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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 782363, 7 pages
http://dx.doi.org/10.1155/2013/782363
Research Article

Solvability of a Fourth-Order Boundary Value Problem with Integral Boundary Conditions

Department of Mathematics, Beihua University, Jilin City 132013, China

Received 21 December 2012; Accepted 13 February 2013

Academic Editor: Meng Fan

Copyright © 2013 Hui Li 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. R. P. Agarwal and Y. M. Chow, “Iterative methods for a fourth order boundary value problem,” Journal of Computational and Applied Mathematics, vol. 10, no. 2, pp. 203–217, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. Z. Bai, “The upper and lower solution method for some fourth-order boundary value problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 6, pp. 1704–1709, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. Cabada and F. M. Minhós, “Fully nonlinear fourth-order equations with functional boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 239–251, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. A. Del Pino and R. F. Manásevich, “Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition,” Proceedings of the American Mathematical Society, vol. 112, no. 1, pp. 81–86, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Du and M. Cui, “Constructive proof of existence for a class of fourth-order nonlinear BVPs,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 903–911, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. Ehme, P. W. Eloe, and J. Henderson, “Upper and lower solution methods for fully nonlinear boundary value problems,” Journal of Differential Equations, vol. 180, no. 1, pp. 51–64, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. D. Franco, D. O'Regan, and J. Perán, “Fourth-order problems with nonlinear boundary conditions,” Journal of Computational and Applied Mathematics, vol. 174, no. 2, pp. 315–327, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H. Feng, D. Ji, and W. Ge, “Existence and uniqueness of solutions for a fourth-order boundary value problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 10, pp. 3561–3566, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. R. Graef and L. Kong, “A necessary and sufficient condition for existence of positive solutions of nonlinear boundary value problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 66, no. 11, pp. 2389–2412, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. R. Grossinho and St. A. Tersian, “The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 41, no. 3-4, pp. 417–431, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. C. P. Gupta, “Existence and uniqueness theorems for the bending of an elastic beam equation,” Applicable Analysis, vol. 26, no. 4, pp. 289–304, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. T. Jankowski, “Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 5, pp. 1289–1299, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. D. Jiang, W. Gao, and A. Wan, “A monotone method for constructing extremal solutions to fourth-order periodic boundary value problems,” Applied Mathematics and Computation, vol. 132, no. 2-3, pp. 411–421, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. P. Kang, Z. Wei, and J. Xu, “Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 245–256, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Y. Li, “Positive solutions of fourth-order boundary value problems with two parameters,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 477–484, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. X.-L. Liu and W.-T. Li, “Existence and multiplicity of solutions for fourth-order boundary value problems with three parameters,” Mathematical and Computer Modelling, vol. 46, no. 3-4, pp. 525–534, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Y. Liu, “Multiple positive solutions of nonlinear singular boundary value problem for fourth-order equations,” Applied Mathematics Letters, vol. 17, no. 7, pp. 747–757, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Lu and S. Jin, “Existence of periodic solutions for a fourth-order P-Laplacian equation with a deviating argument,” Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 513–520, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  19. T. F. Ma, “Positive solutions for a beam equation on a nonlinear elastic foundation,” Mathematical and Computer Modelling, vol. 39, no. 11-12, pp. 1195–1201, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. R. Ma and J. Xu, “Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 1, pp. 113–122, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. R. Ma and T. Chen, “Existence of positive solutions of fourth-order problems with integral boundary conditions,” Boundary Value Problems, vol. 2011, Article ID 297578, 17 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. F. Minhós, T. Gyulov, and A. I. Santos, “Lower and upper solutions for a fully nonlinear beam equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 1-2, pp. 281–292, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. P. K. Palamides, “Uniqueness of monotone positive solutions for singular boundary value problems,” Communications on Applied Nonlinear Analysis, vol. 9, no. 3, pp. 79–89, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. P. Pietramala, “A note on a beam equation with nonlinear boundary conditions,” Boundary Value Problems, vol. 2011, Article ID 376782, 14 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. Pei and S. K. Chang, “Monotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1260–1267, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. J. R. L. Webb, G. Infante, and D. Franco, “Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions,” Proceedings of the Royal Society of Edinburgh A, vol. 138, no. 2, pp. 427–446, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. Z. Wei, “A class of fourth order singular boundary value problems,” Applied Mathematics and Computation, vol. 153, no. 3, pp. 865–884, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. Q. Yao, “Positive solutions for eigenvalue problems of fourth-order elastic beam equations,” Applied Mathematics Letters, vol. 17, no. 2, pp. 237–243, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. X. Zhang and W. Ge, “Positive solutions for a class of boundary-value problems with integral boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 2, pp. 203–215, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. A. Boucherif, S. M. Bouguima, N. Al-Malki, and Z. Benbouziane, “Third order differential equations with integral boundary conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 12, pp. e1736–e1743, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. D. O'Regan, “Fixed-point theory for the sum of two operators,” Applied Mathematics Letters, vol. 9, no. 1, pp. 1–8, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet