Fixed points of set-valued mappings in Menger probabilistic metric spaces endowed with an amorphous binary relation




fixed point, set-valued mapping, Bernstein operator


In this paper, we prove the existence of fixed point results for set-valued mappings in Menger probabilistic metric spaces equipped with an amorphous binary relation and a Hadžić -type t-norm. For the usability of such findings we present a Kelisky-Rivlin type result for a class of Bernstein type special operators introduced by Deo et. al. [Appl. Math. Comput. 201, (2008), 604-612 ] on the space C([ 0, n/n+1]). In this way, these investigations extend, modify and generalize some prominent recent fixed point results of the existing literature.


Download data is not yet available.

Author Biographies

Gopi Prasad, Dr. Shivanand Nautiyal Government Post Graduate College

Department of Mathematics

Sheetal Deshwal, Dr. Shivanand Nautiyal Government Post Graduate College

Department of Mathematics

Rupesh K. Srivastav, Dr. Shivanand Nautiyal Government Post Graduate College

Department of Mathematics


A. Alam and M. Imdad, Relation-theoretic contraction principle, J. Fixed Point Theory Appl. 17, no. 4 (2015), 693-702.

A. Alam, R. George and M. Imdad, Refinements to relation-theoretic contraction principle, Axioms 11, no. 7 (2022), 316.

H. Argoubi, M. Jleli and B. Samet, The study of fixed points for multivalued mappings in a Menger probabilistic metric space endowed with a graph, Fixed Point Theory Appl. 2015 (2015), 113.

S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations intgrales, Fund. Math. 3 (1922), 133-181.

S. K. Bhandari, D. Gopal and P. Konar, Probabilistic α-min Ciric type contraction results using a control function, AIMS Mathematics 5, no. 2 (2020), 1186-1198.

N. Deo, M. A. Noor and M. A. Siddiqui, On approximation by a class of new Bernstein type operators, Appl. Math. Comput. 201 (2008), 604-612.

T. Dinevari and M. Frigon, Fixed point results for multivalued contractions on a metric space with a graph, J. Math. Anal. Appl. 405 (2013), 507-517.

J.-X. Fang, Fixed point theorems of local contraction mappings on Menger spaces, Appl. Math. Mech. 12 (1991), 363-372.

J.-X. Fang, A note on fixed point theorems of Hadžić, Fuzzy Sets Syst. 48 (1992), 391-395.

J.-X. Fang, Common fixed point theorems of compatible and weakly compatible maps in Menger spaces, Nonlinear Anal. 71 (2009), 1833-1843.

O. Hadžić, Fixed point theorems for multivalued mappings in probabilistic metric spaces, Fuzzy Sets Syst. 88 (1997), 219-226.

O. Hadžić and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic, Dordrecht (2001).

J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), 1359-1373.

T. Kamran, M. Samreen and N. Shahzad, Probabilistic G-contractions, Fixed Point Theory Appl. 2013 (2013), 223.

R. P. Kelisky and T. J. Rivlin, Iterates of Bernstein polynomials, Pac. J. Math. 21 (1967), 511-520.

B. Kolman, R. C. Busby and S. Ross, Discrete mathematical structures, Third Edition, PHI Pvt. Ltd., New Delhi, 2000.

S. Lipschutz, Schaum's Outlines of Theory and Problems of Set Theory and Related Topics, McGraw-Hill, New York, (1964).

J. J. Nieto and R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), 223-239.

S. B. Jr. Nadler, Multivalued contraction mappings. Pac. J. Math. 30 (1969), 475-487.

G. Prasad, Coincidence points of relational ψ-contractions and an application, Afrika Mathematica 32, no. 6-7 (2021), 1475-1490.

G. Prasad, Fixed points of Kannan contractive mappings in relational metric spaces, J. Anal. 29, no. 3 (2021), 669-684.

G. Prasad and H. Işık, On solution of boundary value problems via weak contractions, J. Funct. Spaces 2022 (2022), Article ID 6799205.

A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132, no. 5 (2004), 1435-1443.

B. Samet and M. Turinici, Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications, Commun. Math. Anal. 13, no. 2 (2012), 82-97.

B. Schweizer, A. Sklar and E. Thorp, The metrization of statistical metric spaces, Pac. J. Math. 10 (1960), 673-675.

B. Schweizer and Sklar, Probabilistic Metric Spaces, North-Holland, New York (1983).

M. Turinici, Fixed points for monotone iteratively local contractions, Demonstr. Math. 19, no. 1 (1986), 171-180.

M. Turinici, Ran and Reuring's theorems in ordered metric spaces, J. Indian Math. Soc. 78 (2011), 207-214.




How to Cite

G. Prasad, S. Deshwal, and R. K. Srivastav, “Fixed points of set-valued mappings in Menger probabilistic metric spaces endowed with an amorphous binary relation”, Appl. Gen. Topol., vol. 24, no. 2, pp. 307–322, Oct. 2023.



Regular Articles