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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (6): 909-919.doi: 10.3969/j.issn.1005-3085.2015.06.012

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Banach空间脉冲发展方程初值问题解的单调迭代方法及应用

李   莹,   李   剑   

  1. 宝鸡文理学院数学系，宝鸡 721013
  • 收稿日期:2014-07-17 接受日期:2015-03-11 出版日期:2015-12-15 发布日期:2016-02-15
  • 基金资助:
    国家自然科学基金 (11071193; 11371031)；教育部新世纪优秀人才支持计划项目 (NCET-11-1041)；陕西省教育厅科研计划项目 (13JK0572; 14JK1035)；宝鸡文理学院重点科研项目 (Zk0689).

The Monotone Iterative Method for the Initial Value Problems for Impulsive Evolution Equations in Banach Space and Its Applications

LI Ying,   LI Jian   

  1. Department of Mathematics, Baoji University of Arts and Sciences, Baoji 721013
  • Received:2014-07-17 Accepted:2015-03-11 Online:2015-12-15 Published:2016-02-15
  • Supported by:
    The National Natural Science Foundation of China (11071193; 11371031); the Project for New Century Excellent Talents of Ministry of Education (NCET-11-1041); the Foundations of Shaanxi Educational Committee (13JK0572; 14JK1035); the Foundations of Baoji University of Arts and Sciences (Zk0689).

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摘要： 本文研究Banach空间中一阶脉冲发展方程初值问题的解．首先在有限区间上，运用单调迭代方法，在不要求上下解存在以及半群等度连续的条件下，得到了非脉冲发展方程正mild解的存在唯一性．其次在不假定脉冲函数连续以及单调的条件下，通过逐段延拓得到无穷区间上含脉冲的发展方程正mild解的存在唯一性，推广了已有结果．最后，我们将所得抽象结果运用到抛物型偏微分方程上，说明所得定理在应用中的有效性．

关键词: 单调迭代方法, 正mild解, 初值问题, 正$C_0$-半群

Abstract:

This paper investigates the solution of the initial value problem for first-order impulsive evolution equations in Banach space. Firstly, by using the monotone iterative method, we obtain the existence and uniqueness of the positive mild solution to the non-impulsive evolution equations on a finite interval without assuming the existence of upper and lower solutions and the equicontinuity of semigroup. Secondly, without the continuity and monotonicity of the impulsive function, we establish the existence and uniqueness of the positive mild solution to the impulsive evolution equations on an infinite interval by extending the finite interval, which improve the existing results. Finally, the gained abstract results are applied to the parabolic partial differential equations, which illustrates the validity of the obtained theorem in the application.

Key words: monotone iterative method, positive mild solution, initial value problem, positive $C_0$-semigroup

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