Probabilistic fractals and attractors in Menger spaces

Probabilistic fractals are considered as self-similar sets for contraction mappings in hyperspaces of the Menger probabilistic metric spaces. The completeness of hyperspaces of all closed, or closed and bounded, or compact nonempty subsets with respect to the probabilistic Hausdorff metric is studied in some general conditions of triangular norm. Using the properties of hyperspaces and triangular norms of H-type, several existence results of attractors are obtained for a finite family and a countable family of probabilistic nonlinear contractions, respectively. Our results are generalizations in many aspects.