一类非线性弱奇异积分不等式组中未知函数的估计

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H?lder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.

Estimation of Unknown Functions in a Class of Nonlinear Weakly Singular Integral Inequalities

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.