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Multiple Solutions for Systems of Sturm–Liouville Boundary Value Problems

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Abstract

In this paper, the authors discuss the existence of multiple solutions to a class of second-order Sturm–Liouville boundary value systems. Their proofs are based on variational methods and critical point theory.

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References

  1. Averna D., Bonanno G.: A mountain pass theorem for a suitable class of functions. Rocky Mt. J. Math. 39, 707–727 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Averna D., Bonanno G.: A three critical points theorem and its applications to the ordinary Dirichlet problem. Topol. Methods Nonlinear Anal. 22, 93–103 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Averna D., Buccellato S.M., Tornatore E.: On a mixed boundary value problem involving the p-Laplacian. Matematiche (Catania) 66, 93–104 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Benmezai A., Graef J.R., Kong L.: Positive solutions to a two point singular boundary value problem. Differ. Equ. Appl. 3, 347–373 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Bonanno G., Candito P.: Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities. J. Differ. Equ. 244, 3031–3059 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonanno G., Chinnì A.: Existence of three solutions for a perturbed two-point boundary value problem. Appl. Math. Lett. 23, 807–811 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonanno G., D’Aguì G.: A Neumann boundary value problem for the Sturm–Liouville equation. Appl. Math. Comput. 208, 318–327 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Bonanno, G., D’Aguì, G.: Multiplicity results for a perturbed elliptic Neumann problem. Abstr. Appl. Anal. 2010, 10 (2010). doi:10.1155/2010/564363

  9. Bonanno G., Marano S.A.: On the structure of the critical set of non-differentiable functions with a weak compactness condition. Appl. Anal. 89, 1–10 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brézis, H.: Analyse Functionelle—Théorie et Applications. Masson, Paris (1983)

  11. D’Agu, G.: Multiplicity of solutions for Sturm–Liouville problems. Boll. Unione Mat. Ital. 6(9), 725–734 (2013)

  12. D’Agu, G.: Multiplicity results for nonlinear mixed boundary value problem. Bound. Value Probl. 2012, 134 (2012)

  13. D’Agu, G.: Infinitely many solutions for a double Sturm–Liouville problem. J. Glob. Optim. 54, 619–625 (2012)

  14. D’Agu, G.: Existence results for a mixed boundary value problem with Sturm–Liouville equation. Adv. Pure Appl. Math. 2, 237–248 (2011)

  15. Davis J.M., Erbe L.H., Henderson J.: Multiplicity of positive solutions for higher order Sturm–Liouville problems. Rocky Mt. J. Math. 31, 169–184 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. García-Melián J., Sabinade Lis J.: Maximum and comparison principles for operators involving the p-Laplacian. J. Math. Anal. Appl. 218, 49–65 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Graef J.R., Heidarkhani S., Kong L.: A critical points approach for the existence of multiple solutions of a Dirichlet quasilinear system. J. Math. Anal. Appl. 388, 1268–1278 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Graef, J.R., Heidarkhani, S., Kong, L.: Infinitely many solutions for systems of Sturm–Liouville boundary value problems. Results Math. 66, 327–341 (2014)

  19. Graef J.R., Heidarkhani S., Kong L.: Nontrivial solutions for systems of Sturm–Liouville boundary value problems. Differ. Equ. Appl. 6, 255–265 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Hai D.D.: On singular Sturm–Liouville boundary-value problems. Proc. R. Soc. Edinb. Sect. A 140, 49–63 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Heidarkhani, S., Henderson, J.: Critical point approaches to quasilinear second order differential equations depending on a parameter. Topol. Methods Nonlinear Anal. 44, 177–198 (2014)

  22. Henderson J., Ntouyas S.K.: Positive solutions for systems of nonlinear boundary value problems. Nonlinear Stud. 15, 51–60 (2008)

    MathSciNet  MATH  Google Scholar 

  23. Sun J., Li W.: Multiple positive solutions to second-order Neumann boundary value problems. Appl. Math. Comput. 146, 187–194 (2003)

    MathSciNet  MATH  Google Scholar 

  24. Tian Y., Ge W.: Multiple solutions for a second-order Sturm–Liouville boundary value problem. Taiwan. J. Math. 11, 975–988 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Tian Y., Ge W.: Second-order Sturm–Liouville boundary value problem involving the one-dimensional p-Laplacian. Rocky Mt. J. Math. 38, 309–327 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tian, Y., Ge, W., O’Regan, D.: Sign-changing and multiple solutions of impulsive boundary value problems via critical point methods. J. Dyn. Control Syst. doi:10.1007/s10883-014-9243-6. (to appear)

  27. Zhao Z.: Property of positive solutions and its applications for Sturm–Liouville singular boundary value problems. Nonlinear Anal. 69, 4514–4520 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN, 37403, USA

    John R. Graef & Lingju Kong

  2. Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah, 67149, Iran

    Shapour Heidarkhani

Authors
  1. John R. Graef
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  2. Shapour Heidarkhani
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  3. Lingju Kong
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Correspondence to John R. Graef.

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Graef, J.R., Heidarkhani, S. & Kong, L. Multiple Solutions for Systems of Sturm–Liouville Boundary Value Problems. Mediterr. J. Math. 13, 1625–1640 (2016). https://doi.org/10.1007/s00009-015-0595-2

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  • DOI: https://doi.org/10.1007/s00009-015-0595-2

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