This work is for giving the probabilistic aspect to the known b-metric spaces (Czerwik in Atti Semin. Mat. Fis. Univ. Modena 46(2):263-276, 1998), which leads to studying the fixed point property for nonlinear contractions in this new class of spaces.MSC: Primary 54E70; 54H25; secondary 47S50; 34B15

Abstract.We give a probabilistic generalization of the theory of generalized metric spaces [2]. Then, we prove a fixed point theorem for a self-mapping of a probabilistic generalized metric space, satisfying the very general nonlinear contraction condition without the assumption that the space is Hausdorff.

We prove, in b-Menger spaces [9] the existence of common fixed point for nonexpansive mappings in fully convex b-Menger space by using the normal structure property. We provide examples to analyze and illustrate our main results.

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In this work we have shown that an affirmative answer was already given in [1, 5] to the question raised in [4] and have extended a fixed point theorem by L.Ćirić [4] to a larger class of PM spaces. In the final part of the paper we have shown that the result can be yet improved by a common fixed point theorem for a semigroups of ϕ-probabilistic contractions.