Skip to main content
SpringerLink
Log in

Existence Results for Singular Boundary Value Problems on Unbounded Domains in Banach Spaces

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, the Krasnosels’kii fixed point theorem in cones for strict set-contractions is used to investigate the existence of single and twin positive solutions for a class of a two-point boundary value problem of second-order nonlinear differential equations posed on an infinite interval. The nonlinearity, which may have a time-singularity, takes values in a general Banach space and has at most polynomial growth with respect to the unknown.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agarwal, R., O’Regan, D.: Singular Differential and Integral Equations with Applications. Kluwer Academic Publishers, Dordrecht (2003)

  2. Banas, J., Goebel, K.: Measure of Noncompactness in Banach spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Marcel Dekker, New York (1980)

  3. Banas J., Knap Z.: Measure of noncompactness and nonlinear integral equations of convolution type. J. Math. Anal. Appl. 146, 353–362 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brézis, H.: Analyse Fonctionnelle. Théorie et Applications, Masson (1983)

  5. Cac N.P., Gatica J.A.: Fixed point theorems for mapping in ordered Banach space. J. Math. Anal. Appl. 71, 545–557 (1979)

    Article  Google Scholar 

  6. Corduneanu, C.: Integral Equations and Stability of Feedback Systems. Academic Press, New York (1973)

  7. Deimling, K.: Ordinary Differential Equations in Banach Spaces. Springer, Berlin (1977)

  8. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

  9. Djebali S., Mebarki K.: Multiple positive solutions for singular BVPs on the positive half-line. Compu. Math. Appl. 55(12), 2940–2952 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Djebali S., Mebarki K.: On the singular generalized Fisher-like equation with derivative depending nonlinearity. Appl. Math. Comput. 205(1), 336–351 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Djebali S., Moussaoui T.: A class of second order bvps on infinite intervals. Electron. J. Qual. Theory Differ. Equ. 4, 1–19 (2006)

    Google Scholar 

  12. Djebali S., Saifi O.: Positive solutions for singular BVPs on the positive half-line with sign changing and derivative depending nonlinearity. Acta Appl. Math. 110(2), 639–665 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Djebali S., Saifi O., Baoqiang Y.: . Acta Math. Sci. 32(2), 672–694 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Feng, M., Zhang, X., Ge, W.: Positive fixed point of strict set contraction operators on ordered Banach spaces and applications, Abst. Appl. Anal. 2010, Article ID 439137. doi:10.1155/2010/439137

  15. Guo D.: A class of second-order impulsive integro-differential equations on unbounded domain in a Banach space. App. Math. Comput. 125, 59–77 (2002)

    Article  MATH  Google Scholar 

  16. Guo, D., Lakshmikantham, V., and Liu, X.: Nonlinear Integral Equations in Abstract Spaces. Math. and Appl., vol. 373. Kluwer Academic Publishers, Dordrecht (1996)

  17. Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering, vol. 5. Academic Press, Boston (1988)

  18. Jiang W., Wang B.: Positive solutions for second-order multi-point boundaryvalue problems at resonance in Banach spaces. Electron. J. Differ. Equ 22, 1–11 (2011)

    Google Scholar 

  19. Ladas, G.E., Lakshmikantham, V.: Differential Equations in Abstract Spaces. Academic Press, New York (1972)

  20. Lakshmikantham, V., Leela, S.: Nonlinear Differential Equations in Abstract Spaces. Pergamon, Oxford (1981)

  21. Li P., Chen H., Wu Y.: Existence of solutions of n-point boundary value problems on the half-line in Banach spaces. Acta Appl. Math. 110, 785–795 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Liu B.: Positive solutions of second-order three-point boundary value problems with change of sign in Banach spaces. Nonlinear Anal. 64, 1336–1355 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Liu Y.: Boundary value problems for second order differential equations on unbounded domains in a Banach space. Appl. Math. Comput. 135, 569–583 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Potter A.J.B.: A fixed point theorem for positive K-set contractions. Proc. Edinb. Math. Soc. 19, 93–102 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rachunkova, I., Stanek, S., Tvrdy, M.: Solvability of Nonlinear Singular Problems for Ordinary Differential Equations, Contemporary Mathematics and Its Applications, vol. 5. Hindawi Publishing Corporation, New York (2008)

  26. Väth, M.: Integration Theory. A Second Course. World Scientific, Singapore (2000)

  27. Zeidler, E.: Nonlinear Functional Analysis and Its Applications, vol. I. Fixed Point Theorems. Springer, New York (1986)

  28. Zhang X.: Existence of positive solutions for multi-point boundary value problems on infinite intervals in Banach spaces. Appl. Math. Comput. 6(2), 932–941 (2008)

    Article  Google Scholar 

  29. Zhang X., Feng M., Ge W.: . Nonlinear Anal. 69, 3310–3321 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Zhao Y., Chen H., Xu C.: Existence of multiple solutions for three-point boundary-value problems on infinite intervals in Banach spaces. Electron. J. Differ. Equ 44, 1–11 (2012)

    MathSciNet  Google Scholar 

  31. Hao X., Liu L., Wu Y., Xu N.: . Comput. Math. Appl. 61, 1880–1890 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Zhao Y.L., Chen H.B.: Existence of multiple positive solutions for m-point boundary value problems in Banach spaces. J. Comput. Appl. Math. 215, 79–90 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory “Théorie du Point Fixe and Applications”, École Normale Supérieure, BP 92, 16050, Kouba, Algiers, Algeria

    Smaïl Djebali

  2. Laboratory of Applied Mathematics, Faculty of Exact Sciences, A. Mira University, 06000, Bejäia, Algeria

    Fadila Madjidi & Karima Mebarki

Authors
  1. Smaïl Djebali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fadila Madjidi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Karima Mebarki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Smaïl Djebali.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Djebali, S., Madjidi, F. & Mebarki, K. Existence Results for Singular Boundary Value Problems on Unbounded Domains in Banach Spaces. Mediterr. J. Math. 11, 45–74 (2014). https://doi.org/10.1007/s00009-013-0362-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-013-0362-1

Mathematics Subject Classification

Keywords

Search

Navigation