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

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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(uσ(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.

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References

  1. 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.

  2. 2.

    Anderson DR: Eigenvalue intervals for a second-order Sturm-Liouville dynamic equations. to appear in International Journal of Nonlinear Differential Equations

  3. 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.

  4. 4.

    Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Massachusetts; 2001:x+358.

  5. 5.

    Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2003:xii+348.

  6. 6.

    Erbe LH, Peterson AC: Eigenvalue conditions and positive solutions. Journal of Difference Equations and Applications 2000,6(2):165–191. 10.1080/10236190008808220

  7. 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)00154-0

  8. 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.

  9. 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. 10.1080/10236190008808238

  10. 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.

  11. 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, Munich

  12. 12.

    Hilger S: Ein Masskettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, M.S. thesis. Universität Würzburg, Würzburg; 1988.

  13. 13.

    Krasnosel'eskiĭ MA, Zabreĭko PP: Geometrical Methods of Nonlinear Analysis, Fundamental Principles of Mathematical Sciences. Volume 263. Springer, Berlin; 1984:xix+409.

  14. 14.

    Lakshmikantham V, Sivasundaram S, Kaymakçalan B: Dynamic Systems on Measure Chains, Mathematics and Its Applications. Volume 370. Kluwer Academic, Dordrecht; 1996:x+285.

  15. 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.1979.28.28046

  16. 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.2003.04.008

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Correspondence to Basant Karna.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Karna, B., Lawrence, B.A. An existence result for a multipoint boundary value problem on a time scale. Adv Differ Equ 2006, 063208 (2006) doi:10.1155/ADE/2006/63208

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Keywords

  • Boundary Condition
  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis