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Lower and upper functions in a singular Dirichlet problem with ø-Laplacian

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Abstract

The paper investigates the Dirichlet problem with ø-Laplacian of the form

$(\varphi (u'))' + f(t,u,u') = 0,u(0) = u(T) = 0.$

An existence principle which can be used for problems where f(t, x, y) may have singularities at t = 0, t = T and also at x = 0, y = 0, is proved here. As an application of this principle, new conditions that guarantee the solvability of the above problem are found.

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References

  1. R. P. Agarwal, H. Lü, and D. O’Regan, “An upper and lower solution method for one-dimensional singular p-Laplacian,” Memoirs on Differential Equations and Math. Phys. 28, 13–31 (2003).

    MATH  MathSciNet  Google Scholar 

  2. R. P. Agarwal and D. O’Regan, “Singular boundary value problems for superlinear second order ordinary and delay differential equations,” J. Differential Equations 130, 333–355 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  3. R. P. Agarwal and D. O’Regan, “Nonlinear superlinear singular and nonsingular second order boundary value problems,” J. Differential Equations 143, 60–95 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  4. R. P. Agarwal and D. O’Regan, “A Survey of Recent Results for Initial and Boundary Value Problems Singular in the Depend Variable,” in Handbook of Differential Equations, Ordinary Differential Equations, Vol. 1, Ed. by A. Caňada, P. Drá bek, and A. Fonda (Elsevier, North Holland, Amsterdam 2004), pp. 1–68.

    Article  Google Scholar 

  5. R. P. Agarwal, D. O’Regan, and V. Lakshmikantham, “Existence of positive solutions for singular initial and boundary value problems via the classical upper and lower solution approach,” Nonlinear Anal., Theory Methods Appl. 50, 215–222 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  6. R. P. Agarwal, K. Perera, and D. O’Regan, “Multiple positive solutions of singular problems by variational methods,” Proc. Amer. Math. Soc. 134 (3), 817–824 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  7. J. V. Baxley, “Some singular nonlinear boundary value problems,” SIAM J. Math. Anal. 22, 463–479 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Chu and D. O’Regan, “Multiplicity results for second order non-autonomous singular Dirichlet systems,” Acta Appl. Math. 105 (3), 323–338 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  9. P. Habets and F. Zanolin, “Upper and lower solutions for a generalized Emded-Fowler equation,” J. Math. Anal. Appl. 181, 684–700 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Q. Jiang, “Upper and lower solutions method and a singular superlinear boundary value problem for the one-dimensional p-Laplacian,” Comp.Math. Appl. 42, 927–940 (2001).

    Article  MATH  Google Scholar 

  11. D. Q. Jiang, “Upper and lower solutionsmethod and a superlinear singular boundary value problems,” Comp. Math. Appl. 44, 323–337 (2002).

    Article  MATH  Google Scholar 

  12. I. T. Kiguradze, On Some Singular Boundary-Value Problems for Ordinary Differential Equations (Tbilisi Univ. Press, Tbilisi 1975) [in Russian].

    Google Scholar 

  13. I. T. Kiguradze and B. L. Shekhter, “Singular boundary value problems for second order ordinary differential equations,” Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Noveishie Dostizh. 30, 105–201 (1987) [in Russian].

    MathSciNet  Google Scholar 

  14. A. Lomtatitdze, “Positive solutions of boundary value problems for second order differential equations with singular points,” Differentsial’nye Uravneniya 23, 1685–1692 (1987) [in Russian].

    Google Scholar 

  15. A. Lomtatidze and P. Torres, “On a two-point boundary value problem for second order singular equations,” Czechoslovak Math. J. 53, 19–43 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  16. D. O’Regan, Theory of Singular Boundary Value Problems (World Scientific, Singapore, 1994).

    Book  MATH  Google Scholar 

  17. I. Rachůnková and S. Staněk, “Sign-changing solutions of singular Dirichlet boundary value problems,” Archives of Inequal. Appl. 1, 11–30 (2003).

    MATH  Google Scholar 

  18. I. Rachůnková and S. Staněk, “Connections between types of singularities in differential equations and smoothness of solutions of Dirichlet BVPs,” Dyn. Contin. Discrete Impulsive Syst. 10, 209–222 (2003).

    MATH  Google Scholar 

  19. I. Rachůnková and J. Stryja, “Singular Dirichlet BVP for second order ODE,” Georgian Math. J. 14, 325–340 (2007).

    MATH  MathSciNet  Google Scholar 

  20. I. Rachůnková and J. Stryja, “Dirichlet problem with Ø-Laplacian and mixed singularities,” Nonlinear Oscillations 11, 81–95 (2008).

    MathSciNet  Google Scholar 

  21. S. Staněk, “Positive solutions of singular positone Dirichlet boundary value problems,” Math. Comp. Modelling 33, 341–351 (2001).

    Article  MATH  Google Scholar 

  22. S. Staněk, “Positive solutions of the Dirichlet problem with state-dependent functional differential equations,” Funct. Diff. Equations 11, 563–586 (2004).

    MATH  Google Scholar 

  23. S. Staněk, “Positive solutions of singular Dirichlet boundary value problems with time and space singularities,” Nonlinear Analysis 71, 4893–4905 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  24. J. Sun and J. Chu, “Positive solutions of singular Dirichlet problems via variational methods,” J. Korean Math. Soc. 50 (4), 797–811 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  25. S. D. Taliaferro, “A nonlinear singular boundary value problem,” Nonlinear Anal., Theory Methods Appl. 3, 897–904 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  26. A. Tineo, “Existence theorems for a singular two-point Dirichlet problem,” Nonlinear Anal., Theory Methods Appl. 19, 323–333 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  27. J. Y. Wang and W. Gao, “A singular boundary value problem for the one-dimensional p-Laplacian,” J. Math. Anal. Appl. 201, 851–866 (1996).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Palacký University, Olomouc, Czech Republic

    I. Rachůnková

  2. VŠB - Technical University of Ostrava, Ostrava, Czech Republic

    J. Stryja

Authors
  1. I. Rachůnková
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  2. J. Stryja
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Correspondence to I. Rachůnková.

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The article was submitted by the authors for the English version of the journal.

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Rachůnková, I., Stryja, J. Lower and upper functions in a singular Dirichlet problem with ø-Laplacian. Math Notes 97, 588–597 (2015). https://doi.org/10.1134/S0001434615030293

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