An Essential Remark on Relation-Theoretic Metrical Fixed Point Results

Tanusri Senapati, Lakshmi Kanta Dey


In this short note, we notice that the relation-theoretic metrical fixed point results are equivalent with the fixed point results in \(\alpha\)-complete metric spaces. We observe that any arbitrary binary relation on a non empty set \(X\) can be defined in terms of an arbitrary real valued function defined on \(X\times X\). Consequently we show that the results of Alam and Imdad (J. Fixed Point Theory Appl. 17(4) (2015)) and Ahmadullah et al. (arXiv) do not contribute anything new in the literature.


Complete metric space; Binary relation; Fixed point

Full Text:



M. Ahmadullah, M. Imdad and R. Gubran, Relation-theoretic metrical fixed point theorems under nonlinear contractions, arXiv:1611.04136 (2016).

A. Alam and M. Imdad, Relation-theoretic contractive principle, J. Fixed Point Theory Appl. 17(4) (2015), 693–702.

N. Hussain, M.H. Shah, A.A. Harandi and Z. Akhtar, Common fixed point theorem for generalized contractive mappings with applications, Fixed Point Theory Appl. 2013 (2013), Article ID 169, 1 – 17.

B. Kolman, R.C. Busby and S. Ross, Relation algebras, Studies in Logic and Foundations of Mathematics, 150, Elsevier B.V., Amsterdam (2006).

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

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(5) (2004), 1435–1443.

B. Samet, C. Vetro and P. Vetro, Fixed point theorems for (alpha)-(psi)-contractive type mappings, Nonlinear Anal. 75 (2012), 2154 – 2165.

M. Turinici, Abstract comparison principles and multivariabe Gronwall-Bellman inequalities, J. Math. Anal. Appl. 117(1) (1986), 100 – 127.

M. Turinici, Fixed points for monotone iteratively local contractions, Demonstr. Math. 19(1) (1986), 171 – 180.



  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905