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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 594931, 9 pages
http://dx.doi.org/10.1155/2014/594931
Review Article

On Types of Solutions of the Second Order Nonlinear Boundary Value Problems

1Institute of Mathematics and Computer Science, Rainis Boulevard 29, Riga LV-1459, Latvia
2Daugavpils University, Parades Street 1, Daugavpils LV-5400, Latvia

Received 28 April 2014; Accepted 20 June 2014; Published 12 August 2014

Academic Editor: Tongxing Li

Copyright © 2014 Maria Dobkevich 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. S. R. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York, NY, USA, 1974. View at MathSciNet
  2. Yu. A. Klokov and N. I. Vasilyev, Foundations of the Theory of Nonlinear Boundary Value Problems, Zinatne, Riga, Latvia, 1978 (Russian).
  3. I. T. Kiguradze, Singular Boundary Value Problems for Ordinary Differential Equations, Tbilissi St. Univ. Press, Tbilissi, Georgia, 1975, (Russian). View at MathSciNet
  4. R. P. Agarwal and D. O'Regan, An Introduction to Ordinary Differential Equations, Universitext, Springer, New York, NY, USA, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  5. W. G. Kelley and A. C. Peterson, The Theory of Differential Equations: Classical and Qualitative, Pearson Education, New Jersey, NJ, USA, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  6. P. Amster, Topological Methods in the St udy of Boundary Value Problems, Springer, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  7. L. H. Erbe, “A uniqueness theorem for second order differential equations,” Mathematische Zeitschrift, vol. 109, no. 2, pp. 92–96, 1969. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. C. de Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, Elsevier, Amsterdam, The Netherlands, 2006. View at MathSciNet
  9. L. A. Lepin, “On the concepts of lower and upper functions,” Differential Equations, vol. 16, no. 10, pp. 1750–1759, 1980 (Russian). View at MathSciNet
  10. L. A. Lepin, “Generalizaded solutions and the solvability of boundary value problems for a second order differential equation,” Differential Equations, vol. 18, no. 8, pp. 925–931, 1982 (Russian).
  11. H. Epheser, “Über die existenz der lösungen von randwertaufgaben mit gewöhnlichen, nichtlinearen differentialgleichungen zweiter ordnung,” Mathematische Zeitschrift, vol. 61, pp. 435–454, 1955. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. F. Sadyrbaev, “Ljapunov functions and the solvability of the first boundary value problem for an ordinary second-order differential equation,” Differentsial'nye Uravneniya, vol. 16, no. 4, pp. 629–634, 1980 (Russian). View at MathSciNet
  13. H. W. Knobloch, “Comparison theorems for nonlinear second order differential equations,” Journal of Differential Equations, vol. 1, pp. 1–26, 1965. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. H. Knobloch, “Second order differential equalities and a nonlinear boundary value problem,” Journal of Differential Equations, vol. 5, pp. 55–71, 1969. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. L. K. Jackson and K. W. Schrader, “Comparison theorems for nonlinear differential equations,” Journal of Differential Equations, vol. 3, pp. 248–255, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. L. H. Erbe, “Nonlinear boundary value problems for second order differential equations,” Journal of Differential Equations, vol. 7, pp. 459–472, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  17. M. Dobkevich, “Non-monotone convergence schemes,” Mathematical Modelling and Analysis, vol. 17, no. 4, pp. 589–597, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. M. Dobkevich and F. Sadyrbaev, “Approximation schemes and types of solutions for the Neumann BVP,” Proceedings of IMCS of University of Latvia, vol. 13, pp. 12–23, 2013.
  19. R. Conti, “Equazioni differenziali ordinarie quasilineari con condizioni lineari,” Annali di Matematica Pura ed Applicata, vol. 57, pp. 49–61, 1962. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. I. Yermachenko and F. Sadyrbaev, “Types of solutions and multiplicity results for two-point nonlinear boundary value problems,” Nonlinear Analysis, vol. 63, pp. e1725–e1735, 2005.
  21. I. Yermachenko and F. Sadyrbaev, “Quasilinearization and multiple solutions of the Emden-Fowler type equation,” Mathematical Modelling and Analysis, vol. 10, no. 1, pp. 41–50, 2005. View at MathSciNet · View at Scopus
  22. I. Yermachenko and F. Sadyrbaev, “Types of solutions of the second order Neumann problem: multiple solutions,” Proceedings of IMCS of University of Latvia, vol. 4, pp. 5–21, 2004, http://www.lumii.lv/Pages/sbornik1/neumann.pdf.
  23. I. Yermachenko and F. Sadyrbaev, “Types of solutions and multiplicity results for second order nonlinear boundary value problems. Discrete and continuous dynamical systems,” in Proceedings of AIMS Conference, vol. 2007, pp. 1061–1069, Poitier, France, 2006.
  24. I. Yermachenko and F. Sadyrbaev, “Quasilinearization technique for Φ-Laplacian type equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 975760, 11 pages, 2012. View at MathSciNet
  25. I. Yermachenko, “Two-point boundary value problems at resonance,” Mathematical Modelling and Analysis, vol. 14, no. 2, pp. 247–257, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. N. Sveikate and F. Sadyrbaev, “Quasilinearization for resonant boundary value problems with mixed boundary conditions,” Nonlinear Oscillations, vol. 17, no. 1, pp. 112–126, 2014.
  27. N. Sveikate, “Resonant problems by quasilinearization,” International Journal of Differential Equations, vol. 2014, Article ID 564914, 8 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet