Open Access

An existence result for a multipoint boundary value problem on a time scale

Advances in Difference Equations20062006:063208

https://doi.org/10.1155/ADE/2006/63208

Received: 31 January 2006

Accepted: 19 April 2006

Published: 12 September 2006

Abstract

We will expand the scope of application of a fixed point theorem due to Krasnosel'skiĭ and Zabreiko to the family of second-order dynamic equations described by uΔΔ(t) = f(u σ (t)), , with multipoint boundary conditions u(0) = 0, , and for the purpose of establishing existence results. We will determine sufficient conditions on our function f such that the assumptions of the fixed point theorem are satisfied, which in return gives us the existence of solutions.

[1234567891011121314151612345678910111213141516]

Authors’ Affiliations

(1)
Department of Mathematics, Marshall University

References

  1. Anderson DR: Eigenvalue intervals for a second-order mixed-conditions problem on time scales. International Journal of Nonlinear Differential Equations 2002,7(1–2):97–104.Google Scholar
  2. Anderson DR: Eigenvalue intervals for a second-order Sturm-Liouville dynamic equations. to appear in International Journal of Nonlinear Differential EquationsGoogle Scholar
  3. Avery RI, Henderson J: Two positive fixed points of nonlinear operators on ordered Banach spaces. Communications on Applied Nonlinear Analysis 2001,8(1):27–36.MathSciNetMATHGoogle Scholar
  4. Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Massachusetts; 2001:x+358.View ArticleMATHGoogle Scholar
  5. Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2003:xii+348.MATHGoogle Scholar
  6. Erbe LH, Peterson AC: Eigenvalue conditions and positive solutions. Journal of Difference Equations and Applications 2000,6(2):165–191. 10.1080/10236190008808220MathSciNetView ArticleMATHGoogle Scholar
  7. Erbe LH, Peterson AC: Positive solutions for a nonlinear differential equation on a measure chain. Mathematical and Computer Modelling 2000,32(5–6):571–585. 10.1016/S0895-7177(00)00154-0MathSciNetView ArticleMATHGoogle Scholar
  8. Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Massachusetts; 1988:viii+275.MATHGoogle Scholar
  9. Henderson J: Multiple solutions for 2 m th order Sturm-Liouville boundary value problems on a measure chain. Journal of Difference Equations and Applications 2000,6(4):417–429. 10.1080/10236190008808238MathSciNetView ArticleMATHGoogle Scholar
  10. Henderson J: Nontrivial solutions to a nonlinear boundary value problem on a time scale. Communications on Applied Nonlinear Analysis 2004,11(1):65–71.MathSciNetMATHGoogle Scholar
  11. Henderson J, Lawrence BA: Existence of solutions for even ordered boundary value problems on a time scale. Proceedings of the International Conference on Difference Equations, Special Functions, and Applications, 2006, MunichGoogle Scholar
  12. Hilger S: Ein Masskettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, M.S. thesis. Universität Würzburg, Würzburg; 1988.MATHGoogle Scholar
  13. Krasnosel'eskiĭ MA, Zabreĭko PP: Geometrical Methods of Nonlinear Analysis, Fundamental Principles of Mathematical Sciences. Volume 263. Springer, Berlin; 1984:xix+409.View ArticleGoogle Scholar
  14. Lakshmikantham V, Sivasundaram S, Kaymakçalan B: Dynamic Systems on Measure Chains, Mathematics and Its Applications. Volume 370. Kluwer Academic, Dordrecht; 1996:x+285.View ArticleMATHGoogle Scholar
  15. Leggett RW, Williams LR: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana University Mathematics Journal 1979,28(4):673–688. 10.1512/iumj.1979.28.28046MathSciNetView ArticleMATHGoogle Scholar
  16. Sun J-P, Li WT: A new existence theorem for right focal boundary value problems on a measure chain. Applied Mathematics Letters 2005,18(1):41–47. 10.1016/j.aml.2003.04.008MathSciNetView ArticleMATHGoogle Scholar

Copyright

© B. Karna and B.A. Lawrence. 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.