• Miao, Chunmei (Department of Applied Mathematics, Beijing Institute of Technology, College of Science, Changchun University) ;
  • Ge, Weigao (Department of Applied Mathematics, Beijing Institute of Technology)
  • Published : 2009.05.31


In this paper, the singular three-point boundary value problem $$\{{{u"(t)\;+\;f(t,\;u)\;=\;0,\;t\;{\in}\;(0,\;1),}\atop{u(0)\;=\;0,\;u(1)\;=\;{\alpha}u(\eta),}}\$$ is studied, where 0 < $\eta$ < 1, $\alpha$ > 0, f(t,u) may be singular at u = 0. By mixed monotone method, the existence and uniqueness are established for the above singular three-point boundary value problems. The theorems obtained are very general and complement previous know results.


  1. V. A. Il'in, E. I. Moiseev,Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differential quations 23(1987) 803-810.
  2. V. A. Il'in, E. I. Moiseev, Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator, Differential Equations 23(1987) 979-987.
  3. C. P. Gupta, A sharper condition for the solvability of a three-point second order boundary value problem, J. Math. Anal. and Applications 205(1997), 579-586.
  4. J. A. Gatica, V. Oliker and P. Waltman, Singular nonlinear boundary value problems for second-order ordinary differential equations, J. Differential Equations 79(1989), 62-78. https://doi.org/10.1016/0022-0396(89)90113-7
  5. W. Feng, J. R. L. Webb, Solvability of a three-point nonlinear boundary value problems at resonance, Nonlinear Analysis 30(1997), 3227-3238. https://doi.org/10.1016/S0362-546X(96)00118-6
  6. J. R. L. Webb, Positive solutions of some three-point boundary value problems via fixed point index theory, Nonlinear Analysis 47(2001), 4319-4332. https://doi.org/10.1016/S0362-546X(01)00547-8
  7. W. Feng, J. R. L. Webb, Solvability of a m-point boundary value problems with nonlinear growth, J. Math. Anal. and Applications 212(1997), 467-480. https://doi.org/10.1006/jmaa.1997.5520
  8. Z. B. Bai, W. G. Ge, Y. F. Wang, Multiplicity results for some second-order four-point boundary value problems, Nonlinear Analysis 60(2005), 491-500.
  9. P. K. Singh, A second order singular three-point boundary value problem, Appl. Math. Letters 17(2004), 969-976. https://doi.org/10.1016/j.aml.2003.08.011
  10. X. Xu, Multiplicity results for positive solutions of some semi-positive three-point boundary value problems, J. Math. Anal. and Applications 291(2004), 673-689. https://doi.org/10.1016/j.jmaa.2003.11.037
  11. D. X. Ma, J. X. Han, X. G. Chen, Positive solution of three-point boundary value problem for the one-dimensional p-Laplacian with singularities, J. Math. Anal. and pplications 324(2006), 118-133. https://doi.org/10.1016/j.jmaa.2005.11.063
  12. X. N. Lin, D. Q. Jiang, X. Y. Li, Existence and uniqueness of solutions for singular fourth-order boundary value problems, J. Comp. and Applied Mathematics 196(2006), 155-161. https://doi.org/10.1016/j.cam.2005.08.016
  13. D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988.
  14. D. Guo, Fixed point of mixed monotone operators and applications, Application Analysis 31(1998), 215-224.
  15. D. Guo, The Order Methods in Nonlinear Analysis, (in Chinese), Shandong Technical and Science Press, Jinan, 2000.