Reflexive Space Research Articles

Vector calculus on weighted reflexive Banach spaces

We study first-order Sobolev spaces on reflexive Banach spaces via relaxation, test plans, and divergence. We show the equivalence of the different approaches to the Sobolev spaces and to the related tangent bundles.

Integral-Type Representations for the Subdifferential of Suprema

AbstractAn integral-like representation is provided for the $$\varepsilon $$ ε -subdifferential of the supremum of an arbitrary family of convex functions. Our characterizations are expressed through appropriate discrete sums performed on the data functions $$f_{t}$$ f t ’s together with specific singular measures operating on them. Moreover, as long as the underlying space is a reflexive Banach space or a separable normed space, we substitute these additional measures with related limits that involve the data functions. All the objects involved in our characterizations rely intrinsically on the data functions that are (almost) active at the reference point.

Designing an Intervention to Improve Medication Safety for Nursing Home Residents Based on Experiential Knowledge Related to Patient Safety Culture at the Nursing Home Front Line: Cocreative Process Study.

Despite years of attention, avoiding medication-related harm remains a global challenge. Nursing homes provide essential health care for frail older individuals, who often experience multiple chronic diseases and polypharmacy, increasing their risk of medication errors. Evidence of effective interventions to improve medication safety in these settings is inconclusive. Focusing on patient safety culture is a potential key to intervention development as it forms the foundation for overall patient safety and is associated with medication errors. This study aims to develop an intervention to improve medication safety for nursing home residents through a cocreative process guided by integrated knowledge translation and experience-based codesign. This study used a cocreative process guided by integrated knowledge translation and experience-based co-design principles. Evidence on patient safety culture was used as an inspirational source for exploration of medication safety. Data collection involved semistructured focus groups to generate experiential knowledge (stage 1) to inform intervention design in a multidisciplinary workshop (stage 2). Research validation engaging different types of research expertise and municipal managerial representatives in finalizing the intervention design was essential. Acceptance of the final intervention for evaluation was aimed for through contextualization focused on partnership with a municipal advisory board. An abductive, rapid qualitative analytical approach to data analysis was chosen using elements from analyzing in the present, addressing the time-dependent, context-bound aspects of the cocreative process. Experiential knowledge was represented by three main themes: (1) closed systems and gaps between functions, (2) resource interpretation and untapped potential, and (3) community of medication safety and surveillance. The main themes informed the design of preliminary intervention components in a multidisciplinary workshop. An intervention design process focused on research validation in addition to contextualization resulted in the Safe Medication in Nursing Home Residents (SAME) intervention covering (1) campaign material visualizing key roles and responsibilities regarding medication for nursing home residents and (2) "Medication safety reflexive spaces" focused on social and health care assistants. The cocreative process successfully resulted in the multifaceted SAME intervention, grounded in lived experiences shared by some of the most important (but often underrepresented in research) stakeholders: frontline health care professionals and representatives of nursing home residents. This study brought attention toward closed systems related to functions in medication management and surveillance, not only informing the SAME intervention design but as opportunities for further exploration in future research. Evaluation of the intervention is an important next step. Overall, this study represents an important contribution to the complex field of medication safety. RR2-10.2196/43538.

Existence Results of Some Nonlinear Minimization Problems with Application to a System of PDE

A new class of non-self mappings, called proximal condensing operators, is introduced and used to investigate the existence of best proximity points for non-self mappings in reflexive Banach spaces by applying an appropriate measure of noncompactness. In this way, we present generalizations of well-known Schauder and Darbo fixed point problems. We then define the notion of best proximity points for multiplication of two non-self mappings in the setting of Banach algebras and present some sufficient conditions to ensure the existence of such points. Finally, we renorm a Banach space consists of all real valued and continuous functions with two variables to obtain the strict convexity condition and then we survey the existence of an optimal solution for a system of partial differential equations.

A Relaxed Tseng’s Method-based Algorithm for Solving Split Variational Inequalities with Multiple Output Sets

In this paper, the split variational inequality problem with multiple output sets is considered in a more general framework of reflexive Banach spaces. We introduce a relaxed Tseng extragradient method for approximating the solution of this problem when the cost operators are pseudomonotone and non-Lipschitz. Moreover, our proposed algorithm employs the inertial technique and a new self-adaptive step size to guarantee high rate of convergence. Strong convergence of the algorithm is proved in this case without the need for the sequential weak continuity condition. Finally, some numerical experiments are given to show the applicability of our proposed algorithm.

Analytical Solutions for a Nonlinear Quadratic Delayed Functional Integral Inclusion With Feedback Control on the Real Half-Line

Let E be a reflexive Banach space. In this research, we are interested in the solvability of the nonlinear quadratic delayed functional integral inclusion (NQDFII) with a feedback control condition on the real half-axis. Our examination is found within the space BC(R+, E) of bounded continuous functions on the real half-axis R+ and takes values in a reflexive Banach space E beneath the assumption that the set-valued function G satisfy Lipschitz condition in E. The base we depend on in this study is the procedure related to a measure of noncompactness in the space BC(R+, E) by a given norm of continuity and applying Darbo’s fixed point theorem. Moreover, the asymptotic stability of the solution and the asymptotic dependency of the solution on the set of selections SG will be examined. Also, we give an example to illustrate the adequacy and esteem of our comes about.

Best approximation theorems via relatively u-continuous and Jaggi nonexpansive mappings

In this paper, we introduced a new type of cyclic (noncyclic) mappings, called Jaggi relatively nonexpansive mappings and use to investigate the existence of best proximity points (pairs) in the framework of reflexive (strictly convex) Banach spaces by using a geometric concept of proximal normal structure. We also present a best proximity version of Schauder’s fixed point problem for non-self mappings. Finally, we prove the existence of a common best proximity point for an arbitrary family of cyclic relatively u-continuous mappings which are compact and then we obtain a generalization of Markov-Kakutani’s theorem.

On split equality monotone variational inclusion and fixed point problems in reflexive Banach spaces

In this paper, motivated by the works of Akbar and Shahrosvand [Filomat {\bf 32} (2018), no.\ 11, 3917-3932], Ogbuisi and Izuchukwu [Numer. Funct. Anal. Optim. {\bf 41} (2020), no.\ 2, 322-343], and some other related results in the literature, we introduce a Halpern iterative algorithm and employ a Bregman distance approach for approximating a solution of split equality monotone variational inclusion problem and fixed point problem of Bregman relatively nonexpansive mapping in reflexive Banach spaces. Under suitable condition, we state and prove a strong convergence result for approximating a common solution of the aforementioned problems. Furthermore, we give an application of our main result to variational inequality problems and provide some numerical examples to illustrate the convergence behavior of our result. The result presented in this paper extends and complements many related results in literature.

Modi ed inertia Halpern method for split null point problem in Banach spaces

Abstract. In this paper, we study split null point problem in reflexive Banach spaces. Using the Bregman technique together with a modified inertial Halpern method, we approximate a solution of split null point problem. Also, we establish a strong convergence result for approximating the solution of the aforementioned problems. It is worth mentioning that the iterative algorithm employ in this study is design in such a way that it does not require prior knowledge of operator norm. We display some numerical examples to illustrate the performance of the proposed iterative method. The result discuss in this paper extends and complements many related results in literature. Mathematics Subject Classification (2010): 47H06, 47H09, 47J05, 47J25. Keywords: Monotone variational inclusion problem, split feasibility problem, firmly nonexpansive-type mapping, fixed point problem, inertial method.

2-Rotund norms for unconditional and symmetric sequence spaces

A reflexive Banach space with an unconditional basis admits an equivalent 1-unconditional 2R norm and embeds into a reflexive space with a 1-symmetric 2R norm. Partial results on 1-symmetric 2R renormings of spaces with a symmetric basis are obtained.

SPECTRAL THEORY AND FUNCTIONAL CALCULI IN THE REFLEXIVE BANACH SPACES

The design and assessment of closed loop control systems must include accurate and stable accuracy analysis and simulation. An overview of the significance of closed loop control systems and its uses in several engineering disciplines is given in this paper. The paper then explores the different techniques used for evaluating the accuracy and stability of closed loop control systems, including steady-state error analysis, stability analysis, and simulation techniques. It also discusses the design and tuning of control algorithms to optimize the performance of the system, and identifies the different factors that can affect the accuracy and stability of closed loop control systems, such as sensor noise, model inaccuracies, and environmental disturbances.

Betwixt and between in an ‘unreadable city’: Chinese students' urban citizenship and belonging in pandemic-era Amsterdam

As newcomers to Amsterdam, Chinese international students arriving during 2020 – 2021 had to navigate a peculiar urban landscape shaped by the Covid-19 pandemic. Rather than the pandemic's direct impacts, we highlight the unique spatiotemporal context it mediated – a period characterized by its liminality, or in-betweenness, further marked by ambiguity and uncertainty. The study draws on narrative interviews to investigate the students' urban experiences and socio-spatial engagement, focusing on their sense of belonging and urban citizenship. Citizenship is conceptualized as a ‘lived’ that is practiced and contested, making it affective and contextual. We argue that citizenship is inextricably linked to belonging and center feeling belongingness and familiarity as proxies claiming urban citizenship. The interplay of individual subjectivities, socio-spatial practices, and transnational identities helped produce reflexive spaces of home and recontextualized Chinese dynamics in Amsterdam, demonstrating the transcendent nature of the ‘urban’ beyond physical boundaries. We find that interlocutors often attached meanings to micro-spaces rather than the city as a whole. Experiencing liminal transience and racialized (micro)aggressions impacted, to varying degrees, the participants' desire and ability to become familiar with their urban environments and build citizenship.

Integrable Solutions and Continuous Dependence of a Nonlinear Singular Integral Inclusion of Fractional Orders and Applications

Let E be a reflexive Banach space. In this article we study the existence of integrable solutions in the space of all lesbesgue integrable functions on E, L1([0,T],E), of the nonlinear singular integral inclusion of fractional orders beneath the assumption that the multi-valued function G has Lipschitz selection in E. The main tool applied in this work is the Banach contraction fixed point theorem. Moreover, the paper explores a qualitative property associated with these solutions for the given problem such as the continuous dependence of the solutions on the set of selections S1G(τ,x(τ)). As an application, the existence of integrable solutions of the two nonlocal and weighted problems of the fractional differential inclusion is investigated. We additionally provide an example given as a numerical application to demonstrate the effectiveness and value of our results.

Inertial Halpern-type methods for variational inequality with application to medical image recovery

In this paper, we propose inertial Halpern-type algorithms involving a quasi-monotone operator for approximating solutions of variational inequality problems which are fixed points of quasi-nonexpansive mappings in reflexive Banach spaces. We use Bregman distance functions to enhance the efficiency of our algorithms and obtain strong convergence results, even in cases where the Lipschitz constant of the operator involved is unknown a priori. Furthermore, we illustrate the practical applicability of our methods through numerical experiments. Notably, our algorithms excel when compared to recent techniques in the literature. Of particular significance is their successful application in restoring computed tomography medical images that have been affected by motion blur and random noise. Our algorithms consistently outperform established state-of-the-art methods in all conducted experiments, showcasing their competitiveness and potential to advance variational inequality problem-solving, especially in the field of medical image recovery.

Envelopes in Banach spaces

We introduce the notion of isometric envelope of a subspace in a Banach space, establishing its connections with several key elements: (a) we explore its relation to the mean ergodic projection on fixed points within a semigroup of contractions, (b) we draw parallels with Korovkin sets from the 1970s, (c) we investigate its impact on the extension properties of linear isometric embeddings. We use this concept to address the recent conjecture that the Gurarij space and the spaces Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}, p∉2N+4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \ otin 2{\\mathbb {N}}+4$$\\end{document} are the only separable approximately ultrahomogeneous Banach spaces (a certain multidimensional transitivity of the action of the linear isometry group). The similar conjecture for Fraïssé Banach spaces (a strengthening of the approximately homogeneous property) is also considered. We characterize the Hilbert space as the only separable reflexive space in which any closed subspace coincides with its envelope; and we show that the Gurarij space satisfies the same property. We compute some envelopes in the case of Lebesgue spaces, showing that the reflexive Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}-spaces are the only reflexive rearrangement invariant spaces on [0, 1] for which all 1-complemented subspaces are envelopes. We also identify the isometrically unique “full” quotient space of Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document} by a Hilbertian subspace, for appropriate values of p, as well as the associated topological group embedding of the unitary group into the isometry group of Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}.

Nash’s Existence Theorem for Non-Compact Strategy Sets

In this paper, we apply the classical FKKM lemma to obtain the Ky Fan minimax inequality defined on nonempty non-compact convex subsets in reflexive Banach spaces, and then we apply it to game theory and obtain Nash’s existence theorem for non-compact strategy sets, which can be regarded as a new, simple but interesting application of the FKKM lemma and the Ky Fan minimax inequality, and we can also present another proof about the famous John von Neumann’s existence theorem in two-player zero-sum games. Due to the results of Li, Shi and Chang, the coerciveness in the conclusion can be replaced with the P.S. or G.P.S. conditions.

An inertial parallel iterative method for solving generalized mixed equilibrium problems and common fixed point problem in reflexive Banach spaces

An inertial parallel iterative method for solving generalized mixed equilibrium problems and common fixed point problem in reflexive Banach spaces

Weak solutions for the coupled system of Riemann-Stieltjes integral equations

We present an existence theorem for at least one weak solution for a coupled system of Riemann-Stieltjes integral equations in a reflexive Banach space . As an application, we study the existence of weak solutions for the coupled system of Hammerstien-Stieltjes integral equations in

Collaborative research practices in the study of diasporisation: chronotopes of dispersion among Hungarians in Catalonia

ABSTRACT Diasporas and diasporisation have lately been at the centre of interest in sociolinguistic inquiry. Literature on the inclusion of diasporic speakers in academic knowledge production in connection with diasporas is scarce, although this would be of crucial importance to achieve ‘thinking diaspora from below’ (Rosa and Trivedi 2017. “Diaspora and Language.” In The Routledge Handbook of Migration and Language, edited by Suresh Canagarajah, 330–346. London: Routledge) when researching sociolinguistic phenomena. In this article, I discuss an ethnographically informed critical sociolinguistic study of diasporisation among Hungarians in Catalonia between 2018 and 2022 in which I applied collaborative research practices with the key participants. These practices consisted of the joint formulation of questions, the creation of a reflexive diasporic space, and feedback provided in the post-fieldwork phase. I address how these methods contributed to the emergence of a chronotopic approach to diasporisation. I argue that the inclusion of the emic perspectives of diasporic speakers through their involvement in the research process is a fruitful way to approach diasporas and diasporisation. Moreover, by paying attention to the needs and interests of the participants, this way of researching on, for, and with the participants also contributes to better understandings of diasporisation.

Strong convergence theorem of a new modified Bregman extragradient method to solve fixed point problems and variational inequality problems in general reflexive Banach spaces

In this paper, we introduce a new modified self-adaptive extragradient method with the inertial technique for solving the variational inequality problem of a pseudomonotone mapping and the fixed point problem of a Bregman relatively nonexpansive operator in a general reflexive Banach space. We prove a strong convergence theorem of our algorithm under some suitable assumptions. Finally, some numerical experiments are provided to demonstrate the effectiveness of the suggested iterative method.