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### 高阶非线性分数阶微分方程系统的多个正解

1. 1.山西财经大学应用数学学院, 山西 太原 030031;2.山西大学数学科学学院, 山西 太原 030006
• 收稿日期:2017-07-31 出版日期:2017-12-20 发布日期:2017-12-22
• 通讯作者: 翟成波(1977— ),男,博士,教授,研究方向为非线性泛函微分方程. E-mail:cbzhai@sxu.edu.cn E-mail:seastar1981@126.com
• 作者简介:冯海星(1981— ),女,硕士,讲师,研究方向为非线性泛函微分方程. E-mail:seastar1981@126.com
• 基金资助:
国家自然科学基金青年科学基金资助项目(11201272);山西省自然科学基金资助项目(2015011005);2015山西省131人才项目;山西财经大学青年基金资助项目(2014026)

### 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.

• 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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