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

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

Quasi-Triangular Spaces, Pompeiu-Hausdorff Quasi-Distances, and Periodic and Fixed Point Theorems of Banach and Nadler Types

Let C={Cα}α∈A∈[1;∞)A, A-index set. A quasi-triangular space (X,PC;A) is a set X with family PC;A={pα:X2→[0,∞),  α∈A} satisfying ∀α∈A  ∀u,v,w∈X  {pα(u,w)≤Cα[pα(u,v)+pα(v,w)]}. For any PC;A, a left (right) family JC;A generated by PC;A is defined to be JC;A={Jα:X2→[0,∞),  α∈A}, where ∀α∈A  ∀u,v,w∈X  {Jα(u,w)≤Cα[Jα(u,v)+Jα(v,w)]} and furthermore the property ∀α∈A  {limm→∞pα(wm,um)=0}  (∀α∈A  {limm→∞pα(um,wm)=0}) holds whenever two sequences (um:m∈N) and (wm:m∈N) in X satisfy ∀α∈A  {limm→∞supn>mJα(um,un)=0 and limm→∞Jα(wm,um)=0}  (∀α∈A  {limm→∞supn>mJα(un,um)=0 and limm→∞Jα(um,wm)=0}). In (X,PC;A), using the left (right) families JC;A generated by PC;A (PC;A is a special case of JC;A), we construct three types of Pompeiu-Hausdorff left (right) quasi-distances on 2X; for each type we construct of left (right) set-valued quasi-contraction T:X→2X, and we prove the convergence, existence, and periodic point theorem for such quasi-contractions. We also construct two types of left (right) single-valued quasi-contractions T:X→X and we prove the convergence, existence, approximation, uniqueness, periodic point, and fixed point theorem for such quasi-contractions. (X,PC;A) generalize ultra quasi-triangular and partiall quasi-triangular spaces (in particular, generalize metric, ultra metric, quasi-metric, ultra quasi-metric, b-metric, partial metric, partial b-metric, pseudometric, quasi-pseudometric, ultra quasi-pseudometric, partial quasi-pseudometric, topological, uniform, quasi-uniform, gauge, ultra gauge, partial gauge, quasi-gauge, ultra quasi-gauge, and partial quasi-gauge spaces)

Contractivity of Leader type and fixed points in uniform spaces with generalized pseudodistances

AbstractRecently, 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

Sense of coherence as a personality predictor of the quality of life in men and women after myocardial infarction

Background: The vast majority of research on the quality of life (QoL) after myocardial infarction (MI) concentrates on such factors as: the type and course of MI, methods and stage of treatment or the patient&#8217;s occupational and family status. Drawing from general psychological knowledge we may assume that some individual factor, especially personality, is also a significant contributor. The present study focused on a specific personality dimension: sense of coherence (SOC). It is defined as a global life orientation to perceive life as comprehensible (rational, predictable and structured), manageable (adequate and sufficient resources to overcome adversities are perceived as available) and meaningful (the demands created by adversities are seen as challenges and worthy of engagement). Aim: To compare the QoL one year after MI in men and women and to examine the role of SOC as a predictor of the QoL one year after MI, in groups of men and women. Methods: The study group consisted of 83 participants (including 34 women), aged 35&#8211;59 (50.2 +- 6.2) years. They had a history of uncomplicated MI and were referred for post-hospitalisation cardiac rehabilitation in the sanatorium setting. SOC was measured with the Polish version of SOC-13 by A. Antonovsky. The QoL was evaluated with the MacNew questionnaire by N.B. Oldridge and L. Lim. The SOC was assessed during the stay at the heart centre. One year after their MI the participants completed the QoL questionnaires (sent to them by post). Results: Men in comparison to women demonstrated stronger SOC (p < 0.004) and a better QoL in all dimensions: physical (p < 0.001), emotional (p < 0.001), social (p < 0.001) and as a global score (p < 0.001). The SOC turned out to be a significant predictor of the QoL one year after MI even after controlling for demographic and medical factors. Its predictive value was higher for women. Conclusions: Research on the QoL in patients after MI should take into account personality factors. The SOC is a significant predictor, especially in women. Persons after MI scoring low on SOC at the early stage of rehabilitation should receive psychological intervention.Background: The vast majority of research on the quality of life (QoL) after myocardial infarction (MI) concentrates on such factors as: the type and course of MI, methods and stage of treatment or the patient&#8217;s occupational and family status. Drawing from general psychological knowledge we may assume that some individual factor, especially personality, is also a significant contributor. The present study focused on a specific personality dimension: sense of coherence (SOC). It is defined as a global life orientation to perceive life as comprehensible (rational, predictable and structured), manageable (adequate and sufficient resources to overcome adversities are perceived as available) and meaningful (the demands created by adversities are seen as challenges and worthy of engagement). Aim: To compare the QoL one year after MI in men and women and to examine the role of SOC as a predictor of the QoL one year after MI, in groups of men and women. Methods: The study group consisted of 83 participants (including 34 women), aged 35&#8211;59 (50.2 &#177; 6.2) years. They had a history of uncomplicated MI and were referred for post-hospitalisation cardiac rehabilitation in the sanatorium setting. SOC was measured with the Polish version of SOC-13 by A. Antonovsky. The QoL was evaluated with the MacNew questionnaire by N.B. Oldridge and L. Lim. The SOC was assessed during the stay at the heart centre. One year after their MI the participants completed the QoL questionnaires (sent to them by post). Results: Men in comparison to women demonstrated stronger SOC (p < 0.004) and a better QoL in all dimensions: physical (p < 0.001), emotional (p < 0.001), social (p < 0.001) and as a global score (p < 0.001). The SOC turned out to be a significant predictor of the QoL one year after MI even after controlling for demographic and medical factors. Its predictive value was higher for women. Conclusions: Research on the QoL in patients after MI should take into account personality factors. The SOC is a significant predictor, especially in women. Persons after MI scoring low on SOC at the early stage of rehabilitation should receive psychological intervention

Vergence eye movements in bipolar disorder

Aim. With respect to bipolar disorder (BD), previous research have demonstrated saccadic eye movements abnormalities, manifested mainly as an increase in reaction time (latency) in both prosaccadic and antisaccadic task. So far, there were no studies related to vergence eye movements in subjects with BD. Our primary aim was to evaluate vergence tracking performance in this clinical group.  Methods. 30 patients with BD in remission and 23 healthy controls were enrolled. Subjects underwent optometric examination where near point of convergence was measured by the use of Wolff Wand. Instrumented convergence measurements were performed using infrared eye tracker and dedicated vergence stimuli generator. Results. BD patients presented significantly higher average error between eyes’ convergence and convergence required to fixate the target and higher number of saccadic intrusions compared with healthy controls group. Principal component analysis performed on oculometric parameters revealed differences between BD patients and healthy controls. Significant correlations between the vergence disturbances and saccadic intrusions were found. Conclusions. BD patients showed the alterations of the vergence eye movements similar to the disturbances of eye movements in the fronto-parallel plane. While the abnormalities of vergence eye movements in some mental disorders have been reported, we have for the first time objectively measured this phenomenon in BD