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In quasi-pseudometric spaces (not necessarily sequentially complete), we continue the research on the quasi-generalized pseudodistances. We introduce the concepts of semiquasiclosed map and contraction of Nadler type with respect to generalized pseudodistances. Next, inspired by Abkar and Gabeleh we proved new best proximity point theorem in a quasi-pseudometric space. 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: Leaderʼs fixed point theorem – being more general as some Banach, Boyd and Wong, Browder, Burton, Caccioppoli, Dugundji and Granas, Geraghty, Krasnoselʼskiĭ et al., Matkowski, Meir and Keeler, Mukherjea, Rakotch, Tasković, Walter and othersʼ results – have played a great role in metric fixed point theory; in the literature the investigations of periodic points of contractions of Leader or Leader type are not known. We want to show how the introduced here generalized quasi-pseudodistances in quasi-gauge spaces can be used, in a natural way, to define contractions of Leader type and to obtain, for these contractions, the periodic and fixed point theorems without Hausdorff and sequentially complete assumptions about these spaces and without complete graph assumptions about these contractions, which was not done in the previous publications on this subject. The definitions, results and methods presented here are new for maps in quasi-gauge, topological, quasi-pseudometric and quasi-metric spaces. Examples are provided. [Copyright &y& Elsevier]

Abstract: Recently, Jachymski and Jóźwik proved that among various classes of contractions which are introduced and studied in the metric fixed point theory, the Leader contractions are greatest general contractions. In this article, we want to show how generalized pseudodistances in uniform spaces can be used to obtain new and general results of Leader type without complete graph assumptions about maps and without sequentially complete assumptions about spaces, which was not done in the previous publications on this subject. The definitions, results and methods presented here are new for maps in uniform and locally convex spaces and even in metric spaces. Examples showing a difference between our results and the well-known ones are given. [Copyright &y& Elsevier]

Abstract: In uniform spaces, not necessarily sequentially complete, using the concept of the -family of generalized pseudodistances, the fixed point theorem of Subrahmanyam type is established. The result and method of investigations presented here are new for maps in uniform and locally convex spaces and even in metric spaces. [ABSTRACT FROM AUTHOR]

Abstract: The concepts of cone pseudometrics, cone uniform spaces generated by these pseudometrics, cone pseudodistances, set-valued quasi-asymptotic contractions with respect to these pseudodistances and cone closed maps are introduced and studied. Conditions guaranteeing the existence and uniqueness of endpoints (stationary points) of these contractions and conditions ensuring that for each starting point each generalized sequence of iterations of these contractions (in particular, each dynamic process) converges and then the limit is an endpoint are all established. Also the concept of the cone locally convex space as a special case of the cone uniform space is introduced and examples of quasi-asymptotic contractions in cone metric spaces are constructed. The definitions, results, ideas and methods are new for set-valued dynamic systems in cone uniform, cone locally convex and cone metric spaces and even for single-valued maps in these spaces. [Copyright &y& Elsevier]

Abstract: For set-valued dynamic systems in uniform spaces we introduce the concept of quasi-asymptotic contractions with respect to some generalized pseudodistances, describe a method which we use to establish general conditions guaranteeing the existence and uniqueness of endpoints (stationary points) of these contractions and exhibit conditions such that for each starting point each generalized sequence of iterations (in particular, each dynamic process) converges and the limit is an endpoint. The definition, result, ideas and techniques are new for set-valued dynamic systems in uniform, locally convex and metric spaces and even for single-valued maps. [Copyright &y& Elsevier]

Abstract: In this paper, the concept of the set-valued dynamical systems of contractions of Meir–Keeler type in uniform spaces is introduced and conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions are established. The definition and the result presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our result and the well-known ones. [Copyright &y& Elsevier]