About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 951251, 12 pages
http://dx.doi.org/10.1155/2012/951251
Research Article

Necessary and Sufficient Condition for the Existence of Solutions to a Discrete Second-Order Boundary Value Problem

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 20 December 2011; Accepted 22 February 2012

Academic Editor: Yuriy Rogovchenko

Copyright © 2012 Chenghua Gao. 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. D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, Oxford Applied Mathematics and Computing Science Series, Clarendon Press, Oxford, UK, 1977. View at Zentralblatt MATH
  2. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1979. View at Zentralblatt MATH
  3. B. J. Lazan, Damping of Meterials in Structural Mechanics, Pergamon Press, Elmsford, NY, USA, 1968.
  4. N. Minorsky, Nonlinear Oscillations, Van Nostrand, Princeton, NJ, USA, 1962. View at Zentralblatt MATH
  5. E. M. Landesman and A. C. Lazer, “Nonlinear perturbations of linear elliptic boundary value problems at resonance,” Journal of Mathematics and Mechanics, vol. 19, pp. 609–623, 1969/1970.
  6. R. Iannacci and M. N. Nkashama, “Unbounded perturbations of forced second order ordinary differential equations at resonance,” Journal of Differential Equations, vol. 69, no. 3, pp. 289–309, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. R. Kannan, R. K. Nagle, and K. L. Pothoven, “Remarks on the existence of solutions of x+x+arctan(x)=p(t); x(0)=x(π)=0,” Nonlinear Analysis: Theory, Methods & Applications, vol. 22, no. 6, pp. 793–796, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. A. Cañada and P. Drábek, “On semilinear problems with nonlinearities depending only on derivatives,” SIAM Journal on Mathematical Analysis, vol. 27, no. 2, pp. 543–557, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. P. Habets and L. Sanchez, “A two-point problem with nonlinearity depending only on the derivative,” SIAM Journal on Mathematical Analysis, vol. 28, no. 5, pp. 1205–1211, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. P. Drábek, P. Girg, and F. Roca, “Remarks on the range properties of certain semilinear problems of Landesman-Lazer type,” Journal of Mathematical Analysis and Applications, vol. 257, no. 1, pp. 131–140, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. N. Del Toro and F. Roca, “Existence and multiplicity of solutions for certain Dirichlet problems with nonlinearity depending on the derivative,” Nonlinear Analysis: Theory, Methods & Applications, vol. 55, no. 7-8, pp. 827–843, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. J. Rodriguez, “Nonlinear discrete Sturm-Liouville problems,” Journal of Mathematical Analysis and Applications, vol. 308, no. 1, pp. 380–391, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. R. Ma, “Nonlinear discrete Sturm-Liouville problems at resonance,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 11, pp. 3050–3057, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. R. Ma and H. Ma, “Unbounded perturbations of nonlinear discrete periodic problem at resonance,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp. 2602–2613, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. R. Ma and H. Ma, “Existence of sign-changing periodic solutions of second order difference equations,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 463–470, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. P. J. Y. Wong and R. P. Agarwal, “Fixed-sign solutions of a system of higher order difference equations,” Journal of Computational and Applied Mathematics, vol. 113, no. 1-2, pp. 167–181, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. R. P. Agarwal and D. O'Regan, “Nonpositone discrete boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 39, no. 2, pp. 207–215, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. D. Bai and Y. Xu, “Nontrivial solutions of boundary value problems of second-order difference equations,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 297–302, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. J. Henderson, “Positive solutions for nonlinear difference equations,” Nonlinear Studies, vol. 4, no. 1, pp. 29–36, 1997. View at Zentralblatt MATH
  20. J. Henderson and H. B. Thompson, “Existence of multiple solutions for second-order discrete boundary value problems,” Computers & Mathematics with Applications, vol. 43, no. 10-11, pp. 1239–1248, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. R. Ma and Y. N. Raffoul, “Positive solutions of three-point nonlinear discrete second order boundary value problem,” Journal of Difference Equations and Applications, vol. 10, no. 2, pp. 129–138, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. J. Mawhin, “Topological degree and boundary value problems for nonlinear differential equations,” in Topological Methods for Ordinary Differential Equations, P. M. Fitzpatric, M. Matelli, J. Mawhin, and R. Nussbaum, Eds., vol. 1537 of Lecture Notes in Math., pp. 74–142, Springer, Berlin, Germany, 1993. View at Zentralblatt MATH