{"title":"C^2中包含二阶偏微分和差分的复非线性函数方程的解","authors":"H. Xu, Goutam Haldar","doi":"10.58997/ejde.2023.43","DOIUrl":null,"url":null,"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","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Solutions of complex nonlinear functional equations including second order partial differential and difference in C^2\",\"authors\":\"H. Xu, Goutam Haldar\",\"doi\":\"10.58997/ejde.2023.43\",\"DOIUrl\":null,\"url\":null,\"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\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-06-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.58997/ejde.2023.43\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.58997/ejde.2023.43","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
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
Solutions of complex nonlinear functional equations including second order partial differential and difference in C^2
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