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摘要：为了研究一类带p-Laplacian 算子的Caputo分数阶微分方程边值问题正解的存在性，通过计算得到该问题的格林函数，并讨论其性质。运用单调迭代方法，得到该边值问题至少存在2个正解，最后通过实例验证了此类方程边值问题正解的存在性。
关键词：常微分方程其他学科；Caputo分数阶微分；正解；单调迭代方法；边值问题
中图分类号：O175.1 MSC（2010）主题分类：34B15 文献标志码：A
Abstract： In order to investigate the existence of positive solutions to a class of Caputo fractional differential equation boundary value problems with p-Laplacian operator， the Green’s function is obtained by calculus， and its properties are discussed. By using monotone iterative technique， at least two positive solutions are obtained for the boundary value problems. An example is given to illustrate the existence of positive solutions to this kind of equation boundary value problems.
Keywords：ordinary differential equation； Caputo fractional derivative；positive solution；monotone iteratiation；boundary value problems
参考文献/References：
[1] AHMAD B， MATAR M， AGARWAL R. Existence results for fractional differential equations of arbitrary order with nonlocal integral boundary conditions[J]. Boundary Value Problem，2015，220：1-13.
[2] ZHANG Lihong， AHMAD B， WANG Guotao，et al. Nonlinear fractional integro-differential equations on unbounded domains in a Banach space[J]. Journal of Computational and Applied Mathematics，2013，240：51-56.
[3] ALBERTO C， HAMDI Z. Nonlinear fractional differential equations with integral boundary value conditions[J]. Applied Mathematics and Computation，2014，228：251-257.
[4] VONG S. Positive solutions of singlular fractional differential equations with integral boundary conditions[J]. Mathematical and Computer Modelling，2013，57： 1053-1059.
[5] SOTIRIS K N， SINA E. On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions[J]. Applied Mathematics and Computation，2015，266： 235-243.
[6] KOU Chunhai， ZHOU Huacheng， YAN Ye. Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis[J]. Nonlinear Analysis，2011，74： 5975-5986.
[7] CUI Yujun. Uniqueness of solution for boundary value problems for fractional differential equations[J]. Applied Mathematics Letters，2016，51：48-54.
[8] ZHANG Lihong，AHMAD B， WANG Guotao. The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative[J]. Applied Mathematics Letters，2014，31：1-6.
[9] LI Yunhong， LI Guogang. Positive solutions of p-Laplacian fractional differential equations with integral boundary value conditions[J]. Journal of Nonlinear Science and Applications，2016，9（3）：717-726. [10]ZHAI Chengbo， XU Li. Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter[J]. Communications in Nonlinear Science Numerical Simulation，2014，19：2820-2827.
[11] XU Xiaojie，FEI Xiangli. The positive properties of Green’s function for three point boundary value problems of nonlinear fractional differential equations and its applications[J]. Communications in Nonlinear Science Numerical Simulation，2012，17（4）：1555-1565.
[12] NTOUYAS S， ETEMAD S. On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions[J]. Applied Mathematics and Computation，2015，266：235-243.
[13] 李云红，李艳. 带p-Laplacian算子的分数阶微分方程的正解[J].河北科技大学学报，2015，36（6）：593-597.
LI Yunhong，LI Yan. A positive solution for the fractional differential equation with a p-Laplacian operator[J]. Journal of Hebei University of Science and Technology， 2015，36（6）：593-597.
[14] ZHANG Xinguang， LIU Lisan， WIWATANAPATAPHEE B， et al. The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition[J]. Applied Mathematics and Computation，2014，235：412-422.
[15]AMKO S G，KILBAS A A，MARICHEV O I. Fractional Integrals and Derivatives： Theory and Applications[M]. Switzerland： Gordon and Breach，1993.
[16]PODLUNY I. Fractional Differential Equations， Mathematics in Science and Engineering[M]. New York： Academic Press，1999.
关键词：常微分方程其他学科；Caputo分数阶微分；正解；单调迭代方法；边值问题
中图分类号：O175.1 MSC（2010）主题分类：34B15 文献标志码：A
Abstract： In order to investigate the existence of positive solutions to a class of Caputo fractional differential equation boundary value problems with p-Laplacian operator， the Green’s function is obtained by calculus， and its properties are discussed. By using monotone iterative technique， at least two positive solutions are obtained for the boundary value problems. An example is given to illustrate the existence of positive solutions to this kind of equation boundary value problems.
Keywords：ordinary differential equation； Caputo fractional derivative；positive solution；monotone iteratiation；boundary value problems
参考文献/References：
[1] AHMAD B， MATAR M， AGARWAL R. Existence results for fractional differential equations of arbitrary order with nonlocal integral boundary conditions[J]. Boundary Value Problem，2015，220：1-13.
[2] ZHANG Lihong， AHMAD B， WANG Guotao，et al. Nonlinear fractional integro-differential equations on unbounded domains in a Banach space[J]. Journal of Computational and Applied Mathematics，2013，240：51-56.
[3] ALBERTO C， HAMDI Z. Nonlinear fractional differential equations with integral boundary value conditions[J]. Applied Mathematics and Computation，2014，228：251-257.
[4] VONG S. Positive solutions of singlular fractional differential equations with integral boundary conditions[J]. Mathematical and Computer Modelling，2013，57： 1053-1059.
[5] SOTIRIS K N， SINA E. On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions[J]. Applied Mathematics and Computation，2015，266： 235-243.
[6] KOU Chunhai， ZHOU Huacheng， YAN Ye. Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis[J]. Nonlinear Analysis，2011，74： 5975-5986.
[7] CUI Yujun. Uniqueness of solution for boundary value problems for fractional differential equations[J]. Applied Mathematics Letters，2016，51：48-54.
[8] ZHANG Lihong，AHMAD B， WANG Guotao. The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative[J]. Applied Mathematics Letters，2014，31：1-6.
[9] LI Yunhong， LI Guogang. Positive solutions of p-Laplacian fractional differential equations with integral boundary value conditions[J]. Journal of Nonlinear Science and Applications，2016，9（3）：717-726. [10]ZHAI Chengbo， XU Li. Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter[J]. Communications in Nonlinear Science Numerical Simulation，2014，19：2820-2827.
[11] XU Xiaojie，FEI Xiangli. The positive properties of Green’s function for three point boundary value problems of nonlinear fractional differential equations and its applications[J]. Communications in Nonlinear Science Numerical Simulation，2012，17（4）：1555-1565.
[12] NTOUYAS S， ETEMAD S. On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions[J]. Applied Mathematics and Computation，2015，266：235-243.
[13] 李云红，李艳. 带p-Laplacian算子的分数阶微分方程的正解[J].河北科技大学学报，2015，36（6）：593-597.
LI Yunhong，LI Yan. A positive solution for the fractional differential equation with a p-Laplacian operator[J]. Journal of Hebei University of Science and Technology， 2015，36（6）：593-597.
[14] ZHANG Xinguang， LIU Lisan， WIWATANAPATAPHEE B， et al. The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition[J]. Applied Mathematics and Computation，2014，235：412-422.
[15]AMKO S G，KILBAS A A，MARICHEV O I. Fractional Integrals and Derivatives： Theory and Applications[M]. Switzerland： Gordon and Breach，1993.
[16]PODLUNY I. Fractional Differential Equations， Mathematics in Science and Engineering[M]. New York： Academic Press，1999.