Positive Solutions of a Fourth-Order Differential Equation with Multipoint Boundary Conditions

Abstract

This paper is devoted to the study of the following fourth-order multipoint boundary value problem:

$$\left\{\begin{array}{ll} x^{(4)}(t) = \lambda f(t,x(t),x^{\prime }(t) ,x^{\prime \prime}(t)),\quad 0 < t < 1,& \\ x^{(2k+1)}(0)=0,x^{(2k)}(1)=\sum_{i=1}^{m-2}\alpha_{ki}x^{(2k)}(\eta_{ki}),&(k = 0, 1). \end{array}\right. $$

We obtain some sufficient conditions for the existence of at least one or triple positive solutions by using the fixed point theory in cone.

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References

  1. 1.

    Cabada, A., Cid, J.A., Infante, G.: New criteria for the existence of non-trivial fixed points in cones. Fixed Point Theory Appl. 2013, 125 (2013)

    Article  MathSciNet  Google Scholar 

  2. 2.

    Cabada, A., Pouso, R.L., Minhos, F.M.: Extremal solutions to fourth-order functional boundary value problems including multipoint conditions. Nonlinear Anal.: Real World Appl. 10, 2157–2170 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Eggesperger, M., Kosmatov, N.: Positive solutions of a fourth-order multi-point boundary value problem. Commun. Math. Anal. 6, 22–30 (2009)

    MathSciNet  Google Scholar 

  4. 4.

    Henderson, J., Kosmatov, N.: The existence and mutiplicity of constant sign solutions to a three-point boundary value problem. Commun. Appl. Nonlinear Anal. 14(3), 63–78 (2007)

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Il’in, V.A., Moiseev, E.I.: Nonlocal boundary-value problem of the first kind for a Sturm-Liouville operator. Differ. Equations 23, 979–987 (1987). Transl. Differ. Uravn. 23, 1422–1431 (1987)

    MathSciNet  Google Scholar 

  6. 6.

    Kelevedjiev, P.S., Palamides, P.K., Popivanov, N.I.: Another understanding of fourth-order four-point boundary-value problems. Electron. J. Differ. Eqns. 2008(47), 1–15 (2008)

    Google Scholar 

  7. 7.

    Krasnosel’skiı̆, M.A.: Positive Solution of Operator Equations. Noordhoff, Groningen (1964)

    Google Scholar 

  8. 8.

    Liu, X.-J., Jiang, W.-H., Guo, Y.-P.: Multi-point boundary value problems for higher order differential equations. Appl. Math. E-notes 4, 106–113 (2004)

    MATH  MathSciNet  Google Scholar 

  9. 9.

    Leggett, R.W., Williams, L.R.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28, 673–688 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Ma, R.: Existence results of a m-point boundary value problem at resonance. J. Math. Anal. Appl. 294, 147–157 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Pietramala, P.: A note on a beam equation with nonlinear boundary conditions. Bound. Value Probl. 2011(376782), 14 (2011)

    MathSciNet  Google Scholar 

  12. 12.

    Sun, Y.: Positive solutions of nonlinear second-order m-point boundary value problem. Nonlinear Anal. 61, 1283–1294 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Truong, L.X., Ngoc, L.T.P., Long, N.T.: Positive solutions for an m-point boundary value problem. Electron. J. Differ. Eqns. 2008(111), 1–11 (2008)

    Google Scholar 

  14. 14.

    Truong, L.X., Phung, P.D.: Existence of positive solutions for a multi-point four-order boundary-value problem. Electron. J. Differ. Eqns. 2011(119), 1–10 (2011)

    Google Scholar 

  15. 15.

    Webb, J.R.L., Infante, G.: Nonlocal boundary value problems of arbitrary order. J. Lond. Math. Soc. 79, 238–258 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to express their sincere thanks to the referees for their valuable suggestions and remarks leading to improvement of the original manuscript.

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Correspondence to Phan Dinh Phung.

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Phung, P.D. Positive Solutions of a Fourth-Order Differential Equation with Multipoint Boundary Conditions. Vietnam J. Math. 43, 93–104 (2015). https://doi.org/10.1007/s10013-014-0072-4

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Keywords

  • Boundary value problem
  • Positive solution
  • Leggett–Williams fixed point theorem
  • Guo–Krasnosel’skii fixed point theorem

Mathematics Subject Classification (2010)

  • 34B07
  • 34B10
  • 34B18
  • 34B27