Probabilistic Generalized Metric Spaces and Nonlinear Contractions

What is it about?

In 1942 K. Menger, who had played a major role in the development of the theory of metric spaces proposed a probabilistic generalization of this theory. Specifically, he proposed replacing the number $d(p, q)$ by a real function $F_{pq}$ whose value $F_{pq}(x)$ for any real number $x$ is interpreted as the probability that the distance between $p$ and $q$ is less than $x$. The probabilistic metric spaces are assumed to satisfy axioms which are quite similar to the axioms metric spaces. In the sequel to such developments, different probabilistic generalization of the triangle inequality had been and the study of these triangle inequalities has been a central theme in the development of the theory of probabilistic metric spaces. This exciting context intrigued and pushed us to introduce the concept of probabilistic generalized metric space by replacing the triangle inequality with a weaker assumption ‘’quadrilateral inequality’’ . So that the results obtained for such rich spaces become more viable in different directions of applications.

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Why is it important?

In a probabilistic generalized metric space (pgms) $(M, F, τ )$ even though $τ$ is continuous the following assertions hold: (A) in general, a pgms is not a Hausdorff space ; (B) in general, the concepts convergent sequence and Cauchy sequence are independent ; (C) in general, a pgm $F$ is not continuous; (D) in general, a pgm $F$ does not induce a topology on $M$.

Perspectives