JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2020, Vol. 55 ›› Issue (10): 95-103.doi: 10.6040/j.issn.1671-9352.0.2019.833

Previous Articles     Next Articles

Existence of positive solutions for singular boundary value problems of Hadamard fractional differential equations in abstract space

SUN Yan-yan, LIU Yan-sheng   

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, Shandong, China
  • Online:2020-10-20 Published:2020-10-07

PDF (PC)

434

Abstract: The existence of positive solutions of singular boundary value problems in Banach space is investigated. By constructing a new cone and using the fixed point index theory of strict set compression operator, it is established that there are at least two positive solutions for the corresponding approximation of the boundary value problem. Then, using Ascoli-Arzela theorem and the relative compactness of the sequence of solutions, a sufficient condition is obtained for the existence of multiple positive solutions to boundary value problem.

Key words: Hadamard-type fractional derivative, cone, singular boundary value problem

CLC Number: 

  • O175.8
[1] KILBAS A A, TRUJILLO J J. Differential equations of fractional order: methods, results and problems(Ⅰ)[J]. Applicable Analysis, 2001, 78(1):153-192.
[2] KILBAS A A, TRUJILLO J J. Differential equations of fractional order: methods, results and problems(Ⅱ)[J]. Applicable Analysis, 2002, 81(2):435-493.
[3] GUO D J, LAKSHMIKANTHAM V. Nonlinear problems in abstract cones[M]. New York: Academic Press, 1988.
[4] GUO D J, LAKSHMIKANTHAM V, LIU X Z. Nonlinear integrals equations in abstract spaces[M]. Dordrecht: Kluwer Academic Publishers, 1996.
[5] STANEK S. The existence of positive solutions of singular fractional boundary value problems[J]. Computers and Mathematics with Applications, 2011, 62(3):1379-1388.
[6] 刘峰, 魏毅强. 一类分数阶奇异微分方程边值问题正解的存在性[J]. 中北大学学报(自然科学版), 2014, 35(5):515-520. LIU Feng, WEI Yiqiang. The existence of positive solutions for the boundary value problem of a singular fractional differential equation[J]. Journal of North University of China(Natural Science Edition), 2014, 35(5):515-520.
[7] XU Xiaojie, ZHANG Huina. Multiple positive solutions to singular positone and semipositone m-point boundary value problems of nonlinear fractional differential equations[J]. Boundary Value Problems, 2018, 34:2-18.
[8] 郭大钧. 非线性泛函分析[M]. 济南: 山东科学技术出版社, 1985. GUO Dajun. Nonlinear function analysis[M]. Jinan: Shandong Science and Technology Publishing House, 1985.
[9] AHMAD B, ALSAEDI A, NTOUYAS S K, et al. Hadamard-type fractional differential equations, inclusions and inequalities[M]. New York: Springer International Publishing, 2017.
[10] KILBAS A A. Hadamard-type fractional calculus[J]. Journal of the Korean Mathematical Society, 2001, 38(6):1191-1204.
[11] YUAN Chengjun. Multiple positive solutions for(n-1,n)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations[J]. Electronic Journal of Qualitative Theory of Differential Equations, 2010, 36(1):1-12.
[12] JIANG Jiqiang, LIU Weiwei, WANG Hongchuan. Positive solutions to singular Dirichlet-type boundary value problems of nonlinear fractional differential equations[J]. Advances in Difference Equations, Springer, 2018, 169:2-14.
[13] 王家玉, 刘衍胜. 抽象空间中二阶非线性奇异边值问题的正解[J]. 工程数学学报, 2009, 26(1):113-117. WANG Jiayu, LIU Yansheng. Positive solutions to boundary value problems for second-order nonlinear singular differential equations in abstract space[J]. Chinese Journal of Engineering Mathematics, 2009, 26(1):113-117.
[14] LIU Yansheng. Multiple positive solutions to fourth-order singular boundary value problems in abstract spaces[J]. Electronic Journal of Differential Equations, 2004, 120:1-13.
[1] MA Man-tang. Existence of positive solutions for a class of periodic boundary value problems of nonlinear second-order systems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 88-95.
[2] ZHANG Huan, LI Yong-xiang. Positive periodic solutions of higher-order ordinary differential equations with delayed derivative terms [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(4): 29-36.
[3] QI Ting-ting, ZHANG Zhen-fu, LIU Yan-sheng. Existence of positive solutions for fractional differential system with coupled integral boundary conditions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(2): 71-78.
[4] . Uniqueness of solution for singular boundary value problems of fourth-order differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(2): 73-76.
[5] XU Hai-li, LONG Pan-feng. Interaction between alkyl polyglycosides and other different surfactants [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(1): 9-14.
[6] YUE Yuan, TIAN Shuang-liang, CHEN Xiu-ping. An equivalence checking optimization algorithm for partial implementation combinational circuits [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(3): 116-121.
[7] GUO Li-jun. Existence of positive solutions for a third-order three-point boundary value problem of nonlinear differential equations [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(12): 47-53.
[8] XU Yan-chao. Static output feedback robust H∞ control for continuous-time positive systems [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(12): 87-94.
[9] ZHENG Chun-hua, LIU Wen-bin. Existence of positive solutions of the boundary value problem for a class of fractional differential equations with delay [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(03): 73-79.
[10] XU Yi-hong, HAN Qian-qian, WANG Tao, L Qiang. Lagrange optimality conditions for set-valued optimization problem in the sense of ε-strongly efficient elements [J]. J4, 2013, 48(3): 80-83.
[11] FAN Jin-jun, ZHANG Xue-ling, LIU Yan-sheng. Existence of positive solutions of the m-point boundary value problem with p-Laplace operator on time scales [J]. J4, 2012, 47(6): 16-19.
[12] WANG Minggao1,2, TANG Qiuyun3. Existence of multiple solutions for boundary value problem
of a singular differential system on the half-line [J]. J4, 2011, 46(6): 64-66.
[13] TIAN Ying-qiang, JIA Mei, LIU Shu. Positive solutions for third-order boundary value problems with integral boundary conditions [J]. J4, 2011, 46(2): 22-28.
[14] TANG Jing-yong1,2, HE Guo-ping3. Primal-dual infeasible interior-point algorithm for  second-order cone programming [J]. J4, 2011, 46(11): 96-100.
[15] WANG Xin-hua, ZHANG Xing-qiu. Existence of positive solutions for fourth order singular differential equations with Sturm-Liouville boundary conditions [J]. J4, 2010, 45(8): 76-80.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] HE Hai-lun， CHEN Xiu-lan* . Circular dichroism detection of the effects of denaturants and buffers on the conformation of cold-adapted protease MCP-01 and  mesophilic protease BP01[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2013, 48(1): 23 -29 .
[2] ZHAO Jun1, ZHAO Jing2, FAN Ting-jun1*, YUAN Wen-peng1,3, ZHANG Zheng1, CONG Ri-shan1. Purification and anti-tumor activity examination of water-soluble asterosaponin from Asterias rollestoni Bell[J]. J4, 2013, 48(1): 30 -35 .
[3] SUN Xiao-ting1, JIN Lan2*. Application of DOSY in oligosaccharide mixture analysis[J]. J4, 2013, 48(1): 43 -45 .
[4] XUE Qiu-fang1,2, GAO Xing-bao1*, LIU Xiao-guang1. Several equivalent conditions for H-matrix based on the extrapolated GaussSeidel iterative method[J]. J4, 2013, 48(4): 65 -71 .
[5] WU Yun1,2, HAN Ya-die1. Existence of nontrivial solutions of nonlinear elliptic equation with critical potential[J]. J4, 2013, 48(4): 91 -94 .
[6] LI Shi-long,ZHANG Yun-feng . Error analysis of the rational interpolation based on arithmetic average difference quotient[J]. J4, 2007, 42(10): 106 -110 .
[7] MAO Ai-qin1,2, YANG Ming-jun2, 3, YU Hai-yun2, ZHANG Pin1, PAN Ren-ming1*. Study on thermal decomposition mechanism of  pentafluoroethane fire extinguishing agent[J]. J4, 2013, 48(1): 51 -55 .
[8] YANG Ying, JIANG Long*, SUO Xin-li. Choquet integral representation of premium functional and related properties on capacity space[J]. J4, 2013, 48(1): 78 -82 .
[9] LI Yong-ming1, DING Li-wang2. The r-th moment consistency of estimators for a semi-parametric regression model for positively associated errors[J]. J4, 2013, 48(1): 83 -88 .
[10] WU Zhi-jun,SHEN Dan-dan. Architecture and key technologies of network-enabled next generation global flight tracking based on information integration and sharing[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(11): 1 -6 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd