This work is for giving the probabilistic aspect to the knownb-metric spaces (Czerwik in Atti Semin. Mat. Fis. Univ. Modena 46(2):263-276, 1998), which leads to studying the fixed point property for nonlinear contractions in this new class of spaces.

We give a probabilistic generalization of the theory of generalized metric spaces [2]. Then, we prove a fixed point theorem for a self-mapping of a probabilistic generalized metric space, satisfying the very general nonlinear contraction condition without the assumption that the space is Hausdorff.

In this article, we extend the concept of multivalued $\omega$-contraction mappings within the framework of b-Menger spaces. We introduce a novel fixed point theorem specific to these mappings, significantly broadening the existing body of knowledge in this area. Our primary result directly leads to the derivation of an equivalent fixed point theorem for fuzzy b-metric spaces, showcasing the versatility and applicability of our findings in more generalized and complex metric structures. Furthermore, we provide a concrete application of the principal theorem in the context of ordinary b-metric spaces, demonstrating the practical implications and effectiveness of our theoretical advancements. This research contributes to the deeper understanding and potential applications of fixed point theory in various scientific and engineering domains, where probabilistic and fuzzy structures play a critical role.

In this paper, we study the problem of the existence of fixed points for weak-Hicks contractions in generalized Menger spaces. We will provide some examples to demonstrate the applicability and effectiveness of the theories and methods presented in the paper. 2010 Mathematics Subject Classification. Primary 54E70, 54H25. Secondary 47S50, 34B15.

This article introduces quiver spaces, a class of structures that generalize metric spaces and multiplicative metric spaces . They provide a representation framework that is both faithful and general for the principle of the identity of indiscernibles, avoiding the limitations of the classical metric and emphasizing a purely relational and algebraic perspective. Several topologies naturally associated with quiver spaces are also introduced and studied.