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

Kazimierz Włodarczyk,Robert Plebaniak

doi:10.1155/2015/942814

Abstract

In uniform spacesX, Dwith symmetric structures determined by theD-families of pseudometrics which define uniformity in these spaces, the new symmetric and asymmetric structures determined by theJ-families of generalized pseudodistances onXare 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 byJ-families. Results are new also in locally convex and metric spaces. Examples are provided.

Highlights

The concepts of the symmetric and asymmetric structures became established and investigated in mathematics and in theoretical computer science and are some creative ideas in fixed point theory by which some fascinating results have been achieved
Let (X, D) be a uniform space with uniformity defined by a saturated family D = {dα : α ∈ A} of pseudometrics dα : X2 → [0; ∞), α ∈ A, uniformly continuous on X2 (D-family, for short); here A is a nonempty index set
It was discovered that the J-families of generalized pseudodistances defined below generalize: metrics d, distances of Tataru [1], w-distances of Kada et al [2], τ-distances of Suzuki [3], and τ-functions of Lin and Du [4] in metric spaces (X, d) and D-families of pseudometrics and distances of Valyi [5] in uniform spaces (X, D)

Summary

Introduction

The concepts of the symmetric and asymmetric structures became established and investigated in mathematics and in theoretical computer science and are some creative ideas in fixed point theory by which some fascinating results have been achieved. We use notations and auxiliary Theorems 17 and 20 above in proving the following basic fixed point and endpoint theorem for set-valued contractions with respect to J ∈ J(X,D) (of Nadler-type) in uniform spaces (X, D). As a corollary of the above Theorems 17, 20, and 21 we have the following new fixed point and endpoint theorem for set-valued contractions with respect to D-families (of Nadletype) in uniform spaces (X, D). We use notations above and Theorem 21 in proving the following new fixed point theorem for single-valued contractions with respect to J ∈ J(X,D) (of Banach-type) in uniform spaces (X, D). As a corollary from Theorem 31 and its proof we get the following fixed point theorem for single-valued contractions with respect to J ∈ J(X,d) (of Banach-type) in metric spaces (X, d).

Examples Illustrating the Results

Comparisons of Our Results with Nadler’s and Banach’s Results