Skip to main content
SpringerLink
Log in

Fixed points of Perov type contractive mappings on the set endowed with a graphic structure

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

Abstract

The aim of this paper is to present fixed point results of Perov type contractive mappings in the framework of cone metric spaces endowed with a graphic structure. Some examples are presented to support the results proved herein. We also provide an example to show that our results are substantial generalization of comparable results in the existing literature. As an application of our results, we obtain fixed point results of Perov type cyclic contraction mappings in cone metric spaces.

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. Abbas, M., Nazir, T.: Common fixed point of a power graphic contraction pair in partial metric spaces endowed with a graph. Fixed Point Theory Appl. 2013, 20 (2013). doi:10.1186/1687-1812-2013-20

    Article  MathSciNet  MATH  Google Scholar 

  2. Aleomraninejad, S.M.A., Rezapour, Sh, Shahzad, N.: Some fixed point results on a metric space with a graph. Topol. Appl. 159, 659–663 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alghamdi, M.A., Alnafei, S.H., Radenović, S., Shahzad, N.: Fixed point theorems for convex contraction mappings on cone metric spaces. Math. Comput. Model. 54, 2020–2026 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Altun, I., Damjanović, B., Djorić, D.: Fixed point and common fixed point theorems on ordered cone metric spaces. Appl. Math. Lett. 23, 310–316 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Altun, I., Durmaz, G.: Some fixed point theorems on ordered cone metric spaces. Rendiconti del Circolo Matematico di Palermo 58, 319–325 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Banach, S.: Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fund. Math. 3, 133–181 (1922)

    Article  MATH  Google Scholar 

  7. Bojor, F.: Fixed point theorems for Reich type contractions on metric spaces with graph. Nonlinear Anal. 75, 3895–3901 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chatterjea, S.K.: Fixed point theorems. comtes Rendus de l’ Academie Bulgare des Sciences 25, 727–730 (1972)

  9. Cvetković, M., Rakočević, V.: Quasi-contraction of Perov type. Appl. Math. Comput. 235, 712–722 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Du, W.: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 72, 2259–2261 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Du, W., Karapınar, E.: A note on cone b-metric and its related results: generalizations or equivalence? Fixed Point Theory Appl. 2013, 210 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Edelstein, M.: On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 37, 74–79 (1962)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. Hardy, G.E., Rogers, T.D.: A generalization of a fixed point theorem of Reich. Can. Math. Bull. 16, 201–206 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136, 1359–1373 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Janković, S., Kadelburg, Z., Radojević, S.: On cone metric spaces: a survey. Nonlinear Anal. 74, 2591–2601 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jungck, G., Radenović, S., Radojević, S., Rakočević, V.: Common fixed point theorems for weakly compatible pairs on cone metric spaces. Fixed Point Theory Appl. (2009)

  18. Kannan, R.: Some results on fixed points. Bull. Calcutta Math. Soc. 10, 71–76 (1968)

    MathSciNet  MATH  Google Scholar 

  19. 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 

  20. Guang, H.L., Xian, Z.: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332, 1468–1476 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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 

  22. Petric, M.A.: Some remarks concerning C̀irić-Reich-Rus operators. Creat. Math. Inf. 18, 188–193 (2009)

  23. Petrusel, A., Rus, I.A.: Fixed point theorems in ordered L-spaces. Proc. Am. Math. Soc. 134, 411–418 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Perov, A.I.: On cauchy problem for a system of ordinary differential equations. Pviblizhen. Met. Reshen. Differ. Uravn. 2, 115–134 (1964)

    Google Scholar 

  25. Precup, R.: The role of matrices that are convergent to zero in the study of semilinear operator system. Math. Comput. Model. 49, 703–708 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Rezapour, S., Hamlbarani, R.: Some notes on the paper Cone metric spaces and fixed apoint theorems of contractive mappings. J. Math. Anal. Appl. 345, 719–724 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank very much the reviewers for their useful suggestions and remarks that contributed a lot to the improvement of the manuscript. The second author is supported By Grant No. 174025 of the Ministry of Science, Technology and Development, Republic of Serbia.

Author information

Authors and Affiliations

  1. Nonlinear Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    M. Abbas

  2. Department of Mathematics, Faculty of Sciences and Mathematics, University of Niš, Nis, 18000, Serbia

    V. Rakočević

  3. Department of Mathematics, Syed Babar Ali School of Science and Engineering (SBASSE), (LUMS), Lahore, 54792, Pakistan

    A. Iqbal

Authors
  1. M. Abbas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Rakočević
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Iqbal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Rakočević.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abbas, M., Rakočević, V. & Iqbal, A. Fixed points of Perov type contractive mappings on the set endowed with a graphic structure. RACSAM 112, 209–228 (2018). https://doi.org/10.1007/s13398-016-0373-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-016-0373-4

Keywords

Mathematics Subject Classification

Search

Navigation