Advertisement

Positivity

, Volume 22, Issue 5, pp 1269–1279 | Cite as

Existence of positive solutions for periodic boundary value problem with sign-changing Green’s function

  • D. D. Hai
Article
  • 88 Downloads

Abstract

We prove the existence of positive solutions for the boundary value problem
$$\begin{aligned} \left\{ \begin{array}{ll} y^{\prime \prime }+m^{2}y=\lambda g(t)f(y), &{}\quad 0\le t\le 2\pi , \\ y(0)=y(2\pi ), &{}\quad y^{\prime }(0)=y^{\prime }(2\pi ), \end{array} \right. \end{aligned}$$
for certain range of the parameter \(\lambda >0\), where \(m\in (1/2,1/2+\varepsilon )\) with \(\varepsilon >0\) small, and f is superlinear or sublinear at \(\infty \) with no sign-conditions at 0 assumed.

Keywords

Periodic BVP Sign changing Green’s function Positive solutions 

Mathematics Subject Classification

34B15 34B27 

References

  1. 1.
    Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18(4), 620–709 (1976)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Atici, F.M., Guseinov, G.S.: On the existence of positive solutions for nonlinear differential equations with periodic conditions. J. Comput. Appl. Math. 132, 341–356 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cabada, A., Cid, J.A.: Existence and multiplicity of solutions for a periodic Hill’s equation with parametric dependence and singularities. Abstr. Appl. Anal. 2011, 19 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cabada, A., Cid, J.A., Tvrdý, M.: A generalized anti-maximum principle for the periodic one-dimensional p-Laplacian with sign-changing potential. Nonlinear Anal. 72(7–8), 3436–3446 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cac, N.P., Gatica, J.A., Li, Y.: Positive solutions for semilinear problems with coefficient that changes sign. Nonlinear Anal. 37, 501–510 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hai, D.D.: Positive solutions to a class of elliptic boundary value problems. J. Math. Anal. Appl. 227, 195–199 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hai, D.D.: On a superlinear periodic boundary value problem with vanishing Green’s function. Electron. J. Qual. Theory Differ. Equ. 2016(55), 12 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Graef, J.R., Kong, L., Wang, H.: A periodic boundary value problem with vanishing Green’s functions. Appl. Math. Lett. 21, 176–180 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jiang, D., Chu, J., Zhang, M.: Multiplicity of positive solutions to superlinear repulsive singular equations. J. Differ. Equ. 211, 283–302 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jiang, D., Chu, J., O’Regan, Agarwal, R.: Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces. J. Math. Anal. Appl. 28, 563–576 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Li, H.X., Zhang, Y.W.: A second order periodic boundary value problem with a parameter and vanishing Green’s functions. Publ. Math. Debrecen 85, 273–283 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ma, R.: Nonlinear periodic boundary value problems with sign-changing Green’s function. Nonlinear Anal. 74, 1714–1720 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ma, R., Gao, C., Ruipeng, C.: Existence of positive solutions of nonlinear second-order periodic boundary value problems. Bound. Value. Probl. 2010, 626054 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    O’Regan, D., Wang, H.: Positive periodic solutions of systems of second order ordinary differential equations. Positivity 10, 285–298 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Torres, P.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnosel’skii fixed point theorem. J. Differ. Equ. 190, 643–662 (2003)CrossRefGoogle Scholar
  16. 16.
    Webb, J.R.L.: Boundary value problems with vanishing Green’s function. Commun. Appl. Anal. 13, 587–595 (2009)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Zhang, Z., Wang, J.: On existence and multiplicity of positive solutions to periodic boundary value problems for singular second order differential equations. J. Math. Anal. Appl. 281, 99–107 (2003)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhong, S., An, Y.: Existence of positive solutions to periodic boundary value problems with sign-changing Green’s function. Bound. Value Probl. 2011, 8 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMississippi State UniversityMississippi StateUSA

Personalised recommendations