JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (12): 48-57.doi: 10.6040/j.issn.1671-9352.0.2017.380

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Multiple positive solutions of a system of high order nonlinear fractional differential equations

FENG Hai-xing1, ZHAI Cheng-bo2*   

  1. 1. College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030031, Shanxi, China;
    2. School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China
  • Received:2017-07-31 Online:2017-12-20 Published:2017-12-22

Abstract: The existence of multiple positive solutions for a system of high-order nonlinear fractional differential equations is studied. Two or three positive solutions are obtained for the system by using Leggett-Williams fixed point theorem and Krasnoselskiion cone.

Key words: integral boundary value conditions, positive solution, Leggett-Williams fixed point theorem, the system of fractional order differential equation

CLC Number: 

  • O177.91
[1] OLDHAM K B, SPANIER J. The fractional calculus[M]. New York: Academic Press, 1974.
[2] KILBAS A A, SRIVASTAVA H M, TRUJILLO J J. Theory and applications of fractional differential equations[M]. Amsterdam: Elsevier Science, 2006, 204(49-52):2453-2461.
[3] KILBAS A A, MARZAN S A. Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions[J]. Differential Equations, 2005, 41(1):84-89.
[4] METZLER R, SCHICK W, KILIAN H, et al. Relaxation in filled polymers: a fractional calculus approach[J]. Journal of Chemical Physics, 1995, 103(16):7180-7186.
[5] GOODRICH C S. On discrete sequential fractional boundary value problems[J]. Journal of Mathematical Analysis and Applications, 2012, 385(1):111-124.
[6] LAKSHMIKANTHAM V. Theory of fractional functional differential equations[J]. Nonlinear Analysis Theory Methods and Applications, 2008, 69(10):3337-3343.
[7] FENG Wenquan, SUN Shurong, HAN Zhenlai, et al. Existence of solutions for a singular system of nonlinear fractional differential equations[J]. Computers and Mathematics with Applications, 2011, 62(3):1370-1378.
[8] FENG Haixing, ZHAI Chengbo. Existence and uniqueness of positive solutions for a class of fractional differential equation with integral boundary conditions[J]. Nonlinear Analysis Modelling and Control, 2017, 22(2):160-172.
[9] WANG Lin, LU Xinyi. Existence and uniqueness of solutions for a singular system of highter-order nonlinear fractional differential equations with integral boundary conditions[J]. Nonlinear Analysis: Modelling and Control, 2013, 31(31):493-518.
[10] LIANG Sihua, ZHANG Jihui. The existence of three positive solutions for some nonlinear boundary value problems on the half-line[J]. Positivity, 2009, 13(2):443-457.
[11] FERREIRA RAC. Positive solutions for a class of boundary value problems with fractional q-differences[J]. Pergamon Press, Inc, 2011, 61(2):367-373.
[12] YUAN Chengjun. Two positive solutions for(n-1,1)-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations[J]. Communications in Nonlinear Science and Numerical Simulation, 2012, 17(2):930-942.
[13] YANG Chen, ZHAI Chengbo. Uniqueness of positive solutions for a fractional differential equation via a fixed point theorem of a sum operator[J]. Electronic Journal of Differential Equations, 2012, 2012(70):808-826.
[14] ZHAI Chengbo, YAN Weiping, YANG Chen. A sum operator method for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems[J]. Communications in Nonlinear Science and Numerical Simulation, 2013, 18(4):858-866.
[15] 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,2000. GUO Dajun. Partial methods in nonlinear analysis[M]. Jinan: Shandong Science and Technology Press, 2000.
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