, Volume 57, Issue 1, pp 181–190 | Cite as

Weak reciprocal continuity and fixed point theorems



The aim of the present paper is to introduce the notion of weak reciprocal continuity and obtain fixed point theorems by employing the new notion. The new notion is a proper generalization of reciprocal continuity and is applicable to compatible mappings as well as noncompatible mappings. Our results generalize several fixed point theorems.


Fixed point theorems Nonexpansive mapping Compatible maps Noncompatible maps R-weakly commuting mappings Reciprocal continuity R-weakly commuting of type (Af ) and of type (Ag

Mathematics Subject Classification (2000)

47H09 47H10 54H25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aliouche A., Merghadi F.: A common fixed point theorem via a generalized contractive condition. Annales Mathematicae et informaticae 36, 3–14 (2009)MathSciNetGoogle Scholar
  2. 2.
    Chugh R., Kumar S.: Minimal commutativity and common fixed points. J. Indian Math. Soc. 70(1–4), 169–177 (2003)MathSciNetMATHGoogle Scholar
  3. 3.
    Chugh R., Savita: Common fixed points of four R-weakly commuting mappings. J. Indian Math. Soc. 70(1–4), 185–189 (2003)MathSciNetMATHGoogle Scholar
  4. 4.
    Imdad M., Ali J.: Reciprocal continuity and common fixed points of nonself mappings. Taiwan. J. Math. 13(5), 1457–1473 (2009)MathSciNetMATHGoogle Scholar
  5. 5.
    Jungck G.: Commuting mappings and fixed point. Amer. Math. Monthly 83, 261–263 (1976)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Jungck G.: Compatible mappings and common fixed points. Internat. J. Math. Sci. 9, 771–779 (1986)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Kumar S., Chugh R.: common fixed points theorem using minimal commutativity and reciprocal continuity conditions in metric space. Scientiae Mathematicae Japonicae 56(2), 269–275 (2002)MathSciNetMATHGoogle Scholar
  8. 8.
    Mishra U., Randive A.S., Gopal D.: Fixed point theorems via absorbing maps. J. math. 6(1), 49–60 (2008)MathSciNetMATHGoogle Scholar
  9. 9.
    Pant R.P.: Common fixed point theorems for contractive maps. J. Math. Anal. Appl. 226, 251–258 (1998)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Pant R.P.: Common fixed points of four mappings. Bull. Cal. Math. Soc. 90, 281–286 (1998)MathSciNetMATHGoogle Scholar
  11. 11.
    Pant R.P.: Discontinuity and fixed points. J. Math. Anal. Appl. 240, 284–289 (1999)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Pant R.P.: Common fixed points of noncommuting mappings. J. Math. Anal. Appl. 188, 436–440 (1994)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Pant R.P., Joshi P.C., Gupta V.: A Meir-Keeler type fixed point theorem. Indian J. pure appl. Math. 32(6), 779–787 (2001)MathSciNetMATHGoogle Scholar
  14. 14.
    Pant, R.P.: A Meir-Keeler type fixed point theorem and dynamics of functions. Demonstratio Math. 35, 199–206 (2003)MathSciNetGoogle Scholar
  15. 15.
    Pant R.P., Pant V., Lohani A.B.: Reciprocal continuity and common fixed Points. J. Indian Math. Soc. 70(1–4), 157–167 (2003)MathSciNetMATHGoogle Scholar
  16. 16.
    Pant V., Pant R.P.: Common fixed points of conditionally commuting maps. Fixed point theory 11(1), 113–118 (2010)MathSciNetMATHGoogle Scholar
  17. 17.
    Pathak H.K., Cho Y.J., Kang S.M.: Remarks of R-weakly commuting mappings and common fixed point theorems. Bull. Korean Math. Soc. 34, 247–257 (1997)MathSciNetMATHGoogle Scholar
  18. 18.
    Singh S.L., Mishra S.N.: Coincidences and fixed points of reciprocally continuous and compatible hybrid maps. Int. J. Math. Math. Sci. 30(10), 627–635 (2002)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2011

Authors and Affiliations

  1. 1.Department of MathematicsKumaun University NainitalNainitalIndia

Personalised recommendations