引用本文:钟华, 王五生.一类非线性弱奇异积分不等式组中未知函数的估计[J].西南师范大学学报(自然科学版),2019,44(8):45~50
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一类非线性弱奇异积分不等式组中未知函数的估计
钟华, 王五生
河池学院 数学与统计学院, 广西 宜州 546300
摘要:
研究了一类二维非线性弱奇异积分不等式组.该不等式组积分号外有不同的非常数函数因子,不能用向量形式的Gronwall-Bellman型积分不等式进行估计.利用Hölder积分不等式、Gamma函数和Beta函数把弱奇异非线性积分问题转化成没有奇异的非线性积分问题,利用Bernoulli不等式把非线性问题转化成线性问题,利用变量替换技巧和放大技巧研究只含有一个未知函数的积分不等式,接着给出不等式组中两个未知函数的估计.该结果可用于研究积分、微分动力系统解的估计.
关键词:  Gronwall-Bellman型积分不等式  弱奇异积分不等式  二维积分不等式组  显式估计
DOI:10.13718/j.cnki.xsxb.2019.08.009
分类号:O178
基金项目:国家自然科学基金项目(11561019,11161018);广西省自然科学基金项目(2016GXNSFAA380090).
Estimation of Unknown Functions in a Class of Nonlinear Weakly Singular Integral Inequalities
ZHONG Hua, WANG Wu-sheng
School of Mathematics and Statistics, Hechi University, Yizhou Guangxi 546300, China
Abstract:
A class of two-dimensional weakly singular nonlinear integral inequalities have been studied, which include non-constant function factors outside the integral terms, and can not be estimated by Gronwall-Bellman type integral inequalities in vector form. With Hölder integral inequality, Gamma function and Beta function, the weak singular nonlinear integral problem is transformed into no singular nonlinear integral problem; and with Bernoulli inequality, the nonlinear problem is transformed into a linear problem; and with the variable substitution technique and the magnification technique, the integral inequality with only one unknown function is studied, and thenthe estimations of the two unknown functions in the inequality group are given. This result can be used to study the properties of the solutions of the integral and differential dynamical systems.
Key words:  Gronwall-Bellman type integral inequalities  weak singular integral inequalities  two dimensional integral inequalities  explicit estimates
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