An Essential Remark on Relation-Theoretic Metrical Fixed Point Results - Senapati, Tanusri; Dey, Lakshmi Kanta (Communications in Mathematics and Applications, 2018)
In this short note, we notice that the relation-theoretic metrical fixed point results are equivalent with the fixed point results in \(α\)-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 \(X× X\). 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.
Keywords: Complete metric space; Binary relation; Fixed point