Solutions of complex nonlinear functional equations including second order partial differential and difference in C^2

Pub Date : 2023-06-26 DOI:10.58997/ejde.2023.43

H. Xu, Goutam Haldar

引用次数: 0

Abstract

This article is devoted to exploring the existence and the form of finite order transcendental entire solutions of Fermat-type second order partial differential-difference equations $$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2 +f(z_1+c_1,z_2+c_2)^2=e^{g(z_1,z_2)} $$ and $$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2+(f(z_1+c_1,z_2+c_2) -f(z_1,z_2))^2=e^{g(z)}, $$ where $\delta,\eta\in\mathbb{C}$ and $g(z_1,z_2)$ is a polynomial in $\mathbb{C}^2$. Our results improve the results of Liu and Dong [23] Liu et al. [24] and Liu and Yang [25] Several examples confirm that the form of tr

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C^2中包含二阶偏微分和差分的复非线性函数方程的解

本文探讨了费马型二阶偏微分-差分方程$$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2 +f(z_1+c_1,z_2+c_2)^2=e^{g(z_1,z_2)} $$和$$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2+(f(z_1+c_1,z_2+c_2) -f(z_1,z_2))^2=e^{g(z)}, $$的有限阶超越全解的存在性和形式，其中$\delta,\eta\in\mathbb{C}$和$g(z_1,z_2)$是$\mathbb{C}^2$中的一个多项式。我们的结果改进了Liu and Dong [23] Liu et al.[24]和Liu and Yang[25]的结果

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