Fréchet空间上集值微分方程初值问题解的高阶收敛性

王培光, 邢珍钰, 吴曦冉

数学学报 ›› 2021, Vol. 64 ›› Issue (3) : 427-442.

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PDF(523 KB)
数学学报 ›› 2021, Vol. 64 ›› Issue (3) : 427-442. DOI: 10.12386/A2021sxxb0036
论文

Fréchet空间上集值微分方程初值问题解的高阶收敛性

    王培光, 邢珍钰, 吴曦冉
作者信息 +

Higher-Order Convergence of Solutions of Initial Value Problem for Set Differential Equations in Fréchet Space

    Pei Guang WANG, Zhen Yu XING, Xi Ran WU
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文章历史 +

摘要

本文讨论一类Fréchet空间F上的非线性集值微分方程初值问题解的收敛性.基于Fréchet空间F上所有紧致凸子集构成的空间Kc(F)可视为半线性度量空间Kc(Ei)的投影极限和投影极限的性质,通过引入集值函数的Fréchet偏导数以及集值函数的超凸和超凹性定义,应用比较原理和拟线性方法,对所构造的单调迭代序列进行分析,得到了在Kc(F)空间上集值微分方程初值问题的迭代解序列一致且高阶收敛于方程唯一解的判别准则.所得结果发展了Fréchet空间上的微分方程理论.

Abstract

This paper investigates the initial value problem for a class of set differential equations in Fréchet space F. Based on that the set Kc(F) of all compact convex subsets of a Fréchet space F is considered as a projective limit of semilinear metric spaces Kc(Ei), and the properties of projective limit, we introduce the notions of the Fréchet partial derivative, hyperconcave and hyperconvex of set-valued functions. By using the method of quasilinearization and comparison principle, we construct two monotone iterative sequences in Kc(F), and obtain the sequences of approximate solutions which converge uniformly and rapidly to the unique solution of the problem. The obtained results enrich and develop the theory of set-valued differential equations in Fréchet space F.

关键词

Fré / chet空间 / 集值微分方程 / 投影极限 / 拟线性化方法 / 高阶收敛

Key words

Fré / chet space / set differential equations / projective limits / quasilinearization / higher-order convergence

引用本文

导出引用
王培光, 邢珍钰, 吴曦冉. Fréchet空间上集值微分方程初值问题解的高阶收敛性. 数学学报, 2021, 64(3): 427-442 https://doi.org/10.12386/A2021sxxb0036
Pei Guang WANG, Zhen Yu XING, Xi Ran WU. Higher-Order Convergence of Solutions of Initial Value Problem for Set Differential Equations in Fréchet Space. Acta Mathematica Sinica, Chinese Series, 2021, 64(3): 427-442 https://doi.org/10.12386/A2021sxxb0036

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基金

国家自然科学基金资助项目(11771115,11271106)

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