Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2

Axioms

Hong Li¹  Hongyan Xu¹ 

[1] School of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, China;

关键词: Nevanlinna theory;    entire solution;    partial differential difference equation;   

DOI  :  10.3390/axioms10020126

来源: DOAJ

【 摘 要 】

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations $f{(z+c)}^2+2\alpha f\left(z\right)f(z+c)+f{\left(z\right)}^2=e^{g\left(z\right)},$ and the partial differential difference equations $f{(z+c)}^2+2\alpha f(z+c)\frac{\partial f\left(z\right)}{\partial z_1}+{\left(\frac{\partial f\left(z\right)}{\partial z_1}\right)}^2=e^{g\left(z\right)},$$f{(z+c)}^2+2\alpha f(z+c)\left(\frac{\partial f\left(z\right)}{\partial z_1}+\frac{\partial f\left(z\right)}{\partial z_2}\right)+{\left(\frac{\partial f\left(z\right)}{\partial z_1}+\frac{\partial f\left(z\right)}{\partial z_2}\right)}^2=e^{g\left(z\right)}.$ 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.

【 授权许可】

Unknown