Abstract
In this paper, we focus on Menger probabilistic metric spaces of hyperbolic type, extending the notion of hyperbolic metric spaces introduced by W. A. Kirk to the probabilistic framework initiated by Karl Menger and further developed by B. Schweizer and A. Sklar. We explore the fundamental properties of these spaces and provide illustrative examples, including a concrete construction on the positive real line equipped with the Lukasiewicz t-norm. This example shows how hyperbolic-type convexity can be formulated in the context of a Menger probabilistic metric space. A central result of this paper is the proof of a key lemma concerning the asymptotic behavior of two bounded sequences in Menger probabilistic metric spaces of hyperbolic type. As a main application, we introduce the concept of fundamentally nonexpansive mappings and prove a fixed point theorem for such mappings defined on nonempty compact convex subsets of Menger probabilistic metric spaces of hyperbolic type. The obtained results extend classical fixed point theorems from metric and hyperbolic spaces to the probabilistic setting and contribute to the development of nonlinear analysis in probabilistic metric spaces.
Key words: Menger probabilistic metric space, convex structure, hyperbolic type, fixed point.