Open Access

Existence results for φ-Laplacian boundary value problems on time scales

Advances in Difference Equations20062006:021819

https://doi.org/10.1155/ADE/2006/21819

Received: 24 January 2006

Accepted: 1 June 2006

Published: 4 September 2006

Abstract

This paper is devoted to proving the existence of the extremal solutions of a φ-Laplacian dynamic equation coupled with nonlinear boundary functional conditions that include as a particular case the Dirichlet and multipoint ones. We assume the existence of a pair of well-ordered lower and upper solutions.

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Authors’ Affiliations

(1)
Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela

References

  1. Atici FM, Cabada A, Chyan CJ, Kaymakçalan B: Nagumo type existence results for second-order nonlinear dynamic BVPs. Nonlinear Analysis 2005,60(2):209–220.MathSciNetView ArticleMATHGoogle Scholar
  2. Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser, Massachusetts; 2001:x+358.View ArticleMATHGoogle Scholar
  3. Brezis H: Analyse fonctionnelle. Théorie et applications, Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris; 1983:xiv+234.Google Scholar
  4. Cabada A: Extremal solutions for the difference φ -Laplacian problem with nonlinear functional boundary conditions. Computers & Mathematics with Applications 2001,42(3–5):593–601.MathSciNetView ArticleMATHGoogle Scholar
  5. Cabada A: Extremal solutions and Green's functions of higher order periodic boundary value problems in time scales. Journal of Mathematical Analysis and Applications 2004,290(1):35–54. 10.1016/j.jmaa.2003.08.018MathSciNetView ArticleMATHGoogle Scholar
  6. Cabada A, Habets P, Pouso RL: Optimal existence conditions for φ -Laplacian equations with upper and lower solutions in the reversed order. Journal of Differential Equations 2000,166(2):385–401. 10.1006/jdeq.2000.3803MathSciNetView ArticleMATHGoogle Scholar
  7. Cabada A, Otero-Espinar V: Optimal existence results for n th order periodic boundary value difference equations. Journal of Mathematical Analysis and Applications 2000,247(1):67–86. 10.1006/jmaa.2000.6824MathSciNetView ArticleMATHGoogle Scholar
  8. Cabada A, Otero-Espinar V: Existence and comparison results for difference φ -Laplacian boundary value problems with lower and upper solutions in reverse order. Journal of Mathematical Analysis and Applications 2002,267(2):501–521. 10.1006/jmaa.2001.7783MathSciNetView ArticleMATHGoogle Scholar
  9. Cabada A, Otero-Espinar V, Pouso RL: Existence and approximation of solutions for first-order discontinuous difference equations with nonlinear global conditions in the presence of lower and upper solutions. Computers & Mathematics with Applications 2000,39(1–2):21–33. 10.1016/S0898-1221(99)00310-7MathSciNetView ArticleMATHGoogle Scholar
  10. Cabada A, Pouso RL: Extremal solutions of strongly nonlinear discontinuous second-order equations with nonlinear functional boundary conditions. Nonlinear Analysis. Theory, Methods & Applications 2000,42(8):1377–1396. 10.1016/S0362-546X(99)00158-3MathSciNetView ArticleMATHGoogle Scholar
  11. Cabada A, Vivero DR: Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations. Advances in Difference Equations 2004,2004(4):291–310. 10.1155/S1687183904310022MathSciNetView ArticleMATHGoogle Scholar
  12. Dang H, Oppenheimer SF: Existence and uniqueness results for some nonlinear boundary value problems. Journal of Mathematical Analysis and Applications 1996,198(1):35–48. 10.1006/jmaa.1996.0066MathSciNetView ArticleMATHGoogle Scholar
  13. Ladde GS, Lakshmikantham V, Vatsala AS: Monotone Iterative Techniques for Nonlinear Differential Equations, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics. Volume 27. Pitman, Massachusetts; 1985:x+236.Google Scholar
  14. Picard E: Sur l'application des méthodes d'approximations successives a l'étude de certaines équations différentielles ordinaires. Journal de Mathématiques Pures et Appliquées 1893, 9: 217–271.MATHGoogle Scholar
  15. Zhuang W, Chen Y, Cheng SS: Monotone methods for a discrete boundary problem. Computers & Mathematics with Applications 1996,32(12):41–49. 10.1016/S0898-1221(96)00206-4MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Alberto Cabada. 2006

This article is published under license to BioMed Central Ltd. 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.