JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2016, Vol. 51 ›› Issue (6): 78-84.doi: 10.6040/j.issn.1671-9352.0.2015.632

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Positive solutions for some singular fractional differential equation integral boundary value problems with p-Laplacian and a parameter

ZHONG Qiu-yan1, ZHANG Xing-qiu2,3   

  1. 1. Department of Information Technology, Jining Medical College, Jining 272067, Shandong, China;
    2. School of Medical Information Engineering, Jining Medical College, Rizhao 276826, Shandong, China;
    3. School of Mathematics, Liaocheng University, Liaocheng 252059, Shandong, China
  • Received:2015-12-29 Online:2016-06-20 Published:2016-06-15

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Abstract: A special cone is constructed by means of the properties of Green function. By introducing height functions of the nonlinear term on some bounded sets and considering integrations of these height functions, several existence and multiplicity of local positive solutions theorems for some nonlinear fractional differential equation integral boundary value problems with p-Laplacian and a parameter are obtained. The nonlinear term f permits singularities with respect to both the time and space variables.

Key words: fractional differential equation, singularity, p-Laplacian, height functions

CLC Number: 

  • O175.8
[1] SAMKO S G, KILBAS A A, MARICHEV O I. Fractional integral and derivative, in: theory and applications[M]. Switzerland: Gordon and Breach Science Publishers, 1993.
[2] PODLUBNY I. Fractional differential equations, mathematics in science and engineering[M]. San Diego: Academic Press, 1999.
[3] 郭柏灵, 蒲学科, 黄凤辉. 分数阶偏微分方程及其数值解[M]. 北京: 科学出版社, 2011. GUO Boling, PU Xueke, HUANG Fenghui. Fractional partial differential equations and their numerical solutions[M]. Beijing: Science Press, 2011.
[4] AHMAD B, NTOUYAS S K. Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions[J]. Appl Math Comput, 2015, 266:615-622.
[5] WANG Guotao. Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval[J]. Appl Math Lett, 2015, 47:1-7.
[6] ZHANG Xingqiu, WANG Lin, SUN Qian. Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter[J]. Appl Math Comput, 2014, 226:708-718.
[7] HENDERSON J, LUCA R. Positive solutions for a system of fractional differential equations with coupled integral boundary conditions[J]. Appl Math Comput, 2014, 249:182-197.
[8] WANG Yongqing, LIU Lishan, WU Yonghong. Positive solutions for a nonlocal fractional differential equation[J]. Nonlinear Anal, 2011, 74:3599-3605.
[9] JIA Mei, LIU Xiping. Three nonnegative solutions for fractional differential equations with integral boundary conditions[J]. Comput Math Appl, 2011, 62:1405-1412.
[10] ZHANG Xingqiu. Positive solutions for singular higher-order fractional differential equations with nonlocal conditions[J]. J Appl Math Comput, 2015, 49:69-89.
[11] JIANG Weihua. Solvability of fractional differential equations with p-Laplacian at resonance[J]. Appl Math Comput, 2015, 260:48-56.
[12] HAN Zhenlai, LU Hongling, ZHANG Chao. Positive solutions for eigenvalue problems of fractional differential equation with generalized p-Laplacian[J]. Appl Math Comput, 2015, 257:526-536.
[13] YAO Qingliu. Local existence of multiple positive solutions to a singular cantilever beam equation[J]. J Math Anal Appl, 2010, 363:138-154.
[14] YAO Qingliu. Positive solutions of nonlinear beam equations with time and space singularities[J]. J Math Anal Appl, 2011, 374:681-692.
[15] ZHANG Xingqiu. Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions[J]. Appl Math Lett, 2015, 39:22-27.
[16] GUO Dajun, LAKSHMIKANTHAM V. Nonlinear problems in abstract cones[M]. San Diego: Academic Press, 1988.
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