关于非线性积分不等式的估计

在研究各种动力系统时,具有显式估计的不等式是关键的和广泛应用的解析工具之一.这些不等式无需提前知道积分方程的显式解,就能获得这些解的有价值的信息.为了处理新的积分方程,给出了关于一些基本积分不等式的新的显式估计.主要结果为：积分不等式x^p（t）≤a（t）＋m（t）∫0^tf（s）x^p（s）＋g（s）x^q（s）]ds蕴涵x（t）≤a^p^-1（t）＋p^-1a^p^-1-1（t）m（t）h（t）exp∫0^tF（s）ds,t∈R＋.这里p≥1,0〈q≤p,a（t）〉0,x（t）,m（t）,f（t）,g（t）≥0.作为应用,得到一个非线性积分方程解的显式估计.

Estimation on Certain Nonlinear Integral Inequalities

Inequalities with explicit estimates are among the most powerful and widely used analytic tools in the study of various dynamic systems.They enable us to obtain valuable information about solutions of integral equations without the need to know in advance the solutions explicitly.In the present paper,the new explicit estimates on some basic integral inequalities are offered which can be used as tools for handling new equations.The main result as follow： the integral inequality x^p（t）≤a（t）＋m（t）∫0^tf（s）x^p（s）＋g（s）x^q（s）]ds implies x（t）≤a^p^-1（t）＋p^-1a^p^-1-1（t）m（t）h（t）exp∫0^tF（s）ds,t∈R＋. where p≥1,0q≤p,a（t）0,x（t）,m（t）,f（t）,g（t）≥0.As an application of the result,the explicit estimates on the solutions of new nonlinear integral equations are obtained.