JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (2): 66-72.doi: 10.6040/j.issn.1671-9352.0.2016.213

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Existence and uniqueness of solutions for three point boundary value problems of impulsive fractional differential equations

ZHANG Sha, JIA Mei*, LI Yan, LI Xiao-chen   

  1. College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
  • Received:2016-05-16 Online:2017-02-20 Published:2017-01-18

Abstract: The existence and uniqueness of solutions is investigated for a class of three point boundary value problems of fractional differential equations with impulsive. By using the Schauder fixed point theorem and contraction mapping principle, some theorems about the existence and uniqueness of solutions for the boundary value problem are obtained.

Key words: three-point boundary value problem, Caputo derivative, fixed point theorem, existence and uniqueness, impulsive fractional differential equations

CLC Number: 

  • O175.8
[1] 傅希林, 闫宝强, 刘衍胜. 脉冲微分系统引论[M]. 北京:科学出版社, 2005. FU Xilin, YAN Baoqiang, LIU Yansheng. The introduction of impulsive differential systems[M]. Beijing: Science Press, 2005.
[2] 孙继涛, 张瑜, 赵寿为. 脉冲系统的分析与控制[M]. 北京:科学出版社, 2013. SUN Jitao, ZHANG Yu, ZHAO Shouwei. Analysis and control of pulse system[M]. Beijing: Science Press, 2013.
[3] 白占兵. 分数阶微分方程边值问题理论及应用[M]. 北京:中国科学技术出版社, 2012. BAI Zhanbing. Theory and applications of fractional differential equations boundary value problems[M]. Beijing: Science and Technology Press, 2012.
[4] KILBAS A A, SRIVASTAVA H M, TRUJILLO J J. Theory and applications of fractional differential equations[M]. Amsterdam: Elsevier, 2006.
[5] DIETHELM K. The analysis of fractional differential equation[M]. Berlin Heidelberg: Springer-Verlag, 2010.
[6] 郭大钧, 孙经先, 刘兆理. 非线性常微分方程泛函方法[M]. 济南:山东科学技术出版社, 2006. GUO Dajun, SUN Jingxian, LIU Zhaoli. Functional methods for nonlinear ordinary differential equations[M]. Jinan: Shandong Science and Technology Press, 2006.
[7] AHMAD B, SIVASUNDARAM S. Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations[J]. Nonlinear Analysis Hybrid Systems, 2009, 3(3):251-258.
[8] 窦丽霞, 刘锡平, 金京福, 等. 分数阶积分微分方程多点边值问题解的存在性和唯一性[J]. 上海理工大学学报, 2012, 34(1):51-55. DOU Lixia, LIU Xiping, JIN Jingfu, et al. Existence and uniqueness of solutions of multi-point boundary value problems for integro-differential equations of fractional order[J]. Journal of University of Shanghai for Science and Technology, 2012, 34(1):51-55.
[9] WANG Jinrong, ZHOU Yong, MICHAL Feckan. On recent developments in the theory of boundary value problems for impulsive fractional differential equations[J]. Computers and Mathematics with Applications, 2012, 64(10):3008-3020.
[10] LIU Xiping, JIA Mei. Existence of solutions for the integral boundary value problems of fractional order impulsive differential equations[J]. Mathematical Methods in the Applied Sciences, 2015, 39(3):475-487.
[11] ZHOU Jie, FENG Meiqiang. Greens function for sturm-Liouville-type boundary value problems of fractional order impulsive differential equations and its application[J]. Boundary Value Problems, 2014, 2014(1):1-21.
[12] TIAN Yuansheng, BAI Zhanbing. Existence results for the three-point impulsive boundary value problem involving fractional differential equations[J]. Computers and Mathematics with Applications, 2010, 59(8):2601-2609.
[13] LI Xiaoping, CHEN Fulai, LI Xuezhu. Generalized anti-periodic boundary value problems of impulsive fractional differential equations[J]. Communications in Nonlinear Science and Numerical Simulation, 2013, 18(1):28-41.
[14] 陈强, 贾梅, 张海斌. 一类非线性分数阶微分方程四点边值问题解的存在性和唯一性[J]. 山东大学学报(理学版), 2015, 50(4):42-49. CHEN Qiang, JIA Mei, ZHANG Haibin. Existence and uniqueness of solutions for nonlinear fractional four-point boundary value problems[J]. Journal of Shandong University(Natural Science), 2015, 50(4):42-49.
[15] 杨浩, 刘锡平, 吴贵云. 一类分数阶p-Laplace算子微分方程非局部边值问题解的存在性[J]. 山东大学学报(理学版), 2015, 50(4):57-62. YANG Hao, LIU Xiping, WU Guiyun. Existence of the solutions for a type of nonlocal boundary value problems for fractional differential equations with p-Laplacian operator[J]. Journal of Shandong University(Natural Science), 2015, 50(4):57-62.
[16] 刘帅, 贾梅, 秦小娜. 带积分边值条件的分数阶微分方程解的存在性和唯一性[J]. 上海理工大学学报, 2014, 36(5):409-415. LIU Shuai, JIA Mei, QIN Xiaona. Existence and uniqueness of solutions of the fractional differential equation with integral boundary conditions[J]. Journal of University of Shanghai for Science and Technology, 2014, 36(5):409-415.
[17] YANG Liu, ZHANG Weiguo, LIU Xiping. A sufficient condition for the existence of a positive solution for a nonlinear fractional differential equation with the Riemann-Liouville derivative[J]. Applied Mathematics Letters, 2012, 25(25):1986-1992.
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