1985 ◽

Vol 61 (5) ◽

pp. 144-146

Author(s):

Yoshikuni Nakamura

Keyword(s):

Finite Order ◽

Difference Equations ◽

Entire Solutions

Download Full-text

Notes on the Existence of Entire Solutions for Several Partial Differential-Difference Equations

Bulletin of the Iranian Mathematical Society ◽

10.1007/s41980-020-00453-y ◽

2020 ◽

Cited By ~ 1

Author(s):

Hong Yan Xu ◽

Hua Wang

Keyword(s):

Difference Equations ◽

Entire Solutions ◽

Partial Differential

Download Full-text

Zeros, growth and Taylor coefficients of entire solutions of linear q-difference equations

Annales Fennici Mathematici ◽

10.5186/aasfm.2021.4617 ◽

2021 ◽

Vol 46 (1) ◽

pp. 249-277

Author(s):

Walter Bergweiler

Keyword(s):

Difference Equations ◽

Entire Solutions ◽

Taylor Coefficients

Download Full-text

Entire solutions for several second-order partial differential-difference equations of Fermat type with two complex variables

Advances in Difference Equations ◽

10.1186/s13662-020-03201-y ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Hong Yan Xu ◽

Da Wei Meng ◽

Sanyang Liu ◽

Hua Wang

Keyword(s):

Finite Order ◽

Difference Equations ◽

Nevanlinna Theory ◽

Meromorphic Functions ◽

Complex Variables ◽

Second Order ◽

Several Complex Variables ◽

Entire Solutions ◽

Partial Differential

AbstractThis paper is concerned with description of the existence and the forms of entire solutions of several second-order partial differential-difference equations with more general forms of Fermat type. By utilizing the Nevanlinna theory of meromorphic functions in several complex variables we obtain some results on the forms of entire solutions for these equations, which are some extensions and generalizations of the previous theorems given by Xu and Cao (Mediterr. J. Math. 15:1–14, 2018; Mediterr. J. Math. 17:1–4, 2020) and Liu et al. (J. Math. Anal. Appl. 359:384–393, 2009; Electron. J. Differ. Equ. 2013:59–110, 2013; Arch. Math. 99:147–155, 2012). Moreover, by some examples we show the existence of transcendental entire solutions with finite order of such equations.

Download Full-text