Probabilistic B-contraction principles for single-valued mappings

The first fixed point theorem in probabilistic metric spaces was proved by Sehgal and Barucha-Reid [272] for mappings f : S → S, where (S, F, T_M) is a Menger space. Further development of the fixed point theory in a more general Menger space (S, F, T) was connected with investigations of the structure of the t-norm T. Very soon the problem was in some sense completely solved. Namely, if we restrict ourselves to complete Menger spaces (S, F, T), where T is a continuous t-norm, then any probabilistic q-contraction f : S → S has a fixed point if and only if the t-norm is of H-type.