Fixed Points for (ϕ, ψ)-Contractions in Menger Probabilistic Metric Spaces

Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 41)


In this paper, we work out a fixed point result for weakly contractive mapping in Menger probabilistic metric space. A combination of analytic and order theoretic approach is used to establish our main theorem. The main result is illustrated with an example.



The research work of the first author (Vandana Tiwari) is supported by the University Grant Commission (UGC) (No. 19-06/2011(i)EU-IV), India.


  1. 1.
    An, T.V., Chi, K.P., Karapinar, E., Thanh, T.D., An extension of generalized (ψ, ϕ)-weak contractions. Int. J. Math. Math. Sci. Article ID 431872, 11 pages (2012)Google Scholar
  2. 2.
    Aydi, H., Karapinar, E., Shatanawi, W., Coupled fixed point results for (ψ, ϕ)-weakly contractive condition in ordered partial metric spaces. Comp. Math. with Appl. 62, 4449–4460 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aydi, H., Postolache, M., Shatanawi, W., Coupled fixed point results for (ψ, ϕ)-weakly contractive mappings in ordered G-metric spaces. Comp. Math. with Appl. 63, 298–309 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Berinde, V., Approximating fixed points of weak ϕ-contractions. Fixed Point Theory. 4, 131–142 (2003)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Choudhury, B.S., Das, K., A new contraction principle in Menger spaces. Acta Math. Sin. Engl. Ser. 24(8), 1379–1386 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Doric, D., Common fixed point for generalized (ψ, ϕ)-weak contractions. Appl. Math. Lett. 22, 1896–1900 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dutta, P.N., Choudhury, B.S.: A generalization of contraction principle in metric spaces. Fixed Point Theory and Appl. 8, Article ID 406368 (2008)CrossRefGoogle Scholar
  8. 8.
    Dutta, P.N., Choudhury, B.S., Das, K.: Some fixed point results in Menger spaces using a control function. Surv. Math. Appl. 4, 41–52 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fang, J.-X.: On φ-contractions in probabilistic and fuzzy metric spaces. Fuzz. Sets and Sys. 267, 86–99 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jamala, N., Sarwara, M., Imdad, M.: Fixed point results for generalized (ψ, ϕ)-weak contractions with an application to system of non-linear integral equations, Trans. A. Razm. Math. Inst. 171, 182–194 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Latif, A., Mongkolkeha, C., Sintunavarat, W.: Fixed point theorems for generalized α, β-weakly contraction mappings in metric spaces and applications, Sci. World. Article ID 784207, 14 pages (2014)Google Scholar
  12. 12.
    Luo, T.: Fuzzy (ψ, ϕ)-contractive mapping and fixed point theorem. Appl. Math. Sci. 8(148), 7375–7381 (2014)Google Scholar
  13. 13.
    Meneger, K.: Statistical metrics. Proc. Nat. Acad. Sci. USA, 28, 535–537 (1942)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Moradi, S., Farajzadeh, A.: On the fixed point of (ψ, ϕ)-weak and generalized (ψ, ϕ)-weak contraction mappings. Appl. Math. Lett. 25, 1257–1262 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Popescu, O.: Fixed point for (ψ-ϕ)-weak contractions. Appl. Math. Lett. 24, 1–4 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Rhoades, B.E.: Some theorems on weakly contractive maps. Nonlinear Anal. TMA. 47, 2683–2693 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Saha, P., Choudhury, B.S., Das, P.: Weak coupled coincidence point results having a partially ordering in fuzzy metric space. Fuzzy. Inf. Eng. 7, 1–18 (2016)MathSciNetGoogle Scholar
  18. 18.
    Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for (α, ψ)-contractive type mappings. Nonlinear Anal. TMA. 75, 2154–2165 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 313–334 (1960)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. Elsevier, New York, (1983)zbMATHGoogle Scholar
  21. 21.
    Sehgal, V.M., Bharucha-Reid, A.T.: Fixed points of contraction mappings on PM-spaces. Math. Syst. Theory. 6, 97–102 (1972)CrossRefGoogle Scholar
  22. 22.
    Tiwari, V., Som, T.: Fixed points for ϕ-contraction in Menger probabilistic generalized metric spaces, (Accepted in Ann. Fuzzy Math. Inform.)Google Scholar
  23. 23.
    Xiao, J.Z., Zhu, X.H., Cao, Y.F.: Common coupled fixed point results for probabilistic φ-contractions in Menger spaces. Nonlinear Anal. TMA. 74, 4589–4600 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhang, Q., Song, Y.: Fixed point theory for generalized ϕ-weak contractions. Appl. Math. Lett. 22, 75–78 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndian Institute of Technology (BHU)VaranasiIndia

Personalised recommendations