Existence results for a system of nonlinear operator equations and block operator matrices in locally convex spaces

The purpose of this paper is to prove some fixed point results dealing with a system of nonlinear equations defined in an angelic Hausdorff locally convex space ( X , { | ⋅ | p } p ∈ Λ ) (X,\{\lvert\,{\cdot}\,\rvert_{p}\}_{p\in\Lambda}) having the 𝜏-Krein–Šmulian property, where 𝜏 is a weaker Hausdorff locally convex topology of 𝑋. The method applied in our study is connected with a family Φ Λ τ \Phi_{\Lambda}^{\tau} -MNC of measures of weak noncompactness and with the concept of 𝜏-sequential continuity. As a special case, we discuss the existence of solutions for a 2 × 2 2\times 2 block operator matrix with nonlinear inputs. Furthermore, we give an illustrative example for a system of nonlinear integral equations in the space C ⁢ ( R + ) × C ⁢ ( R + ) C(\mathbb{R}^{+})\times C(\mathbb{R}^{+}) to verify the effectiveness and applicability of our main result.

Leray–Schauder Fixed Point Theorems for Block Operator Matrix with an Application

In this paper, we establish some new variants of Leray–Schauder-type fixed point theorems for a 2 × 2 block operator matrix defined on nonempty, closed, and convex subsets Ω of Banach spaces. Note here that Ω need not be bounded. These results are formulated in terms of weak sequential continuity and the technique of De Blasi measure of weak noncompactness on countably subsets. We will also prove the existence of solutions for a coupled system of nonlinear equations with an example.

Continuity in general practice and hospitalization patterns: an observational study

Background High continuity of care is a key feature of strong general practice. This study aimed to assess the effect of a programme for enhancing strong general practice care on the continuity of care in Germany. The second aim was to assess the effect of continuity of care on hospitalization patterns. Methods We performed an observational study in Germany, involving patients who received a strong general practice care programme (n=1.037.075) and patients who did not receive this programme (n=723.127) in the year 2017. We extracted data from a health insurance database. The cohorts were compared with respect to three measures of continuity of care (Usual Provider Index, Herfindahl Index, and the Sequential Continuity Index), adjusted for patient characteristics. The effects of continuity in general practice on the rates of hospitalization, rehospitalization, and avoidable hospitalization were examined in multiple regression analyses. Results Compared to the control cohort, continuity in general practice was higher in patients who received the programme (continuity measures were 12.47 to 23.76% higher, P< 0.0001). Higher continuity of care was independently associated with lowered risk of hospitalization, rehospitalization, and avoidable hospitalization (relative risk reductions between 2.45 and 9.74%, P< 0.0001). Higher age, female sex, higher morbidity (Charlson-index), and home-dwelling status (not nursing home) were associated with higher rates of hospitalization. Conclusion Higher continuity of care may be one of the mechanisms underlying lower hospitalization rates in patients who received strong general practice care, but further research is needed to examine the causality underlying the associations.

Fixed point theorems in WC-Banach algebras and their applications to infinite systems of integral equations

The paper is devoted to prove a few fixed point theorems for operators acting in WC-Banach algebras and satisfying some conditions expressed in terms of a generalized Lipschitz continuity and measures of weak noncompactness. Moreover, the assumptions imposed on the mentioned operators are formulated with help of weak topology and weak sequential continuity. Our fixed point results will be illustrated by proving the existence of solutions of an infinite system of nonlinear integral equations.

Relationship between Continuity of Care in the Multidisciplinary Treatment of Patients with Diabetes and Their Clinical Results

Multidisciplinary treatment and continuity of care throughout treatment are important for ensuring metabolic control and avoiding complications in diabetic patients. This study examines the relationship between continuity of care of the treating disciplines and clinical evolution of patients. Data from 1836 adult patients experiencing type 2 diabetes mellitus were analyzed, in a period between 12 and 24 months. Continuity was measured by using four well known indices: Usual Provider Continuity (UPC), Continuity of Care Index (COCI), Herfindahl Index (HI), and Sequential Continuity (SECON). Patients were divided into five segments according to metabolic control: well-controlled, worsened, moderately decompensated, highly decompensated, and improved. Well-controlled patients had higher continuity by physicians according to UPC and HI indices (p-values 0.029 and <0.003), whereas highly decompensated patients had less continuity in HI (p-value 0.020). Continuity for nurses was similar, with a greater continuity among well-controlled patients (p-values 0.015 and 0.001 for UPC and HI indices), and less among highly decompensated patients (p-values 0.004 and <0.001 for UPC and HI indices). Improved patients had greater adherence to the protocol than those who worsened. The SECON index showed no significant differences across the disciplines. This study identified a relationship between physicians and nurse’s continuity of care and metabolic control in patients with diabetes, consistent with qualitative findings that highlight the role of nurses in treatment.

On G-sequential continuity

Let X be a first countable Hausdorff topological group. The limit of a sequence in X defines a function denoted by lim from the set of all convergent sequences to X. This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing lim with an arbitrary linear functional G defined on a linear subspace of the vector space of all real sequences. Recently ?akall? has extended the concept to the topological group setting and introduced the concepts of G-sequential compactness, G-sequential continuity and sequential connectedness. In this paper we give a further investigation of G-sequential continuity in topological groups.

A Proof of Constructive Version of Brouwer's Fixed Point Theorem with Uniform Sequential Continuity

It is often said that Brouwer's fixed point theorem cannot be constructively proved. On the other hand, Sperner's lemma, which is used to prove Brouwer's theorem, can be constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer's fixed point theorem using Sperner's lemma. They, however, assume uniform continuity of functions. We consider uniform sequential continuity of functions. In classical mathematics, uniform continuity and uniform sequential continuity are equivalent. In constructive mathematics a la Bishop, however, uniform sequential continuity is weaker than uniform continuity. We will prove a constructive version of Brouwer's fixed point theorem in an n-dimensional simplex for uniformly sequentially continuous functions. We follow the Bishop style constructive mathematics.

On convergent sequences and fixed point theorems inD-metric spaces

Examples of completeD-metric spaces are given in which every convergent sequence has at most two limits and in which there are convergent sequences with exactly two limits. Also an example of a completeD-metric space having a convergent sequence with infinitely many limits is given and, using the example, several fixed point theorems inD-metric spaces are shown to be false. Modifications of some of these theorems and their generalizations are obtained either by imposing restrictions on the number of limits of certain convergent sequences in the space or by assuming the sequential continuity of theD-metric in any two variables and the theorems so obtained are illustrated by means of examples.