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《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (6): 57-67.doi: 10.6040/j.issn.1671-9352.0.2022.595

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二阶脉冲发展方程非局部问题mild解的存在性

李莉(),杨和*()   

  1. 西北师范大学数学与统计学院, 甘肃 兰州 730070
  • 收稿日期:2022-11-14 出版日期:2023-06-20 发布日期:2023-05-23
  • 通讯作者: 杨和 E-mail:2430353979@qq.com;yanghe@nwnu.edu.cn
  • 作者简介:李莉(1995—), 女, 硕士研究生, 研究方向为非线性泛函分析. E-mail: 2430353979@qq.com
  • 基金资助:
    国家自然科学基金委地区科学基金资助项目(12061062)

Existence of mild solutions for the nonlocal problem of second-order impulsive evolution equations

Li LI(),He YANG*()   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Received:2022-11-14 Online:2023-06-20 Published:2023-05-23
  • Contact: He YANG E-mail:2430353979@qq.com;yanghe@nwnu.edu.cn

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摘要：

首先通过引入一个Green函数, 给出了含非局部条件$ u(0)=\sum\limits_{k=1}^n C_k u\left(\tau_k\right)$的二阶非线性脉冲发展方程mild解的新定义。其次运用Sadovskii不动点定理证明了该mild解的存在性。最后, 给出了一个具体例子作为抽象结果的应用。

关键词: 二阶脉冲发展方程, 余弦族, 非局部条件, Sadovskii不动点定理, mild解

Abstract:

A new definition of mild solutions of the second-order impulsive evolution equations involving nonlocal condition $ u(0)=\sum\limits_{k=1}^n C_k u\left(\tau_k\right)$ is given by introducing a Green function. Then, the existence of mild solutions of the concerned problem is proved by applying the Sadovskii fixed point theorem. At last, an example is provided as an application of the obtained abstract result.

Key words: second-order impulsive evolution equation, cosine family, nonlocal condition, Sadovskii fixed point theorem, mild solution

中图分类号: 

  • O175.15
1 LAKSHMIKANTHAM V , BAINOV D D , SIMEONOV P S . Theory of impulsive differential equations[M]. Singapore: World Scientific, 1989.
2 SAMOILENKO A M , PERESTYUK N A . Impulsive differential equations[M]. Singapore: World Scientific, 1995.
3 BALL J . Initial boundary value problems for an extensible beam[J]. Math Anal Appl, 1973, 42, 61- 90.
doi: 10.1016/0022-247X(73)90121-2
4 FITZGIBBON W E . Global existence and boundedness of solutions to the extensiblebeam equation[J]. Math Anal, 1982, 13, 739- 745.
5 FATTORINI H O . Second order linear differential equations in Banach spaces[M]. NorthHolland: Elsevier Science, 1985: 24- 38.
6 TRAVIS C C , WEBB G F . Compactness, regularity, and uniform continuity properties of strongly continuous cosine families[J]. Houston J Math, 1977, 3 (4): 555- 567.
7 TRAVIS C C , WEBB G F . Cosine families and abstract nonlinear second order differential equation[J]. Acta Math Acad Sci Hungar, 1978, 32 (3/4): 75- 96.
8 BALACHANDRAN K , PARK D G , ANTHONI S M . Existence of solutions of abstract-nonlinear second-order neutral functional integrodifferential equations[J]. Comput Math Appl, 2003, 46 (8/9): 1313- 1324.
9 HERNANDEZ E . Some comments on: existence of solutions of abstract nonlinear second-order neutral functional integrodifferential equations[J]. Comput Math Appl, 2005, 50 (5/6): 655- 669.
10 AKHMEROV R R . Measures of noncompactness and condensing operators[M]. Boston: Birkhauser, 1992.
11 ALVAREZ J C . Measure of noncompactness and fixed points of nonexpansive condensing mappings in locally convex spaces[J]. Rev Real Acad Cienc Exact Fis Natur Madrid, 1985, 79 (1/2): 53- 66.
12 DENG K . Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions[J]. J Math Anal Appl, 1993, 179 (2): 630- 637.
doi: 10.1006/jmaa.1993.1373
13 AKCA H , COVACHEV V , COVACHEVA Z . Existence theorem for a second-order impulsive functional differential equation with a nonlocal condition[J]. J Nonlinear Convex Anal, 2016, 17 (6): 1129- 1136.
14 HENRIQUEZ H R , HERNANDEZ E . Existence of solutions of a second order abstract functional Cauchy problem with nonlocal conditions[J]. Ann Polon Math, 2006, 88 (2): 141- 159.
doi: 10.4064/ap88-2-5
15 CHEN Pengyu , ZHANG Xuping , LI Yongxiang . Approximate controllability of non-autonomous evolution system with nonlocal conditions[J]. J Dyn Control Syst, 2020, 26 (1): 1- 16.
doi: 10.1007/s10883-018-9423-x
16 KAMENSKII M , OBUKHOVSKII V , ZECCA P . Condensing multivalued maps and semilinear differential inclusions in banach spaces[M]. New York: W. de Gruyter, 2001.
17 HEINZ H R . On the behavior of measure of noncompactness with respect to differentiation and integration of vector-valued functions[J]. Nonlinear Anal, 1983, 7 (12): 1351- 1371.
doi: 10.1016/0362-546X(83)90006-8
18 郭大钧. 非线性分析中的半序方法[M]. 济南: 山东科学技术出版社, 2005.
GUO Dajun . Semiorder methods in nonlinear analysis[M]. Jinan: Shandong Science and Technology Press, 2005.
19 SADOVSKII B N . On a fixed point principle[J]. Funct Anal Appl, 1967, 1 (2): 74- 76.
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