Stavovi o nepokretnoj tački na prostorima sa nedeterminističkom metrikom

Abstract

From Freschet’s definition of metric space in the early twentieth century, the various generalizations of metric spaces such as Menger spaces, fuzzy metric spaces, intuitionistic fuzzy metric spaces, L-fuzzy metric spaces and others, were obtained. On the other hand, the fundamental theorem in the theory of fixed points in metric spaces represents Banach contractive Principle. This theorem, which for some special cases has already been considered by a Liouville, Picard and Goursat, in the past eight decades till today, has found many important applications and has been generalized in many directions. One of them is the introduction of nonlinear contractive condition by Rakotch in 1962, and then by Boyd and Wong in 1969. The field of research in this dissertation belongs to the nonlinear functional analysis, and partly to the topology and applied mathematics. Since the basic structures on which we conduct our research are the spaces with nondeterministic distances, the obtained results have applications in modern scientific research in physics. In this dissertation,the nonlinear contractive condition which is associated with different forms of the boundness of sets in the spaces with nondeterministic distances, will be discussed. In addition, we will be discussing the existence and uniqueness of fixed and common fixed points, under conditions of weak commutativity and compatibility for different types of mappings. The hypothesis of the proceeding is that with qualitatively characterization for the domain of mapping, we can relax the conditions of nonlinear contraction and thereby obtain information about the existence or absence of fixed points for this mapping. A special place in this dissertation takes convexity and topological characterization of completeness and compactness of the spaces with nondeterministic distances. These results are associated with the existence of fixed and common fixed points for non-expansive mappings, with nonlinear contractive condition. The main question is how the convex structure and the previously mentioned results, can be transferred to the spaces with nondeterministic distances.