| 摘要: |
| 常见的Gronwall不等式分为积分形式与微分形式。首先，对于常见的积分型Gronwall不等式，旨在给予一种新的证明方法，不同于以往不等式两端乘以指数函数的证明方法，而是应用最基本的积分公式加以证明，并用该不等式证明了一阶常微分方程解的唯一性；其次，旨在推广微分型Gronwall不等式，应用基本微分型不等式证明了波动方程解的唯一性及热传导方程的解能量估计；再者，应用变量代换、求导公式及基本的微分型Gronwall不等式，把一阶微分型的Gronwall不等式推广为两种情形：右端控制项由一次方升到α(α>0)次方；把一阶微分型的Gronwall不等式推广到二阶微分型的Gronwall不等式，并得到与一阶相似的结论。 |
| 关键词: Gronwall不等式 微分方程 应用 推广 |
| DOI： |
| 分类号: |
| 基金项目: |
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| Application and Generalization of a Class of Differential Gronwall Inequalities |
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NIE Dong-ming, MA Yan-li, PAN Na-na
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Department of Common Courses, Anhui Xinhua University, Hefei 230088, China
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| Abstract: |
| The common Gronwall inequalities are divided into integral and differential forms. Firstly, for the common integral Gronwall inequalities, the purpose is to give a new proof method, which is different from the proof method of exponential function multiplied by the two ends of former inequalities, but is proved by using the basic integral formula, and the uniqueness of the solutions of the ordinary differential equations of the first order type is proved by using this inequality. Secondly, the purpose is to generalize differential Gronwall inequality, and the uniqueness of the solutions of wave equation and the estimation of the solution energy of heat conduction equation are proved by using the basic differential equalities. Furthermore, by using variable substitution, derivation formula and the basic differential Gronwall inequality, the Gronwall inequality of the first order differential type is generalized into two cases: the right end control term rises from 1 to the αth power (α>0); the Gronwall inequality of the first order differential type is extended to the Gronwall inequality of the second order differential type, and the conclusion similar to that of first order differential type is obtained. |
| Key words: Gronwall inequality differential equation application generalization |
