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

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

Hausdorff quasi-distances, periodic and fixed points for Nadler type set-valued contractions in quasi-gauge spaces

In a quasi-gauge space (X,P) with quasi-gauge , using the left (right) -families of generalized quasi-pseudodistances on X (-families on X generalize quasi-gauge ), the left (right) quasi-distances DL−Jη ( DR−Jη) of Hausdorff type on 2X are defined, η∈{1,2,3}, the three kinds of left (right) set-valued contractions of Nadler type are constructed, and, for such contractions, the left (right) -convergence of dynamic processes starting at each point w0∈X is studied and the existence and localization of periodic and fixed point results are proved. As implications, two kinds of left (right) single-valued contractions of Banach type are defined, and, for such contractions, the left (right) -convergence of Picard iterations starting at each point w0∈X is studied, and existence, localization, periodic point, fixed point and uniqueness results are established. Appropriate tools and ideas of studying based on -families and also presented examples showed that the results: are new in quasi-gauge, topological, gauge, quasi-uniform and quasi-metric spaces; are new even in uniform and metric spaces; do not require completeness and Hausdorff properties of the spaces (X,P), continuity of contractions, closedness of values of set-valued contractions and properties DL−Jη(U,V)=DL−Jη(V,U) ( DR−Jη(U,V)=DR−Jη(V,U)) and DL−Jη(U,U)=0 ( DR−Jη(U,U)=0), η∈{1,2,3}, U,V∈2X; provide information concerning localizations of periodic and fixed points; and substantially generalize the well-known theorems of Nadler and Banach types.Publikacja w ramach programu Springer Open Choice/Open Access finansowanego przez Ministerstwo Nauki i Szkolnictwa Wyższego i realizowanego w ramach umowy na narodową licencję akademicką na czasopisma Springer w latach 2010-2014

Multi-valued F-contractions and the solution of certain functional and integral equations

Wardowski [Fixed Point Theory Appl., 2012:94] introduced a new concept of contraction and proved a fixed point theorem which generalizes Banach contraction principle. Following this direction of research, we will present some fixed point results for closed multi-valued F-contractions or multi-valued mappings which satisfy an F-contractive condition of Hardy-Rogers-type, in the setting of complete metric spaces or complete ordered metric spaces. An example and two applications, for the solution of certain functional and integral equations, are given to illustrate the usability of the obtained results