一类三变量非线性和差分不等式中未知函数的估计

Estimation of Unknown Functions in a Class of Nonlinear Sum-Difference Inequalities in Three Independent Variables

Gronwall-Bellman型积分不等式的离散形式及其推广形式是研究和差分方程解的存在性、有界性和唯一性等定性性质的重要工具.研究了一类六重非线性和差分不等式,和号外含非常数因子,和项外包含了非常数项.利用差分算子的性质、求和技巧、变量替换技巧和积分中值定理等分析手段,给出了和差分不等式中未知函数的上界估计,推广了已有结果.最后举例说明所得结果可以用来研究三独立变量差分方程解的定性性质.

The discrete form and generalizations form of Gronwall-Bellman type integral inequality are the important tools to study of existence,boundedness and uniqueness and other qualitative properties of solutions of the difference equations.This paper studies a class of nonlinear sum-difference inequalities,which include a nonconstant factor outside summarizing symbols and a nonconstant term outside summation term.The upper bounds of the unknown function in the sum-difference inequali ty is estimated explicitly using proper ties of difference operators,summing techniques,the techniques of change of variable,the method of amplification,the integral mean value theorem,and other analysis techniques,which generalized some known results.Finally an example is given to illustrate the results can be used to study the qualitative properties of solutions of the difference equations with three independent variables.