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Positive Solutions of a Three-Point Boundary-Value Problem for the p-Laplacian Dynamic Equation on Time Scales

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Ukrainian Mathematical Journal Aims and scope

We consider a three-point boundary-value problem for a p-Laplacian dynamic equation on time scales. We prove the existence of at least three positive solutions of the boundary-value problem by using the Avery and Peterson fixed-point theorem. The conditions used in this case differ from the conditions used in the major part of available papers. As an interesting point, we can mention the fact that the nonlinear term f involves the first derivative of the unknown function. As an application, an example is given to illustrate our results.

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References

  1. R. P. Agarwal and D. O’Regan, “Triple solutions to boundary value problems on time scales,” Appl. Math. Lett., 44, 527–535 (2001).

    MATH  Google Scholar 

  2. R. P. Agarwal and D. O’Regan, “Nonlinear boundary value problems on time scales,” Appl. Math. Lett., 13, 7–11 (2000).

    MathSciNet  MATH  Google Scholar 

  3. D. Anderson, “Solutions to second-order three-point problems on time scales,” J. Difference Equat. Appl., 8, 673–688 (2002).

    Article  MathSciNet  Google Scholar 

  4. D. R. Anderson, R. Avery, and J. Henderson, “Existence of solutions for a one-dimensional p-Laplacian on time scales,” J. Difference Equat. Appl., 10, 889–896 (2004).

    Article  MathSciNet  Google Scholar 

  5. R. I. Avery, C. J. Chyan, and J. Henderson, “Twin solutions of boundary value problems for ordinary differential equations and finite difference equations,” Comput. Math. Appl., 42, 695–704 (2001).

    Article  MathSciNet  Google Scholar 

  6. R. I. Avery and A. Peterson, “Three positive fixed points of nonlinear operators on ordered Banach spaces,” Comput. Math. Appl., 42, 313–322 (2001).

    Article  MathSciNet  Google Scholar 

  7. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Cambridge, MA (2001).

    Book  Google Scholar 

  8. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, Cambridge, MA (2003).

    Book  Google Scholar 

  9. A. Dogan, “On the existence of positive solutions for the one-dimensional p-Laplacian boundary value problems on time scales,” Dynam. Systems Appl., 24, 295–304 (2015).

    MathSciNet  MATH  Google Scholar 

  10. A. Dogan, “Three positive solutions of a three-point boundary value problem for the p-Laplacian dynamic equation on time scales,” Comm. Optim. Theory, 2018, 1–13 (2018).

    Google Scholar 

  11. M. Guo, “Existence of positive solutions for p-Laplacian three-point boundary value problems on time scales,” Math. Comput. Model., 50, 248–253 (2009).

    Article  MathSciNet  Google Scholar 

  12. Z. He, “Double positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales,” J. Comput. Appl. Math., 182, 304–315 (2005).

    Article  MathSciNet  Google Scholar 

  13. Z. He and X. Jiang, “Triple positive solutions of boundary value problems for p-Laplacian dynamic equations on time scales,” J. Math. Anal. Appl., 321, 911–920 (2006).

    Article  MathSciNet  Google Scholar 

  14. Z. He and L. Li, “Multiple positive solutions for the one-dimensional p-Laplacian dynamic equations on time scales,” Math. Comput. Model., 45, 68–79 (2007).

    Article  MathSciNet  Google Scholar 

  15. S. Hong, “Triple positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales,” J. Comput. Appl. Math., 206, 967–976 (2007).

    Article  MathSciNet  Google Scholar 

  16. H. Luo and Q. Z. Ma, “Positive solutions to a generalized second-order three-point boundary value problem on time scales,” Electron. J. Different. Equat., 17, 1–14 (2005).

    MathSciNet  MATH  Google Scholar 

  17. D. O’Regan, Existence Theory for Nonlinear Ordinary Differential Equations, Kluwer Academic Publishers , Dordrecht (1997).

    Book  Google Scholar 

  18. H. Su, B. Wang, and Z. Wei, “Positive solutions of four-point boundary value problems for four-order p-Laplacian dynamic equations on time scales,” Electron. J. Different. Equat., 78, 1–13 (2006).

    MathSciNet  MATH  Google Scholar 

  19. H. R. Sun and W. T. Li, “Positive solutions for nonlinear three-point boundary value problems on time scales,” J. Math. Anal. Appl., 299, 508–524 (2004).

    Article  MathSciNet  Google Scholar 

  20. H. R. Sun, L. T. Tang, and Y. H. Wang, “Eigenvalue problem for p-Laplacian three-point boundary value problems on time scales,” J. Math. Anal. Appl., 331, 248–262 (2007).

    Article  MathSciNet  Google Scholar 

  21. H. R. Sun and W. T. Li, “Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales,” J. Different. Equat., 240, 217–248 (2007).

    Article  MathSciNet  Google Scholar 

  22. H. R. Sun and Y. H. Wang, “Existence of positive solutions for p-Laplacian three-point boundary value problems on time scales,” Electron. J. Different. Equat., 92, 1–14 (2008).

    MathSciNet  Google Scholar 

  23. D. B. Wang, “Three positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales,” Nonlin. Anal., 68, 2172–2180 (2008).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, A. Gul University, Kayseri, Turkey

    A. Dogan

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  1. A. Dogan
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Correspondence to A. Dogan.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 6, pp. 790–805, June, 2020. Ukrainian DOI: 10.37863/umzh.v72i6.646.

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Dogan, A. Positive Solutions of a Three-Point Boundary-Value Problem for the p-Laplacian Dynamic Equation on Time Scales. Ukr Math J 72, 917–934 (2020). https://doi.org/10.1007/s11253-020-01832-8

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  • DOI: https://doi.org/10.1007/s11253-020-01832-8

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