Skip to main content
Log in

A Kannan-type fixed point theorem for multivalued mappings with application

  • Original Research Paper
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

Abstract

In this paper, we present a Kannan-type fixed point theorem for multivalued mappings defined on a complete metric space in terms of a Suzuki-type contraction. As an application, we prove an existence and uniqueness theorem for a functional equation arising in dynamic programming of continuous multistage decision processes.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Banach, S. 1922. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae 3: 133–181.

    Article  MathSciNet  Google Scholar 

  2. Bellman, R., and E.S. Lee. 1978. Functional equations arising in dynamic programming. Aequationes Mathematicae 17: 1–18.

    Article  MathSciNet  Google Scholar 

  3. Bhakta, P.C., and S. Mitra. 1984. Some existence theorems for functional equations arising in dynamic programming. Journal of Mathematical Analysis and Applications 98: 348–362.

    Article  MathSciNet  Google Scholar 

  4. Chandra, N., M.C. Joshi, and N.K. Singh. 2017. Fixed point theorems for generalized multivalued contraction. Journal of Analysis 26: 1–11. https://doi.org/10.1007/s41478-017-0067-0.

    Article  MathSciNet  Google Scholar 

  5. Chatterjee, S.K. 1972. Fixed point theorem. Comptes Rendus de l’Academie Bulgare des Sciences 25: 727–730.

    MathSciNet  Google Scholar 

  6. Connell, E.H. 1959. Properties of fixed point spaces. Proceedings of the American Mathematical Society 10: 974–979.

    Article  MathSciNet  Google Scholar 

  7. Damjanović, B., and D. Dorić. 2011. Multivalued generalizations of the Kannan fixed point theorem. Filomat 25: 125–131.

    Article  MathSciNet  Google Scholar 

  8. D. Dorić and R. Lazović. 2011. Some Suzuki-type fixed point theorems for generalized multivalued mappings and applications. Fixed Point Theory and Applications

  9. Gornicki, J. 2017. Fixed point theorems for Kannan type mappings. Fixed Point Theory and Applications 19: 2145–2152.

    Article  MathSciNet  Google Scholar 

  10. Hardy, G.E., and T.D. Rogers. 1973. A generalization of a fixed point theorem of Reich. Canadian Mathematical Bulletin 16: 201–206.

    Article  MathSciNet  Google Scholar 

  11. Jaskiewicz, A., J. Matkowski, and A.S. Nowak. 2013. Persistently optimal policies in stochastic dynamic programming with generalized discounting. Mathematics of Operations Research 38 (1): 108–121.

    Article  MathSciNet  Google Scholar 

  12. Kaliaj, S.B. 2017. A functional equation arising in dynamic programming. Aequationes Mathematicae 91: 635–645.

    Article  MathSciNet  Google Scholar 

  13. Kannan, R. 1968. Some results on fixed points. Bulletin of the Calcutta Mathematical Society 60: 71–76.

    MathSciNet  MATH  Google Scholar 

  14. Kikkawa, M., and T. Suzuki. 2008. Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Analysis 69: 2942–2949.

    Article  MathSciNet  Google Scholar 

  15. Kikkawa, M., T. Suzuki. 2008. Some similarity between contractions and Kannan mappings. Fixed Point Theory and Applications. Art. ID: 649749.

  16. Kikkawa, M., and T. Suzuki. 2008. Some similarity between contractions and Kannan mappings II. Pure and Applied Mathematics 55: 1–13.

    MathSciNet  MATH  Google Scholar 

  17. Kikkawa, M., and T. Suzuki. 2009. Some notes on fixed point theorems with constants. Pure and Applied Mathematics 56: 11–18.

    MathSciNet  MATH  Google Scholar 

  18. Liu, Z., and S.M. Kang. 2007. Existence and uniqueness of solutions for two classes of functional equations arising in dynamic programming. Acta Mathematicae Applicatae Sinica (English Series) 23: 195–208.

    Article  MathSciNet  Google Scholar 

  19. Liu, Z., Y. Xu, J.S. Ume, and S.M. Kang. 2006. Solutions to two functional equations arising in dynamic programming. Journal of Computational and Applied Mathematics 192: 251–269.

    Article  MathSciNet  Google Scholar 

  20. Popescu, O. 2009. Two fixed point theorems for generalized contractions with constants in complete metric space. Central European Journal of Mathematics 7 (3): 529–538.

    Article  MathSciNet  Google Scholar 

  21. Reich, S. 1971. Some remarks concerning contraction mappings. Canadian Mathematical Bulletin 14: 121–124.

    Article  MathSciNet  Google Scholar 

  22. Reich, S. 1972. Fixed points of contractive functions. Bollettino dell’Unione Matematica Italiana 4 (5): 26–42.

    MathSciNet  MATH  Google Scholar 

  23. Subrahmanyam, V. 1975. Completeness and fixed-points. Monatshefte fur Mathematik 80: 325–330.

    Article  MathSciNet  Google Scholar 

  24. Suzuki, T. 2008. A generalized Banach contraction principle that characterizes metric completeness. Proceedings of the American Mathematical Society 136: 1861–1869.

    Article  MathSciNet  Google Scholar 

  25. Suzuki, T. 2009. A new type of fixed point theorem in metric spaces. Nonlinear Analysis 71 (11): 5313–5317.

    Article  MathSciNet  Google Scholar 

  26. Zamfirescu, T. 1972. Fixed point theorems in metric spaces. Archiv der Mathematik (Basel) 23: 292–298.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sokol Bush Kaliaj.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kaliaj, S.B. A Kannan-type fixed point theorem for multivalued mappings with application. J Anal 27, 837–849 (2019). https://doi.org/10.1007/s41478-018-0135-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41478-018-0135-0

Keywords

Mathematics Subject Classification

Navigation