JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (6): 57-67.doi: 10.6040/j.issn.1671-9352.0.2022.595

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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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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

CLC Number: 

  • 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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