Skip to main content
SpringerLink
Log in

Positive solutions to periodic boundary value problems involving nonlinear operators of Meir-Keeler-type

  • Published:
Rendiconti del Circolo Matematico di Palermo Aims and scope Submit manuscript

Abstract

We establish fixed point theorems for mixed monotone operators of Meir-Keeler type on ordered Banach spaces through projective metric. As applications, we use the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for different classes of nonlinear problems which include two-order two-point boundary value problems and fourth-order two-point boundary value problems for elastic beam equations.

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.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  2. Avery, R.I., Peterson, A.C.: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 42(3–5), 313–322 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, Z.: The upper and lower solution method for some fourth-order boundary value problems. Nonlinear Anal. 67, 1704–1709 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133–181 (1922)

    MATH  Google Scholar 

  5. Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, Y.Z.: A variant of the Meir-Keeler-Type theorem in ordered Banach spaces. J. Math. Anal. Appl. 236, 585–593 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ćirić, Lj.: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 45, 267–273 (1974)

    MATH  Google Scholar 

  8. Samet, B.: Coupled fixed point theorems for a generalized Meir–Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 72, 4508–4517 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. Dugundji, J., Granas, A.: Fixed Point Theory. Springer, Berlin (2003)

    MATH  Google Scholar 

  11. Edelstein, M.: An extension of Banach’s contraction principle. Proc. Am. Math. Soc. 12, 7–10 (1961)

    MathSciNet  MATH  Google Scholar 

  12. Graef, J.R., Yang, B.: Existence and nonexistence of positive solutions of fourth-order nonlinear boundary value problems. Appl. Anal. 74(1–2), 201–214 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo, D.J., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, Boston (1988)

    MATH  Google Scholar 

  14. Jachymski, J.R.: Equivalent conditions and the Meir-Keeler type theorems. J. Math. Anal. Appl. 194, 293–303 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jachymski, J.R.: Equivalence of some contractivity properties over metrical structures. Proc. Am. Math. Soc. 125, 2327–2335 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lakshmikantham, V., Leela, S.: Remarks on first and second order periodic boundary value problem. Nonlinear Anal. 8, 281–287 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Leela, S.: Monotone method for second periodic boundary value problems. Nonlinear Anal. 7, 349–355 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  18. Meir, A., Keeler, E.: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326–329 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nashine, H.K., Samet, B., Vetro, C.: Monotone generalized nonlinear contractions and fixed point theorems in ordered metric spaces. Math. Comput. Model. 54, 712–720 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nieto, J.J.: Nonlinear second order periodic value problems with Caratheodory functions. Appl. Anal. 34, 111–128 (1989)

    Article  MATH  Google Scholar 

  21. Nieto, J.J.: Periodic boundary value problem for second order integro-ordinary differential equations with general kernel and Caratheodory nonlinearities. Int. J. Math. Math. Sci. 18, 757–764 (1985)

    Article  Google Scholar 

  22. Nussbaum, R.D.: Iterated Nonlinear Maps and Hilbert’s Projective Metric. Mem. Amer. Math. Soc., vol. 75. American Mathematical Society, Providence (1988)

    Google Scholar 

  23. Ping, S.Y.: Existence and multiplicity of positive solutions for an elastic beam equation. Appl. Math. J. Chin. Univ. 26(3), 253–264 (2011)

    Article  Google Scholar 

  24. Rakotch, E.: A note on contractive mappings. Proc. Am. Math. Soc. 13, 459–465 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  25. Suzuki, T.: A generalized Banach contraction principle which characterizes metric completeness. Proc. Am. Math. Soc. 136, 1861–1869 (2008)

    Article  MATH  Google Scholar 

  26. Thompson, A.C.: On certain contraction mappings in a partially ordered vector space. Proc. Am. Math. Soc. 14, 438–443 (1963)

    MATH  Google Scholar 

  27. Vetro, C.: On Branciari’s theorem for weakly compatible mappings. Appl. Math. Lett. 23(6), 700–705 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yang, Y.R.: Triple positive solutions of a class of fourth-order two-point boundary value problems. Appl. Math. Lett. 23, 366–370 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yao, Q.: Positive solutions of nonlinear second-order periodic boundary value problems. Appl. Math. Lett. 20, 583–590 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zeidler, E.: Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  31. Zhai, C., Anderson, D.R.: A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations. J. Math. Anal. Appl. 375, 388–400 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ecole Supérieure des Sciences et Techniques de Tunis, Université de Tunis, 5, Avenue Taha Hussein-Tunis, B.P. 56, Bab Menara, 1008, Tunisia

    Maher Berzig & Bessem Samet

Authors
  1. Maher Berzig
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bessem Samet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bessem Samet.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berzig, M., Samet, B. Positive solutions to periodic boundary value problems involving nonlinear operators of Meir-Keeler-type. Rend. Circ. Mat. Palermo 61, 279–296 (2012). https://doi.org/10.1007/s12215-012-0089-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-012-0089-z

Keywords

Mathematics Subject Classification

Search

Navigation