一类带p-Laplacian算子的分数阶耦合系统在共振条件下的边值问题

华东师范大学学报(自然科学版) ›› 2014, Vol. 2014 ›› Issue (3): 30-39.

一类带p-Laplacian算子的分数阶耦合系统在共振条件下的边值问题

程玲玲¹, 刘文斌¹, 叶晴晴²

1. 中国矿业大学~~理学院, 江苏~~徐州 221116;
2. 南京理工大学~~理学院, 南京 210094

收稿日期:2013-04-01 修回日期:2013-07-01 出版日期:2014-05-25 发布日期:2014-07-25

Boundary value problem for a coupled system of fractional differential equations with p-Laplacian operator at resonance

CHENG Ling-ling¹, LIU Wen-bin¹, YE Qing-qing²

1. College of Science, China University of Mining and Technology, Xuzhou Jiangsu 221116, China;
2. School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

Received:2013-04-01 Revised:2013-07-01 Online:2014-05-25 Published:2014-07-25

摘要/Abstract

摘要： 应用葛渭高的Mawhin延拓定理的外延理论,
证明了当dim Ker M = 2 时解的存在性定理,
其中~$M$~为构造的拟线性算子．并给出了例子, 验证这个定理.

关键词: 分数阶微分方程, 边值问题, 迭合度, p-Laplacian算子

Abstract: This paper gave the existence
theorem of solutions with dim Ker M = 2 by the extension of Mawhin's continuation theorem due to Ge, where M is a
quasi-linear operator. And an example to illustrate the theorem was also given.

Key words: fractional differential equation, boundary value problem, coincidence degree, p-Laplacian operator

中图分类号:

O175

程玲玲, 刘文斌, 叶晴晴. 一类带p-Laplacian算子的分数阶耦合系统在共振条件下的边值问题[J]. 华东师范大学学报(自然科学版), 2014, 2014(3): 30-39.

CHENG Ling-ling, LIU Wen-bin, YE Qing-qing. Boundary value problem for a coupled system of fractional differential equations with p-Laplacian operator at resonance[J]. Journal of East China Normal University(Natural Sc, 2014, 2014(3): 30-39.

参考文献

{1} HEGAZI A S, AHMED E, MATOUK A E. On chaos control and synchronization of the commensurate fractional order Liu system [J]. Communications in Nonlinear Science \& Numerical Simulation, 2013, 18: 1193-1202.
{2} SANDERSE B. Energy-conserving Runge-Kutta methods for the incompressible Navier-Stokes equations [J]. Journal of Computational Physics, 2013, 233: 100-131.
{3} KILBAS A A. Theory and Applications of Fractional Differential Equations [M]. Amsterdam: Elsevier Science, 2006.
{4} MILLER K S. An Introduction to the Fractional Calculus and Differential Equations [M]. New York: John \& Wiley, 1993.
{5} LAKSHMIKANTHAM V, VATSALA A S. Basic theory of fractional differential equations [J]. Nonlinear Analysis: Theory, Methods \& Applications, 2008, 69: 2677-2682.
{6} BABAKHANI A, DAFTARDAR G V. Existence of positive solutions for N-term non-autonomous fractional differential equations [J]. Positivity, 2005, 9: 193-206.
{7} BAI C Z. Positive solutions for nonlinear fractional differential equations with coefficient that changes sign [J]. Nonlinear Analysis: Theory, Methods \& Applications, 2006, 64: 677-689.
{8} AGARWAI R P, OREGA D, STANEK S. Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations [J]. Journal of Mathematical Analysis and Applications , 2008, 69: 2677-2682.
{9} BENCHOHRA M, HENDERSON J, NTOUYAS SK. Existence results for positive solutions for fractional order functional differential equations with infinite delay [J]. Journal of Mathematical Analysis and Applications, 2005, 338: 1340-1350.
{10} ZHOU W, PENG J. Existence of solutions to boundary value problem for fractional differential equations [J]. Chinese Journal of engineering Mathematics, 2011, 28: 727-735.
{11} AHMADB, SIVASUNDARAM S. Existence results for nonlinear impulsive hybrid boundary value problem involving fractional differential equations [J]. Nonlinear Analysis: Hybrid Systems, 2009, 3: 251-258.
{12} BAI Z, LV H. Positive solutions for boundary value problem of nonlinear fractional differential equation [J]. Journal of Mathematical Analysis and Applications, 2005, 311: 495-505.
{13} WANG G, LIU W, ZHU S. The existence solutions for a fractional 2m-point boundary value problems [J]. Journal of Applied Mathematics, 2012: 2012: 1-14.
{14} ZHANG Y, BAI Z. Existence of solutions for nonlinear fractional three-point boundary value problems at resonance [J]. Journal of Applied Mathematics and Computing, 2012, 36: 417-440.
{15} ZHANG s, ZHANG J. Positive solutions for boundary-value problems of nonlinear fractional differential equation [J]. Nonlinear Analysis: Theory, Methods \& Applications, 2006, 36: 1-12.
{16} LIANG s, LV H. Positive solutions for boundary value problem of nonlinear fractional differential equation [J]. Journal of Mathematical Analysis and Applications, 2009, 71: 5545-5550.
{17} BENCHOHRA M, HAMANI S, NTOUYAS SK. Boundary value problem for differential equations with fractional order and nonlocal conditions [J]. Nonlinear Analysis: Theory, Methods \& Applications, 2009, 71: 2391-2396.
{18} CHEN T, LIU W, CHENG Y. Antiperiodic solutions for Linard-type differential equation with p-Laplacian operator [J]. Boundary Value Problems, 2010, 2010: 1-12.
{19} LIAN L, GE W. The existence of solutions of m-point \emph{p}-Laplacian boundary value problems at resonance [J]. Acta Mathematicae Applicatae Sinica, 2005, 28: 288-295.
{20} LIU B. On the existence of solutions for the periodic boundary value problems with \emph{p}-Laplacian operator [J]. Jouranl of systems Science Mathematical Sciences, 2003, 23: 76-85.
{21} FENG H, A symmetric solution of a multipoint boundary value problem with one-dimensional p-Laplacian at resonance [J]. Nonlinear Analysis: Theory, Methods \& Applications, 2008, 10: 3964-3972.
{22} SU X. Boundary value problem for a coupled system of nonlinear fractional differential equations [J]. Applied Mathematics Letters, 2009, 22: 64-69.
{23} JIANG W. Solvability for a coupled system of fractional differential equations at resonance [J]. Nonlinear Analysis: Real World Applications, 2012, 13: 2285-2292.
{24} GE W, REN. An extension of Mawhin's continuation theorem and its application to boundary value problems with p-Laplacian . Nonlinear Analysis, 2004, 58: 477-488.
{25} PODLUBNY I. Fractional differential equation [M]. San Diego: Academic Press, 1999.
{26} SAMKO S G. Fractional integrals and derivatives: Theory and Applications [M]. Switzerland: Gordon and Breach, 1993.
{27} LAKSHMIKANTHAM V. Theory of fractional dynamic systems [M]. Cambridge: Cambridge Academic Publishers, 2009.

[1]	王晶晶, 路艳琼. 二阶差分方程周期边值问题正解存在的最优条件[J]. 华东师范大学学报（自然科学版）, 2020, 2020(2): 41-49.
[2]	董凤娇, 胡贝贝. 广义Sasa-Satsuma方程在半直线上的初边值问题[J]. 华东师范大学学报(自然科学版), 2019, 2019(4): 33-41.
[3]	林苏榕, 倪明康. 三阶非线性向量常微分方程边值问题的奇摄动[J]. 华东师范大学学报(自然科学版), 2012, 2012(1): 138-150.
[4]	路艳琼, 高承华. 二阶变系数离散Neumann边值问题正解的存在性[J]. 华东师范大学学报(自然科学版), 2011, 2011(5): 66-72,102.
[5]	邓益军. 四维张量积二次矩形有限元最大模的超逼近[J]. 华东师范大学学报(自然科学版), 2011, 2011(4): 135-141.
[6]	姚庆六. 一类奇异三阶常微分方程的正解存在性与多解性[J]. 华东师范大学学报(自然科学版), 2010, 2010(3): 113-118.
[7]	姚庆六. 非线性Sturm-Liouville问题的一个正解存在定理[J]. 华东师范大学学报(自然科学版), 2009, 2009(1): 32-36.
[8]	耿凤杰;朱德明. 时间尺度上二阶动力方程的特征值问题(英）[J]. 华东师范大学学报(自然科学版), 2007, 2007(1): 7-12.