End point theorems of multivalued operators without continuity satisfying hybrid inequality under two different sets of conditions

  • Binayak S. Choudhury
  • Nikhilesh MetiyaEmail author
  • Sunirmal Kundu


In the fixed point theory of contractive mappings, that is, mappings satisfying contractive inequalities, in partially ordered metric spaces the fixed point results are obtained under the assumption that the contractive inequality condition holds for pairs of points which are related by partial order rather than for arbitrary pairs of points. Alternatives to partial order for the purpose of restricting the contraction conditions in the fixed point results are the admissibility conditions. In this work we define a multivalued hybrid inequality by generalizing and combining two types of contractive inequalities and establish end point results for those operators which satisfy the hybrid inequality defined here under the two separate environments described above. For our purpose we define a new admissibility condition. The results are established in the most general structure of a metric space. The main theorems proved here are in the domain of setvalued analysis. The corresponding singlevalued cases are discussed. There is nowhere any assumption of continuity. The methodology is either a combination of analytic and order theoretic methods or purely analytic. Moreover the newly introduced method of proof in fixed point theory through Pata-type results are followed in the multivalued case. There are supporting examples of the main results.


Metric space Partial order Multivalued cyclic \((\alpha , \beta )\)-admissible mapping \(\delta \)-distance End point 

Mathematics Subject Classification

54H10 54H25 47H10 



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


  1. 1.
    Alizadeh, S., Moradlou, F., Salimi, P.: Some fixed point results for \((\alpha - \beta )\) - \((\psi - \varphi )\)—contractive mappings. Filomat 28(3), 635–647 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Altun, I., Turkoglu, D.: Some fixed point theorems for weakly compatible multivalued mappings satisfying an implicit relation. Filomat 22, 13–21 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Banach, S.: Sur les oprations dans les ensembles abstraits et leurs applications aux quations intgrales. Fund Math. 3, 133–181 (1922)CrossRefGoogle 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.
    Chakraborty, M., Samanta, S.K.: A fixed point theorem for Kannan-type maps in metric spaces, 7331v2 [math. GN] 16 December 2012. arXiv:1211
  6. 6.
    Chatterjea, S.K.: Fixed-point theorems. C.R. Acad. Bulgare Sci. 25, 727–730 (1972)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cho, S.H.: A fixed point theorem for weakly \(\alpha \)-contractive mappings with application. Appl. Math. Sci. 7, 2953–2965 (2013)MathSciNetGoogle Scholar
  8. 8.
    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
  9. 9.
    Choudhury, B.S., Metiya, N.: Coincidence point theorems for a family of multivalued mappings in partially ordered metric spaces. Acta Universitatis Matthiae Belii, series Mathematics 21, 13–26 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    Choudhury, B.S., Kundu, A., Metiya, N.: Fixed point results for Ćirić type weak contraction in metric spaces with applications to partial metric spaces. Filomat 28(7), 1505–1516 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Choudhury, B.S., Metiya, N., Som, T., Bandyopadhyay, C.: Multivalued fixed point results and stability of fixed point sets in metric spaces. Facta Universitatis (NIS) Ser. Math. Inform. 30, 501–512 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Choudhury, B.S., Metiya, N., Bandyopadhyay, C.: Fixed points of multivalued \(\alpha \)-admissible mappings and stability of fixed point sets in metric spaces. Rend. Circ. Mat. Palermo 64, 43–55 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ćirić, L.B., Ume, J.S.: Some common fixed point theorems for weakly compatible mappings. J. Math. Anal. Appl. 314, 488–499 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Damjanović, B., Samet, B., Vetro, C.: Common fixed point theorems for multi-valued maps. Acta Mathematica Scientia 32B(2), 818–824 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dorić, D.: Common fixed point for generalized \((\psi, \varphi )\)-weak contractions. Appl. Math. Lett. 22, 1896–1900 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fisher, B.: Common fixed points of mappings and setvalued mappings. Rostock Math. Colloq. 18, 69–77 (1981)zbMATHGoogle Scholar
  18. 18.
    Gnana Bhaskar, T., Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379–1393 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gordji, M.E., Baghani, H., Khodaei, H., Ramezani, M.: A generalization of Nadler’s fixed point theorem. J. Nonlinear Sci. Appl. 3(2), 148–151 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Harjani, J., Sadarangani, K.: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 71, 3403–3410 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hussain, N., Karapinar, E., Salimi, P., Akbar, F.: \(\alpha \)-admissible mappings and related fixed point theorems. J. Inequal. Appl. 2013, 114 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kadelburg, Z., Radenović, S.: A note on pata-type cyclic contractions. Sarajevo J. Math. 11(2), 235–245 (2015)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kadelburg, Z., Radenović, S.: Fixed point theorems under Pata-type conditions in metric spaces. J. Egypt. Math. Soc. 24, 77–82 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kannan, R.: Some results on fixed points. Bull. Cal. Math. Soc. 60, 71–76 (1968)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Karapinar, E., Samet, B.: Generalized \(\alpha - \psi \) contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012 (2012), Article ID 793486Google Scholar
  26. 26.
    Nadler Jr., S.B.: Multivalued contraction mappings. Pac. J. Math. 30, 475–488 (1969)CrossRefzbMATHGoogle Scholar
  27. 27.
    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
  28. 28.
    Nieto, J.J., Rodrguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pata, V.: A fixed point theorem in metric spaces. J. Fixed Point Theory Appl. 10, 299–305 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    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
  31. 31.
    Reich, S.: Fixed points of contractive functions. Boll. Un. Mat. Ital. 5, 26–42 (1972)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for \(\alpha -\psi \)-contractive type mappings. Nonlinear Anal. 75, 2154–2165 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Shen, M., Hong, S.: Common fixed points for generalized contractive multivalued operators in complete metric spaces. Appl. Math. Lett. 22, 1864–1869 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Turinici, M.: Abstract comparison principles and multivariable Gronwall–Bellman inequalities. J. Math. Anal. Appl. 117, 100–127 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  • Binayak S. Choudhury
    • 1
  • Nikhilesh Metiya
    • 2
    Email author
  • Sunirmal Kundu
    • 1
  1. 1.Department of MathematicsIndian Institute of Engineering Science and TechnologyShibpur, HowrahIndia
  2. 2.Department of MathematicsSovarani Memorial CollegeJagatballavpur, HowrahIndia

Personalised recommendations