Fixed Point Theorems for Multivalued Contractions of Hicks Type in Probabilistic Metric Spaces

A probabilistic semi-metric space (S, F ) is said to be of class H ([5]) if there exists a metric d on S such that, for t > 0, d(p, q) < t ⇔ Fpq(t) > 1− t. We will prove that (S, F ) is of class H iff the mapping K, defined on S × S by K (p, q) = sup{t ≥ 0 | t ≤ 1− Fpq(t)} is a metric on S. Two fixed point theorems for multivalued contractions in probabilistic metric spaces are also proved. Incidentally, the equality of two well-known probabilistic metrics is obtained. 1 Preliminaries. In this section we recall some notions of probabilistic metric spaces theory that will be used in the sequel. For more details on this topic we refer the reader to the books [2] and [13]. The class of distribution functions, denoted by ∆+, is the class of all functions F : [0,∞) → [0, 1] with the properties: a) F (0) = 0; b) F is increasing; c) F is left continuous on (0,∞). A special element of ∆+ is the function ε0, defined by ε0(t) = { 0, if t = 0 1, if t > 0 . If X is a nonempty set, a mapping F : X ×X −→ ∆+ is called a probabilistic distance on X and F (x, y) is denoted by Fxy. Let X be a nonempty set and F be a probabilistic distance. The pair (X,F ) is called a PSM space if the following axioms (PM0) and (PM1) are satisfied: (PM0) : Fxy = ε0 iff x = y (PM1) : Fxy = Fyx ∀x, y ∈ X. A Menger space under the t-norm T ([13]) is a triple (X,F, T ) where (X,F ) is a PSM space and the triangle inequality (PM2M ) Fxy(t+ s) ≥ T (Fxz(t), Fzy(s)), ∀x, y, z ∈ X, ∀t, s > 0