Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 1, pp 315–335 | Cite as

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

  • Abdelhamid BenmezaïEmail author
  • Salima Mellal
Original Paper


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.


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

Mathematics Subject Classification

Primary 34B05 Secondary 34B40 



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


  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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Benmezaï, A., Esserhane, W., Henderson, J.: Sturm–Liouville BVPs with Caratheodory nonlinearities. Electron. J. Differ. Equ. 2016(298), 1–49 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Benmezaï, A.: Fixed point theorems in cones under local conditions. Fixed Point Theory 18, 107–126 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Berestycki, H.: On some non-linear Sturm–Liouville boundary value problems. J. Differ. Equ. 26, 375–390 (1977)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Constantin, A.: On an infinite interval boundary value problem. Ann. Mat. Pura ed Appl. IV, 379–394 (1999)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)MathSciNetCrossRefzbMATHGoogle 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

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