Skip to main content
SpringerLink
Log in

Graphical metric space: a generalized setting in fixed point theory

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Building on recent ideas of Jachymski, we work on the notion of graphical metric space and prove an analogous result for the contraction mapping principle. In particular, the triangular inequality is replaced by a weaker one, which is satisfied by only those points which are situated on some path included in the graphical structure associated with the space. Some consequences, examples and an application to integral equations are presented to confirm the significance and unifying power of obtained generalizations.

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. Bakhtin, I.A.: The contraction mapping principle in quasimetric spaces. Funct. Anal. Unianowsk Gos. Ped. Inst. 30, 26–37 (1989)

    Google Scholar 

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

    MATH  Google Scholar 

  3. Ben-El-Mechaiekh, H.: The Ran-Reurings fixed point theorem without partial order: a simple proof. J. Fixed Point Theory Appl. 16, 373–383 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Branciari, A.: A fixed point theorem of Banach–Caccioppoli type on a class of generalized metric spaces. Publ. Math. Debrecen 57(1–2), 31–37 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Czerwik, S.: The contraction mapping principle in quasimetric spaces. Acta Math. Univ. Ostrav. 1, 5–11 (1993)

    MATH  Google Scholar 

  6. Edelstein, M.: An extension of Banachs contraction principle. Proc. Am. Math. Soc. 12, 7–10 (1961)

    MathSciNet  MATH  Google Scholar 

  7. Fréchet, M.: Sur quelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo 22, 1–74 (1906)

    Article  MATH  Google Scholar 

  8. Gähler, S.: 2-metric Räume and ihre topologische strucktur. Math. Nachr. 26, 115–148 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, L.G., Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332, 1468–1476 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136, 1359–1373 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kirk, W.A., Srinivasan, P.S., Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 4, 79–89 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Matthews, S.G.: Partial metric topology. In: Proc. 8th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci., vol. 728, pp. 183–197 (1994)

  13. Nieto, J.J., Rodríguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some application to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for \(\alpha \)-\(\psi \)-contractive type mappings. Nonlinear Anal. 75(4), 2154–2165 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shukla, S., Abbas, M.: Fixed point results of cyclic contractions in product spaces. Carpathian J. Math. 31, 119–126 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Referee(s) for their valuable comments and suggestions, which were very useful to improve the presentation of the paper. S. Shukla is thankful to Professor M.K. Dube for his regular encouragements and motivation for research.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Shri Vaishnav Institute of Technology and Science, Gram Baroli Sanwer Road, Indore, M.P., 453331, India

    Satish Shukla

  2. Faculty of Mechanical Engineering, University of Belgrade, Belgrade, Serbia

    Stojan Radenović

  3. Department of Mathematics, State University of Novi Pazar, Novi Pazar, Serbia

    Stojan Radenović

  4. Department of Mathematics and Computer Science, Università degli Studi di Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Calogero Vetro

Authors
  1. Satish Shukla
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stojan Radenović
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Calogero Vetro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Satish Shukla.

Additional information

C. Vetro is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shukla, S., Radenović, S. & Vetro, C. Graphical metric space: a generalized setting in fixed point theory. RACSAM 111, 641–655 (2017). https://doi.org/10.1007/s13398-016-0316-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-016-0316-0

Keywords

Mathematics Subject Classification

Search

Navigation