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

Research ArticleIn uniform spaces (...) with symmetric structures determined by the D-families of pseudometrics which define uniformity in these spaces, the new symmetric and asymmetric structures determined by the J-families of generalized pseudodistances on (...) 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 J-families. Results are new also in locally convex and metric spaces. Examples are provided

A two-parameter control for contractive-like multivalued mappings

We propose a general approach to defining a contractive-like multivalued mappings $F$ which avoids any use of the Hausdorff distance between the sets $F(x)$ and $F(y)$. Various fixed point theorems are proved under a two-parameter control of the distance function $d_{F}(x)=dist(x,F(x))$ between a point $x \in X$ and the value F(x) \ss X. Here, both parameters are numerical functions. The first one \a\,:[0,+\i)\rightarrow [1,+\i) controls the distance between $x$ and some appropriate point $y \in F(x)$ in comparison with $d_{F}(x)$, whereas the second one \b\,:[0,+\i)\rightarrow [0,1) estimates $d_{F}(y)$ with respect to $d(x,y)$. It appears that the well harmonized relations between \a and \b are sufficient for the existence of fixed points of $F$. Our results generalize several known fixed-point theorems

On best proximity points for set-valued contractions of Nadler type with respect to b-generalized pseudodistances in b-metric spaces

In this paper, in b-metric space, we introduce the concept of b-generalized pseudodistance which is an extension of the b-metric. Next, inspired by the ideas of Nadler (Pac. J. Math. 30:475-488, 1969) and Abkar and Gabeleh (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 107(2):319-325, 2013), we define a new set-valued non-self-mapping contraction of Nadler type with respect to this b-generalized pseudodistance, which is a generalization of Nadler’s contraction. Moreover, we provide the condition guaranteeing the existence of best proximity points for T : A → 2B. A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the error inf{d(x, y) : y ∈ T(x)}, and hence the existence of a consummate approximate solution to the equation T(x) = x. In other words, the best proximity points theorem achieves a global optimal minimum of the map x → inf{d(x; y) : y ∈ T(x)} by stipulating an approximate solution x of the point equation T(x) = x to satisfy the condition that inf{d(x; y) : y ∈ T(x)} = dist(A; B). The examples which illustrate the main result given. The paper includes also the comparison of our results with those existing in the literature

On metric-preserving functions and fixed point theorems

Kirk and Shahzad have recently given fixed point theorems concerning local radial contractions and metric transforms. In this article, we replace the metric transforms by metric-preserving functions. This in turn gives several extensions of the main results given by Kirk and Shahzad. Several examples are given. The fixed point sets of metric transforms and metric-preserving functions are also investigated.Comment: Submitte