In the present article, first the concept of a T-contraction with respect to a wt-distance in b-metric spaces is introduced and second the equivalence of the existence and uniqueness of fixed points of mappings satisfying usual contractions and their corresponding T-contractions is discussed. Then we extend our discussion into a T-contraction in relation to a generalized c-distance and show in particular that the fixed point results of Nazir et al. with respect to a T-contraction [Afr. Mat. (2024),35:69] are equivalent to their usual contractions although they can give good applications and solve interesting mathematical problems. A few examples, applications and notes are added to illustrate the importance of the main results, as well as subsequent issues on this concept.
