On φ-contractions in probabilistic and fuzzy metric spaces

In this paper, by means of weakening conditions of the gauge function φ, a new fixed point theorem for probabilistic φ-contraction in Menger probabilistic metric spaces with a t-norm of H-type is established. This theorem improves and generalizes the recent results of Ćirić (2010) [4], Jachymski (2010) [12] and Xiao et al. (2013) [25]. By using the theorem, we also obtain some important corollaries and the fixed point theorems for φ-contraction in fuzzy metric spaces.