Dynamic Processes, Fixed Points, Endpoints, Asymmetric Structures, and Investigations Related to Caristi, Nadler, and Banach in Uniform Spaces

Abstract

In uniform spaces $\left(X, \mathcal{D}\right)$ with symmetric structures determined by the $\mathcal{D}$-families of pseudometrics which define uniformity in these spaces, the new symmetric and asymmetric structures determined by the $\mathcal{J}$-families of generalized pseudodistances on $X$ are constructed; using these structures the set-valued contractions of two kinds of Nadler type are defined and the new and general theorems concerning the existence of fixed points and endpoints for such contractions are proved. Moreover, using these new structures, the single-valued contractions of two kinds of Banach type are defined and the new and general versions of the Banach uniqueness and iterate approximation of fixed point theorem for uniform spaces are established. Contractions defined and studied here are not necessarily continuous. One of the main key ideas in this paper is the application of our fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies in uniform spaces with structures determined by $\mathcal{J}$-families. Results are new also in locally convex and metric spaces. Examples are provided.