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Positive Solutions of One-Dimensional p-Laplacian Boundary Value Problems for Fourth-Order Differential Equations with Deviating Arguments

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Abstract

This paper considers the existence of positive solutions of four-point boundary value problems for fourth-order ordinary differential equations with deviating arguments and p-Laplacian. We discuss such problems in the cases when the deviating arguments are delayed or advanced, what may concern optimization issues related to some technical problems. To obtain the existence results, a fixed point theorem for cones due to Avery and Peterson is applied. According to the Author’s knowledge, the results are new. It is a first paper where a fixed point theorem for cones is applied to fourth-order differential equations with deviating arguments and p-Laplacian. An example is included to verify the theoretical results.

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References

  1. Bai, D., Xu, Y.: Existence of positive solutions for boundary value problem of second-order delay differential equations. Appl. Math. Lett. 18, 621–630 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Du, B., Hu, X., Ge, W.: Positive solutions to a type of multi-point boundary value problem with delay and one-dimensional p-Laplacian. Appl. Math. Comput. 208, 501–510 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Jankowski, T.: Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments. Appl. Math. Comput. 197, 179–189 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jankowski, T.: Positive solutions to second order four-point boundary value problems for impulsive differential equations. Appl. Math. Comput. 202, 550–561 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jankowski, T.: Existence of positive solutions to second order four-point impulsive differential problems with deviating arguments. Comput. Math. Appl. 58, 805–817 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jiang, D.: Multiple positive solutions for boundary value problems of second-order delay differential equations. Appl. Math. Lett. 15, 575–583 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Khan, R.A., Webb, J.R.L.: Existence of at least three solutions of a second-order three-point boundary value problem. Nonlinear Anal. 64, 1356–1366 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sun, B., Qu, Y., Ge, W.: Existence and iteration of positive solutions for a multipoint one-dimensional p-Laplacian boundary value problem. Appl. Math. Comput. 197, 389–398 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang, W., Sheng, J.: Positive solutions to a multi-point boundary value problem with delay. Appl. Math. Comput. 188, 96–102 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, Y., Zhao, W., Ge, W.: Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional p-Laplacian. J. Math. Anal. Appl. 326, 641–654 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yang, C., Zhai, C., Yan, J.: Positive solutions of the three-point boundary value problem for second order differential equations with an advanced argument. Nonlinear Anal. 65, 2013–2023 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Graef, J., Kong, L.: Necessary and sufficient conditions for the existence of symmetric positive solutions of singular boundary value problems. J. Math. Anal. Appl. 331, 1467–1484 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Graef, J.R., Qian, C., Yang, B.: A three point boundary value problem for nonlinear fourth order differential equations. J. Math. Anal. Appl. 287, 217–233 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bai, Z., Wang, H.: On the positive solutions of some nonlinear fourth-order beam equations. J. Math. Anal. Appl. 270, 357–368 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wei, Z., Pang, C.: The method of lower and upper solutions for fourth order singular m-point boundary value problems. J. Math. Anal. Appl. 322, 675–692 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang, X., Liu, L.: Positive solutions of fourth-order four-point boundary value problems with p–Laplacian operator. J. Math. Anal. Appl. 336, 1414–1423 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, X., Liu, L.: A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with p-Laplacian. Nonlinear Anal. 68, 3127–3137 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhao, J., Wang, L., Ge, W.: Necessary and sufficient conditions for the existence of positive solutions of fourth order multi-point boundary value problems. Nonlinear Anal. 72, 822–835 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang, X., Feng, M., Ge, W.: Symmetric positive solutions for p-Laplacian fourth-order differential equations with integral boundary conditions. J. Comput. Appl. Math. 222, 561–573 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Differential Equations and Applied Mathematics, Gdansk University of Technology, 11/12 G.Narutowicz Str, 80–233, Gdansk, Poland

    T. Jankowski

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  1. T. Jankowski
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Correspondence to T. Jankowski.

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Communicated by F.E. Udwadia.

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Jankowski, T. Positive Solutions of One-Dimensional p-Laplacian Boundary Value Problems for Fourth-Order Differential Equations with Deviating Arguments. J Optim Theory Appl 149, 47–60 (2011). https://doi.org/10.1007/s10957-010-9774-2

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  • DOI: https://doi.org/10.1007/s10957-010-9774-2

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