Reflexive Space Research Articles

Extension of monotone operators and Lipschitz maps invariant for a group of isometries

Abstract We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in $L^{p}$ -spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.

Novel inertial methods for fixed point problems in reflexive Banach spaces with applications

Novel inertial methods for fixed point problems in reflexive Banach spaces with applications

Approximate solvability method for nonlocal impulsive evolution equation

Abstract In this article, without assuming the compactness of semigroup, we deal with the existence and uniqueness of a mild solution for semilinear impulsive evolution equation with nonlocal condition in a reflexive Banach space by applying the approximate solvability method and Yosida approximations of the infinitesimal generator of C 0-semigroup.

Subgradient Extra-Gradient Algorithm for Pseudomonotone Equilibrium Problems and Fixed-Point Problems of Bregman Relatively Nonexpansive Mappings

In this article, we introduce a new subgradient extra-gradient algorithm to find the common element of a set of fixed points of a Bregman relatively nonexpansive mapping and the solution set of an equilibrium problem involving a Pseudomonotone and Bregman–Lipschitz-type bifunction in reflexive Banach spaces. The advantage of the algorithm is that it is run without prior knowledge of the Bregman–Lipschitz coefficients. Finally, two numerical experiments are reported to illustrate the efficiency of the proposed algorithm.

Stability analysis for set-valued inverse mixed variational inequalities in reflexive Banach spaces

This work is devoted to the analysis for a new class of set-valued inverse mixed variational inequalities (SIMVIs) in reflexive Banach spaces, when both the mapping and the constraint set are perturbed simultaneously by two parameters. Several equivalence characterizations are given for SIMVIs to have nonempty and bounded solution sets. Based on the equivalence conditions, under the premise of monotone mappings, the stability result for the SIMVIs is obtained in the reflexive Banach space. Furthermore, to illustrate the results, an example of the traffic network equilibrium control problem is provided at the end of this paper. The results presented in this paper generalize and extend some known results in this area.

ON SPLIT GENERALIZED EQUILIBRIUM AND FIXED POINT PROBLEMS OF BREGMAN W-MAPPINGS WITH MULTIPLE OUTPUT SETS IN REFLEXIVE BANACH SPACES

In this paper, we introduce a Halpern iteration process for computing the common solution of split generalized equilibrium problem and fixed points of a countable family of Bregman W-mappings with multiple output sets in reflexive Banach spaces. We prove a strong convergence result for approximating the solutions of the aforementioned problems under some mild conditions. It is worth mentioning that the iterative algorithm employ in this article is designed in such a way that it does not require the prior knowledge of operator norm. We also provide some numerical examples to illustrate the performance of our proposed iterative method. The result discuss in this paper extends and complements many related results in literature.

Mediatorship: Exploring transformational conflict narratives that elicit generative processes through interconnected continuums that cultivate conflict wisdom

AbstractNarrative inquiry, narrative conflict resolution, and auto/duoethnography can be seen as sources of agency, fostering emancipatory processes. Autoethnography is a method that can elicit constructive social change by cultivating narrative coherency. These generative methods and practices have the ability to enkindle a cultural shift. We need new stories to initiate transformational processes, resetting individual life trajectories and humanities' course. Writing Homing In: An Adopted Child's Story Mandala of Connecting, Reunion, and Belonging, provided an autoethnographic exploration that encompassed notions of the good life in an autoethnographic process akin to spiritual journey. This adoption and reunion story gave rise to concepts and transformational learning pedagogies that can be used to engender resolution processes that can be understood as “lines of flight” or flyways. The homing in mechanism that was found in cases of adopted children that were able to find their birth parents, is shown to work as a compass, guiding individual life trajectories. Auto/duoethnography are methods that elicit relational transformation while contributing to the development of conflict wisdom. Transformagram Portfolios incorporating these approaches were designed to teach courses within the Cultural and Social Department at Creighton University. Writing conflict and illness narratives elevated personal questing to art form by using future forming research methods to beautify story mandalas. Auto/duoethnography can be developed as an effective conflict resolution tool in response to current “glocal” challenges. Transformational pedagogies have facetted a holistic lens by creating reflexive space. In this context, mediatorship is a vessel of exploration that transports conflict resolution practices, while orienting experiential learning that interconnects the inner and outer journey. Transformagram Portfolios use auto/duoethnography to nurture narrative coherency, living wisdom, and generativity by scaffolding learning processes that provide an innovative pedagogical instrument for conflict resolution.

Construction of 𝑝-energy and associated energy measures on Sierpiński carpets

We establish the existence of a scaling limit E p \mathcal {E}_p of discrete p p -energies on the graphs approximating a generalized Sierpiński carpet for p > d A R C p > d_{\mathrm {ARC}} , where d A R C d_{\mathrm {ARC}} is the Ahlfors regular conformal dimension of the underlying generalized Sierpiński carpet. Furthermore, the function space F p \mathcal {F}_{p} defined as the collection of functions with finite p p -energies is shown to be a reflexive and separable Banach space that is dense in the set of continuous functions with respect to the supremum norm. In particular, ( E 2 , F 2 ) (\mathcal {E}_2, \mathcal {F}_2) recovers the canonical regular Dirichlet form constructed by Barlow and Bass [Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), pp. 225–257] or Kusuoka and Zhou [Probab. Theory Related Fields 93 (1992), pp. 169–196]. We also provide E p \mathcal {E}_{p} -energy measures associated with the constructed p p -energy and investigate its basic properties like self-similarity and chain rule.

Clinical Action Research and Its Radically Reflexive Way: Dialogical Encounters to Face Current Organizational Needs

The paper presents a clinical and radically reflexive approach to action research, capable of nurturing caring relationships and renewed organizational wealth. These outcomes are achieved by creating space for research-oriented, in-the-moment, constitutive, and collective reflexivity. The paper discusses a framework for understanding a clinical and radically reflexive approach to action research. Second, the radically reflexive perspective is illustrated via a case study. Finally, the specific conditions and limits of such an approach are discussed. The radically reflexive perspective we present hopes to return researchers and participants to the center as authors of renewed organizational landscapes, thanks to their collective agency and their capacity to resist the taken-for-granted cultural and social structures whose “sickness” often has a toxic and unconscious impact on people lives. The paper, therefore, contributes to an understanding of radical reflexivity in action research and its implications for productive dialogue between other clinical, dialogical, and reflexive action-research approaches.

Inertial method for solving split equality Hammerstein-type equation problems in reflexive Banach spaces

The purpose of this paper is to introduce an inertial scheme for solving Hammerstein-type equation problems in general reflexive real Banach spaces, where the underlying mappings are generally continuous monotone, not just uniformly continuous monotone. A strong convergence theorem is proved under some mild conditions, and finally numerical examples are provided to demonstrate the applicability of the algorithm.

Queer Connections and Fostering Inclusion: Reflections on Qualitative Focus Group Design

Qualitative social science research has an obligation to resist replicating or reinforcing social harms. This paper explores how researchers can adopt strategies to develop inclusive, safe, and reflexive research spaces with queer participants. Reflecting on a research study that explored the impact of queer representation in young adult literature on young readers, the paper explores the concepts of building queer connections and fostering inclusion as methods to support qualitative research design. Through this reflexive process, it has been found that intentionally using strategies to establish queer connections between participants and promoting inclusion through empathetic listening can significantly enhance participant experiences, primarily in terms of safety, support, and openness to vulnerability. These deliberate strategies also play a crucial role in reinforcing participant engagement and retention. Further, these findings indicate that these methods have additional benefits for research that transitions from face-to-face methods to online and digital formats. We argue that qualitative social science researchers who work with the queer community have an obligation to create inclusive and safe spaces for participants – not only to resist replicating stigmatising social structures, but to move beyond a pedagogy of pathologization in queer research.

An inertial method for solving systems of generalized mixed equilibrium and fixed point problems in reflexive Banach spaces

The Generalized mixed equilibrium problems, which includes the equilibrium and mixed equilibrium problem of monotone type mapping is considered in this paper with the fixed point of Bregman strongly nonexpansive mapping in the framework of real reflexive Banach space. We approximate the common solution of the system of generalized mixed equilibrium and fixed point problems for the finite family of Bregman strongly nonexpansive mappings using an inertial Halpern method. Using our iterative method, we prove a strong convergence result for solving the aforementioned problems. We also present some applications and numerical examples to demonstrate the performance of our iterative method. Our result improves and extends some important results presented by authors in the literature.

Research governance for change: funding project-based measures in the field of responsible research and innovation (RRI) and their potential for organisational learning

PurposeThis paper aims to look at the implementation of project-funded research governance and its potential to induce organisational learning on responsible research and innovation (RRI). This paper analysed what types of organisational learning and change can take place within organisations of an Europe-funded project and to what extent. This paper examined whether and how change occurs and how it is shaped and co-produced with other orderings.Design/methodology/approachThe paper is based on materials and evidence collected while working on the internal evaluation of a Horizon 2020-funded project. Analysis of the results of the mixed methods evaluation design was used to characterise occurrences of organisational learning and change.FindingsThe authors identified different forms of learning (single-loop learning, double-loop learning, reflexive and reflective learning and situational learning). The extent of learning that could lead to long-lasting organisational change was limited. This was due to the project-based and organisational design, the key-based definition of RRI and the indeterminacy of what constitutes learning and change – both at the level of funding and performing the project. For organisational change to occur, the authors argue for governance mechanisms based on reflexive learning that consider a range of structural conditions and measures.Originality/valueOrganisational learning plays an important role in change processes, which has so far been given too little consideration concerning the governance and implementation of RRI through project-based funding. The authors argue for a restructuring of governance and funding mechanisms to create more space for reflexivity and learning.

A topological characterisation of Haar null convex sets

In R d \mathbb {R}^d , a closed, convex set has zero Lebesgue measure if and only its interior is empty. More generally, in separable, reflexive Banach spaces, closed and convex sets are Haar null if and only if their interior is empty. We extend this facts by showing that a closed, convex set in a separable Banach space is Haar null if and only if its weak ∗ ^* closure in the second dual has empty interior with respect to the norm topology. It then follows that, in the metric space of all nonempty, closed, convex and bounded subsets of a separable Banach space, converging sequences of Haar null sets have Haar null limits.

On Non-linear Mappings Preserving the Semi-inner Product

We say that a smooth normed space X has a property (SL), if every mapping f:X rightarrow X preserving the semi-inner product on X is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every finite-dimensional smooth normed space. In this paper, we establish several new results concerning the property (SL). We give a simple example of a smooth and strictly convex Banach space which is isomorphic to the space ell _p, but without the property (SL). Moreover, we provide a characterization of the property (SL) in the class of reflexive smooth Banach spaces in terms of subspaces of quotient spaces. As a consequence, we prove that the space ell _p have the property (SL) for every 1< p < infty . Finally, using a variant of the Gowers–Maurey space, we construct an infinite-dimensional uniformly smooth Banach space X such that every smooth Banach space isomorphic to X has the property (SL).

Strong Convergence Theorem Obtained by a Generalized Projections Method for Solving an Equilibrium Problem and Fixed Point Problems

In this paper we introduce a new projection-type algorithm in a reflexive Banach space. Then, using generalized resolvents operators and generalized projections, we prove a strong convergence theorem for computing a common element of the set of fixed points of a Bregman relatively nonexpansive mapping, solutions of an equilibrium problem, fixed points of a resolvent operator and fixed points of an infinite family of Bregman W-mappings.

ON THE CRISIS OF CULTURE AND MAN IN THE «FAR-FETCHED» TRADITIONALISM OF R.GUENON

The problem of the crisis situation of Western European culture at the beginning of the 19th century became the central problem in the ideological heritage of R. Guenon. In his reflections, the French thinker turns to the consideration of the role and significance of traditions in the life of a person and society. R.Guenon fills the term “tradition” with his semantic load, making this term a key element in his philosophy. The reflexive space of Guénon’s philosophy is the sphere of the philosophy of religion and it is difficult to attribute it to the sphere of religious philosophy. This difficulty lies in the departure of the French thinker from the traditions of the Christian worldview to the ideological foundations of Eastern mysticism, agnosticism and theosophical views. Perhaps it is these preferences of Guénon that lead the researchers of his work to the idea that he is, in a certain sense, the “forerunner” of the postmodern era in Europe. Tradition and the crisis of traditions for the philosopher are considered as the main subject of study. The scale of historical and philosophical analysis is taken by the thinker without embarrassment as wide as possible. Ultimately, the very crisis of culture for Guenon can and must serve as a transition to a new stage in the development of culture. In this new stage, a syncretic creative renewal of world religions on the basis of the Eastern mysteries will take place. The cyclic concept of the author’s history does not retain its traditional connection with the Christian understanding of the future, the loss of the Christian eschatological understanding of man. the Renaissance is declared a break point with the traditional understanding of being. For a philosopher, tradition is not reduced to folk custom. And already at this stage, we are witnessing Guénon’s attempt to give the established philosophical terms their own sound, fitting them to his concept and esoteric context.

Editorial Thinking outside the academic writing box

The title of this editorial is adapted froma lineinthebook review published in this issue of the Journal of Academic Writing (JoAW). The review iswritten by Livingstone,whoargues for the importance of texts that push,“those of us in academia, who have become too fixed in our ways, who are afraid of thinking outside-the-box.”This line reflects a core value of JoAW, as the journal has always endeavoured to serve as a reflexive space for innovation and development for EATAW members and the wider community of researchersand practitionersinterested in academic writing. The various genres JoAWpublishes that go beyond the traditional research article, the formative approach it takes to publishing, and the value it attributes to open-access, practice-oriented researchdemonstratejust some of the waysin which JoAWhasaimedto push boundariesin academic writing research and practice.

An improved subgradient extragradient method with two different parameters for solving variational inequalities in reflexive Banach spaces

An improved subgradient extragradient method with two different parameters for solving variational inequalities in reflexive Banach spaces

On the dominated convergence theorem for the Kurzweil–Stieltjes integral

AbstractThis paper is concerned with a version of the Lebesgue dominated convergence theorem (DCT) which has been stated for the Kurzweil–Stieltjes integral of real functions. Our objective in this work is to analyze the extension of this result to include vector functions with values in Banach spaces. We establish that the mentioned convergence theorem for the Kurzweil–Stieltjes integral can be formulated in weaker versions for reflexive and separable Banach spaces, and spaces having the Schur property, nonetheless it is not verified in the general case.