End point results of generalized setvalued almost contractions in metric spaces endowed with a graph

  • Binayak S. Choudhury
  • D. Khatua
  • N. Metiya
  • P. Saha
Original Article


In this paper, we consider multivalued mappings satisfying two different inequalities and obtain end point results for such mappings in a metric space endowed with a graph. The main theorems are illustrated with an example. The line of research is setvalued analysis in the combined domain of graph theory and metric space. The methodology is a blending of graph theoretic and analytic methods.


Metric space Directed graph Multivalued mappings End point Fixed point 

Mathematics Subject Classification

54H10 54H25 47H10 



The authors gratefully acknowledge the suggestions made by the learned referee.


  1. 1.
    Abbas, M., Nazir, T., Lampert, T. A., Radenovi\(\acute{c}\), S.: Common fixed points of set-valued F-contraction mappings on domain of sets endowed with directed graph. Comp. Appl. Math. 36(4), 1607–1622 (2017)Google Scholar
  2. 2.
    Babu, G.V.R., Sandhya, M.L., Kameswari, M.V.R.: A note on a fixed point theorem of Berinde on weak contractions. Carpath. J. Math. 24(1), 08–12 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Banach, S.: Sur les oprations dans les ensembles abstraits et leurs applications aux equations intgrales. Fund Math. 3, 133–181 (1922)CrossRefzbMATHGoogle Scholar
  4. 4.
    Beg, I., Butt, A.R.: Common fixed point for generalized set valued contractions satisfying an implicit relation in partially ordered metric spaces. Math. Commun. 15, 65–76 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Beg, I., Butt, A. R., Radojevi\(\acute{c}\), S.: The contraction principle for set valued mappings on a metric space with a graph. Comput. Math. Appl. 60, 1214–1219 (2010)Google Scholar
  6. 6.
    Berinde, V.: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 9(1), 43–53 (2004)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Berinde, V., P\(\check{a}\)curar, M.: Fixed points and continuity of almost contractions. Fixed Point Theory 9(1), 23–34 (2008)Google Scholar
  8. 8.
    Bojor, F.: Fixed point of \(\varphi \) -contraction in metric spaces endowed with a graph. An. Univ. Craiova Ser. Mat. Inform. 37(4), 85–92 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bojor, F.: Fixed point theorems for Reich type contractions on metric spaces with a graph. Nonlinear Anal. 75, 3895–3901 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Choudhury, B.S., Kundu, A.: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal. 73, 2524–2531 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Choudhury, B.S., Metiya, N.: Multivalued and singlevalued fixed point results in partially ordered metric spaces. Arab J. Math. Sci. 17, 135–151 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Choudhury, B.S., Metiya, N.: Fixed point theorems for almost contractions in partially ordered metric spaces. Ann. Univ. Ferrara 58, 21–36 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Choudhury, B.S., Metiya, N., Postolache, M.: A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl. 2013, 152 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    \(\acute{C}\)iri\(\acute{c}\), L. B., Ume, J. S.: Some common fixed point theorems for weakly compatible mappings. J. Math. Anal. Appl. 314, 488–499 (2006)Google Scholar
  15. 15.
    \(\acute{C}\)iri\(\acute{c}\), L. B., Abbas, M., Saadati, R., Hussain, N.: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 217, 5784–5789 (2011)Google Scholar
  16. 16.
    Fisher, B.: Common fixed points of mappings and setvalued mappings. Rostock Math. Colloq. 18, 69–77 (1981)zbMATHGoogle Scholar
  17. 17.
    Fisher, B., Sessa, S.: Two common fixed point theorems for weakly commuting mappings. Period. Math. Hungar. 20, 207–218 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bhaskar, T.G., Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379–1393 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Harjani, J., Sadarangani, K.: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 71, 3403–3410 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136(4), 1359–1373 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kannan, R.: Some results on fixed points. Bull. Cal. Math. Soc. 10, 71–76 (1968)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kannan, R.: Some results of fixed points-II. Am. Math. Mon. 76, 405–408 (1969)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Jr, S.B.: Nadler, Multivalued contraction mapping. Pac. J. Math. 30, 475–488 (1969)CrossRefGoogle Scholar
  24. 24.
    Nashine, H.K., Samet, B., Vetro, C.: Monotone generalized nonlinear contractions and fixed point theorems in ordered metric spaces. Math. Comput. Modell. 54, 712–720 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nieto, J.J., López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nieto, J.J., López, R.: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Act. Math. Sin. English Ser. 23(12), 2205–2212 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Shukla, S., Radenović, S., Vetro, C.: Set-valued Hardy-Rogers type contraction in 0-complete partial metric spaces. Int. J. Math. Math. Sci. (2014).
  29. 29.
    Tiammee, J., Suantai, S.: Coincidence point theorems for graph-preserving multi-valued mappings. Fixed Point Theory Appl. (2014).
  30. 30.
    Turinici, M.: Abstract comparison principles and multivariable Gronwall–Bellman inequalities. J. Math. Anal. Appl. 117, 100–127 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zamfirescu, T.: Fixed point theorems in metric spaces. Arch. Mat. (Basel) 23, 292–298 (1972)CrossRefzbMATHGoogle Scholar

Copyright information

© Sociedad Matemática Mexicana 2019

Authors and Affiliations

  • Binayak S. Choudhury
    • 1
  • D. Khatua
    • 1
  • N. Metiya
    • 2
  • P. Saha
    • 1
  1. 1.Department of MathematicsIndian Institute of Engineering Science and TechnologyHowrahIndia
  2. 2.Department of MathematicsSovarani Memorial CollegeHowrahIndia

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