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Abstract and Applied Analysis
Volume 2014, Article ID 518238, 11 pages
http://dx.doi.org/10.1155/2014/518238
Research Article

Existence and Monotone Iteration of Positive Pseudosymmetric Solutions for a Third-Order Four-Point BVP

Department of Mathematics, Beihua University, Jilin 132013, China

Received 19 January 2014; Revised 6 March 2014; Accepted 13 March 2014; Published 8 April 2014

Academic Editor: Bashir Ahmad

Copyright © 2014 Xuezhe Lv et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. Greguš, Third Order Linear Differential Equations, Mathematics and Its Applications, Reidel, Dordrecht, The Netherlands, 1987. View at MathSciNet
  2. A. R. Aftabizadeh, C. P. Gupta, and J.-M. Xu, “Existence and uniqueness theorems for three-point boundary value problems,” SIAM Journal on Mathematical Analysis, vol. 20, no. 3, pp. 716–726, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. R. P. Agarwal, “Existence-uniqueness and iterative methods for third-order boundary value problems,” Journal of Computational and Applied Mathematics, vol. 17, no. 3, pp. 271–289, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. A. Boucherif and N. Al-Malki, “Nonlinear three-point third-order boundary value problems,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1168–1177, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. A. Cabada, M. R. Grossinho, and F. Minhós, “Extremal solutions for third-order nonlinear problems with upper and lower solutions in reversed order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 6, pp. 1109–1121, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. P. W. Eloe and B. Ahmad, “Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions,” Applied Mathematics Letters, vol. 18, no. 5, pp. 521–527, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. D. Li, L. Wang, and M. Pei, “Existence and monotone iteration of positive pseudosymmetric solutions for a third-order four-point BVP with p-Laplacian,” Abstract and Applied Analysis, vol. 2013, Article ID 192509, 12 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. R. Graef and J. R. L. Webb, “Third order boundary value problems with nonlocal boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1542–1551, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. L.-J. Guo, J.-P. Sun, and Y.-H. Zhao, “Existence of positive solutions for nonlinear third-order three-point boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 10, pp. 3151–3158, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. M. R. Grossinho, F. M. Minhós, and A. I. Santos, “Existence result for a third-order ODE with nonlinear boundary conditions in presence of a sign-type Nagumo control,” Journal of Mathematical Analysis and Applications, vol. 309, no. 1, pp. 271–283, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. C. P. Gupta and V. Lakshmikantham, “Existence and uniqueness theorems for a third-order three-point boundary value problem,” Nonlinear Analysis, vol. 16, no. 11, pp. 949–957, 1991. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. J. Henderson, “Best interval lengths for boundary value problems for third order Lipschitz equations,” SIAM Journal on Mathematical Analysis, vol. 18, no. 2, pp. 293–305, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. G. Infante and P. Pietramala, “A third order boundary value problem subject to nonlinear boundary conditions,” Mathematica Bohemica, vol. 135, no. 2, pp. 113–121, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. D. Jiang and R. P. Agarwal, “A uniqueness and existence theorem for a singular third-order boundary value problem on [0, ),” Applied Mathematics Letters, vol. 15, no. 4, pp. 445–451, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. S. Jin and S. Lu, “Existence of solutions for a third-order multipoint boundary value problem with p-Laplacian,” Journal of the Franklin Institute, vol. 347, no. 3, pp. 599–606, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. X. Lin, Z. Du, and W. Liu, “Uniqueness and existence results for a third-order nonlinear multi-point boundary value problem,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 187–196, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  17. R. Ma, “Multiplicity results for a third order boundary value problem at resonance,” Nonlinear Analysis: Theory, Methods & Applications, vol. 32, no. 4, pp. 493–499, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. F. M. Minhós, “On some third order nonlinear boundary value problems: existence, location and multiplicity results,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1342–1353, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. J. J. Nieto, “Periodic solutions for third order ordinary differential equations,” Commentationes Mathematicae Universitatis Carolinae, vol. 32, no. 3, pp. 495–499, 1991. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. D. O'Regan, “Topological transversality: applications to third order boundary value problems,” SIAM Journal on Mathematical Analysis, vol. 18, no. 3, pp. 630–641, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. Pei and S. K. Chang, “Existence and uniqueness of solutions for third-order nonlinear boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 23–35, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. I. Rachůnková, “Periodic boundary value problems for third-order differential equations,” Mathematica Slovaca, vol. 41, no. 3, pp. 241–248, 1991. View at Google Scholar · View at MathSciNet
  23. B. Sun, J. Zhao, P. Yang, and W. Ge, “Successive iteration and positive solutions for a third-order multipoint generalized right-focal boundary value problem with p-Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 220–230, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. Y. Sun, “Positive solutions for third-order three-point nonhomogeneous boundary value problems,” Applied Mathematics Letters, vol. 22, no. 1, pp. 45–51, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. F. Wang and Y. Cui, “On the existence of solutions for singular boundary value problem of third-order differential equations,” Mathematica Slovaca, vol. 60, no. 4, pp. 485–494, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. Y. Wang and W. Ge, “Existence of solutions for a third order differential equation with integral boundary conditions,” Computers and Mathematics with Applications, vol. 53, no. 1, pp. 144–154, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. P. J. Y. Wong, “Constant-sign solutions for a system of third-order generalized right focal problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5–7, pp. e2153–e2163, 2005. View at Publisher · View at Google Scholar · View at Scopus
  28. Q. Yao, “Positive solutions of singular third-order three-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 354, no. 1, pp. 207–212, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  29. P. Zhang, “Iterative solutions of singular boundary value problems of third-order differential equation,” Boundary Value Problems, vol. 2011, Article ID 483057, 10 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  30. C. Zhou and D. Ma, “Existence and iteration of positive solutions for a generalized right-focal boundary value problem with p-Laplacian operator,” Journal of Mathematical Analysis and Applications, vol. 324, no. 1, pp. 409–424, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. H. Pang, M. Feng, and W. Ge, “Existence and monotone iteration of positive solutions for a three-point boundary value problem,” Applied Mathematics Letters, vol. 21, no. 7, pp. 656–661, 2008. View at Publisher · View at Google Scholar · View at Scopus
  32. B. Sun and W. Ge, “Successive iteration and positive pseudo-symmetric solutions for a three-point second-order p-Laplacian boundary value problems,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1772–1779, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. R. Avery and J. Henderson, “Existence of three positive pseudo-symmetric solutions for a one dimensional p-Laplacian,” Journal of Mathematical Analysis and Applications, vol. 277, no. 2, pp. 395–404, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus