Advertisement

Nodal Solutions for Asymptotically Linear Second-Order BVPs on the Half-line

Original Paper
  • 1 Downloads

Abstract

In this article, we prove under eigenvalue criteria, existence result for nodal solutions to the boundary value problem posed on the positive half-line:
$$\begin{aligned} \left\{ \begin{array}{l} -u^{\prime \prime }(t)+q(t)u(t)=u(t)f(t,u(t))\quad t>0, \\ u(0)=\lim _{t\rightarrow +\infty }u(t)=0, \end{array} \right. \end{aligned}$$
where \(q\in C\left( \mathbb {\mathbb {R} }^{+},\mathbb {\mathbb {R}}^{+}\right) \) may be unbounded from above and \( f:{\mathbb {R}}^{+}\times \mathbb {R} \rightarrow \mathbb {R} \) is a continuous function.

Keywords

Sturm–Liouville BVPs on infinite intervals Linear eigenvalue problem Global bifurcation theory 

Mathematics Subject Classification

Primary 34B05 Secondary 34B40 

Notes

Acknowledgements

The authors thank the referee for all his comments and suggestions about this paper.

References

  1. 1.
    Agarwal, R.P., Mustafa, O.G., Rogovchenko, Y.V.: Existence and asymptotic behavior of solutions of a boundary value problem on an infinite interval. Math. Comput. Model. 41, 135–157 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Agarwal, R.P., O’Regan, D.: Boundary value problems of nonsingular type on the semi-infinite interval. Tohoku Math. J. 51, 391–397 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Baxley, J.V.: Existence and uniqueness for nonlinear boundary value problems on infinite intervals. J. Math. Anal. Appl. 147, 122–133 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Benmezaï, A., Esserhane, W., Henderson, J.: Nodal solutions for singular second-order boundary-value problems. Electron. J. Differ. Equ. 2014(156), 1–39 (2014)MathSciNetMATHGoogle Scholar
  5. 5.
    Benmezaï, A., Esserhane, W., Henderson, J.: Sturm–Liouville BVPs with Caratheodory nonlinearities. Electron. J. Differ. Equ. 2016(298), 1–49 (2016)MathSciNetMATHGoogle Scholar
  6. 6.
    Benmezaï, A.: Fixed point theorems in cones under local conditions. Fixed Point Theory 18, 107–126 (2017)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Berestycki, H.: On some non-linear Sturm–Liouville boundary value problems. J. Differ. Equ. 26, 375–390 (1977)CrossRefMATHGoogle Scholar
  8. 8.
    Bobisud, L.E.: Existence of positive solutions to some nonlinear singular boundary value problem on finite and infinite intervals. J. Math. Anal. Appl. 173, 69–83 (1993)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Constantin, A.: On an infinite interval boundary value problem. Ann. Mat. Pura ed Appl. IV, 379–394 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Djebali, S., Mebarki, M.: Existence results for a class of BVPs on the positive half-line. Comm. Appl. Nonlinear Anal. 14, 13–31 (2007)MathSciNetMATHGoogle Scholar
  11. 11.
    Djebali, S., Mebarki, M.: Multiple positive solutions for singular BVPs on the positive half-line. Comput. Math. Appl. 55, 2940–2952 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Eloe, P.W., Kaufman, E.R., Tisdell, C.C.: Multiple solutions of a boundary value problem on an unbounded domain. Dyn. Systems Appl. 15, 53–65 (2006)MathSciNetMATHGoogle Scholar
  13. 13.
    Eloe, P.W., Grimm, L.J., Mashburn, J.D.: A boundary value problem on an unbounded domain. Differ. Equ. Dyn. Syst. 8, 125–140 (2000)MathSciNetMATHGoogle Scholar
  14. 14.
    Granas, A., Guenther, R.B., Lee, J.W., O’Regan, D.: Boundary value problems in infinite intervals and semicondutor devices. J. Math. Anal. Appl. 116, 335–348 (1986)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gross, O.A.: The boundary value problem on an infinite interval: existence, uniqueness, asymptotic behavior of bounded solutions to a class of nonlinear second-order differential equations. J. Math. Anal. Appl. 7, 100–109 (1963)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ma, R., Zhu, B.: Existence of positive solutions for a semipositone boundary value problem on the half line. Comput. Math. Appl. 8, 1672–1686 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Zettl, A.: Sturm–Liouville theory. American Mathematical Society, Mathematical Surveys and Monographs, vol 121 (2005)Google Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Faculty of MathematicsUSTHBAlgiersAlgeria

Personalised recommendations