A GENERALIZATION OF A CONTRACTION PRINCIPLE IN PROBABILISTIC METRIC SPACES. PART II

A fixed point theorem concerning probabilistic contractions satisfying an implicit rela-tion, which generalizes a well-known result of Hadžić, is proved. 1. Preliminaries In this section we recall some useful facts from the probabilistic metric spaces theory. For more details concerning this problematic we refer the reader to the books [1, 3, 9]. 1.1. t-norms. A triangular norm (shortly t-norm) is a binary operationT: [0,1]×[0,1]→ [0,1]: = I which is commutative, associative, monotone in each place, and has 1 as the unit element. Basic examples are TL: I × I → I, TL(a,b) =Max(a+ b − 1,0) (Łukasiewicz t-norm), TP(a,b) = ab, and TM(a,b) =Min{a,b}. We also mention the following families of t-norms: (i) Sugeno-Weber family (TSWλ)λ∈(−1,∞), defined by T SW λ =max(0,(x + y − 1 + λxy)/ (1+ λ)), (ii) Domby family (TDλ)λ∈(0,∞), defined by T D λ = (1 + (((1 − x)/x)λ + ((1 − y)/ y)λ)1/λ)−1, (iii) Aczel-Alsina family (TAAλ)λ∈(0,∞), defined by T AA λ = e−(|logx|λ+|log y|λ)1/λ. Definition 1.1 [2, 3]. It is said that the t-norm T is of Hadžić-type (H-type for short) and T ∈ if the family {Tn}n∈N of its iterates defined, for each x in [0,1], by T0(x) = 1, Tn+1(x) = T(Tn(x),x), ∀n ≥ 0, (1.1) is equicontinuous at x = 1, that is, ∀ε ∈ (0,1) ∃δ ∈ (0,1) such that x> 1 − δ = ⇒ Tn(x)> 1 − ε, ∀n ≥ 1. (1.2) There is a nice characterization of continuous t-norms T of the class [8]