一类非线性Mate-Nevai型离散不等式及其应用  

A Class of Nonlinear Discrete Mate-Nevai Inequalities and Their Applications

作　　者：严勇[1] 

机构地区：[1]四川民族学院数学系,四川康定626001

出　　处：《四川师范大学学报（自然科学版）》2011年第5期677-683,共7页Journal of Sichuan Normal University（Natural Science）

基　　金：四川省教育厅自然科学重点基金(09ZC064)资助项目

摘　　要：Gronwall不等式及其各种线性、非线性推广是研究微分方程和差分方程解的存在性、有界性、唯一性和稳定性的重要工具.而离散的Gronwall不等式在验证微分方程与积分方程数值解的收敛性方面有着十分的重要作用.研究了一类非线性的Mate-Nevai型离散不等式,在B.G.Pachpatte(Tamkang J Math,2001,32:217-223.)的结果的基础上增加了二元函数项,该不等式含有两个无穷和项和一个非常数项.放弃对函数的单调性要求,通过将求和号外的函数作常量化,利用函数的单调化技巧和函数的次可乘性,给出了不等式中的未知函数的估计,进而将所得的不等式的估计用于研究一类非线性和分-差分方程解的估计.Gronwall inequalities and their various linear and nonlinear generalizations play very important roles in the discussion of existence,uniqueness,continuation,boundedness,and stability properties of solutions of differential equations and difference equations.Discrete Gronwall inequality plays an important role in proving convergence of numerical solutions of differential and integral equations.Some nonlinear discrete inequalities of Mate-Nevai type are established.On the basis of B.G.Pachpatte＇s result（Tamkang J Math,2001,32：217-223.）,we add term of function in two variables,two infinite summations and nonconstant term.In the generalized inequalities,the monotonicity of given functions is not required.We use techniques of monotonization,submultiplicative property of function and reduction of the term outside summations to be constant,and estimate the unknown function.Finally,the result is applied to discuss estimates of solutions for sum-difference equations.

关 键 词：非线性离散不等式 和分-差分方程 未知函数估计 

分 类 号：O178[理学—数学]