《山东大学学报(理学版)》 ›› 2020, Vol. 55 ›› Issue (10): 95-103.doi: 10.6040/j.issn.1671-9352.0.2019.833
孙妍妍,刘衍胜
SUN Yan-yan, LIU Yan-sheng
摘要： 研究了Banach空间中奇异边值问题正解的存在性。通过构造一个特殊的锥,利用严格集压缩算子的不动点指数理论,建立了该边值问题的近似问题至少有两个正解的存在性。然后借助Ascoli-Arzela定理,利用近似问题解序列的相对紧性,得到边值问题至少有两个正解的充分条件。
中图分类号:
| [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] | 马满堂. 一类非线性二阶系统周期边值问题正解的存在性[J]. 《山东大学学报(理学版)》, 2019, 54(6): 88-95. |
| [2] | 章欢,李永祥. 含时滞导数项的高阶常微分方程的正周期解[J]. 《山东大学学报(理学版)》, 2019, 54(4): 29-36. |
| [3] | 亓婷婷, 张振福, 刘衍胜. 一类具有耦合积分边值条件的分数阶微分系统正解的存在性[J]. 《山东大学学报(理学版)》, 2019, 54(2): 71-78. |
| [4] | 崔玉军,赵聪. 四阶微分方程奇异边值问题解的唯一性[J]. 山东大学学报（理学版）, 2017, 52(2): 73-76. |
| [5] | 岳园,田双亮,陈秀萍. 部分实现组合电路的等价验证优化算法[J]. 山东大学学报（理学版）, 2016, 51(3): 116-121. |
| [6] | 郭丽君. 非线性微分方程三阶三点边值问题正解的存在性[J]. 山东大学学报（理学版）, 2016, 51(12): 47-53. |
| [7] | 徐言超. 连续时间正系统的静态输出反馈鲁棒H∞控制[J]. 山东大学学报（理学版）, 2016, 51(12): 87-94. |
| [8] | 郑春华, 刘文斌. 一类具有时滞的分数阶微分方程边值问题正解的存在性[J]. 山东大学学报（理学版）, 2015, 50(03): 73-79. |
| [9] | 徐义红, 韩倩倩,汪涛,吕强. 集值优化问题ε-强有效元的Lagrange型最优性条件[J]. J4, 2013, 48(3): 80-83. |
| [10] | 范进军,张雪玲,刘衍胜. 时间测度上带p-Laplace算子的m点边值问题正解的存在性[J]. J4, 2012, 47(6): 16-19. |
| [11] | 王明高1,2, 唐秋云3. 半直线上奇异微分方程组边值问题多个正解的存在性[J]. J4, 2011, 46(6): 64-66. |
| [12] | 田应强,贾梅,柳树. 含积分边界条件的三阶常微分方程边值问题正解的存在性[J]. J4, 2011, 46(2): 22-28. |
| [13] | 汤京永1,2,贺国平3. 二阶锥规划的原始-对偶不可行内点法[J]. J4, 2011, 46(11): 96-100. |
| [14] | 路慧芹. 一类非线性悬臂梁方程正解的存在性[J]. J4, 2011, 46(1): 81-86. |
| [15] | 王新华, 张兴秋. 具有Sturm-Liouville边界条件的四阶奇异微分方程正解的存在性[J]. J4, 2010, 45(8): 76-80. |
|
