Reflexive Space Research Articles

Transforming young adults’ climate learning and actions through a co-created climate communication retreat

This article brings climate communication approaches to transformational climate learning by critically evaluating an experimental climate communication retreat that brought 20 young adults from across Europe together in Austria to co-create climate communications as a constitutive dimension of climate action. Structured around the transformational principles of interdisciplinarity, multidimensionality, collaborative project-based learning, reflexivity and action-oriented, the retreat specifically focused upon the creative co-production, between peers, of climate communication as communicative meaning-making and action. The retreat experience transformed young people’s sociocultural understandings of climate change, and climate communication as meaning-making and action, and increased their self and group efficacy. The key factors contributing to these transformations were: the forging of collective identity, peer-to-peer learning, emotional sharing, reflexive spaces, inspiring learning environment, interdisciplinary learning, multidimensional experiences, and collaborative project-based communications. Situating co-created climate communication within transformational learning can help facilitate collective experiences beyond direct climate action participation, helping create education for social change.

On the Bregman-proximal iterative algorithm for the monotone inclusion problem in Banach spaces

In this paper, we focus on the solution of a class of monotone inclusion problems in reflexive Banach spaces. To reflect the geometry of the space and the operator, a more general proximal point iterative algorithm with Bregman divergence is proposed and some strong convergence results for the proposed scheme under standard assumptions are obtained. Meanwhile, the convex optimization problems and the critical point problems are studied in the applications, and the recovery of the sparse signal is simulated in the numerical experiments.

An analysis on time optimal feedback control for Caputo fractional neutral systems with delay

ABSTRACT This paper aims to provide a systematic approach to investigate the optimal feedback control systems governed by the fractional neutral evolution system with fixed delay in the sense of Caputo fractional derivatives in separable reflexive Banach spaces. Initially, we study the existence result of mild solutions for the system by applying fixed point theorem for a condensing map due to Martelli. The existence of feasible pairs is obtained by using the Filippov theorem and the Cesari property. Furthermore, the existence of optimal feedback control and time optimal control are studied for the corresponding control problem. Finally, an example is given to validate our findings.

An inertial‐like Tseng's extragradient method for solving pseudomonotone variational inequalities in reflexive Banach spaces

A survey of the existing results in the literature shows that several of the results on variational inequality problem were established under some stringent conditions and employed some form of linesearch technique even in the framework of Hilbert spaces. However, due to the loop nature of the linesearch technique, the implementation of such algorithms might be economically nonviable. In this article, we study the pseudomonotone variational inequality problem. We propose a new inertial‐like Tseng's extragradient method with non‐monotonic self‐adaptive step size that does not involve any linesearch technique for finding the solution of the problem in reflexive Banach spaces. Under some mild conditions, we prove a strong convergence result for the proposed method without the sequentially weakly continuity condition often assumed by authors to guarantee convergence when studying the pseudomonotone variational inequality problems. Moreover, we apply our result to study a constrained convex minimization problem. Finally, we present several theoretical and real applications to optimal control numerical experiments to illustrate and show the performance of our method in comparison with related methods in the literature.

An inertial method for solving split equality quasimonotone Minty variational inequality problems in reflexive Banach spaces

An inertial method for solving split equality quasimonotone Minty variational inequality problems in reflexive Banach spaces

Existence results for variational–hemivariational inequality systems with nonlinear couplings

In this paper we investigate a system of coupled inequalities consisting of a variational–hemivariational inequality and a quasi-hemivariational inequality on Banach spaces. The approach is topological, and a wide variety of existence results is established for both bounded and unbounded constraint sets in real reflexive Banach spaces. Applications to Contact Mechanics are provided in the last section of the paper. More precisely, we consider a contact model with (possibly) multivalued constitutive law whose variational formulation leads to a coupled system of inequalities. The weak solvability of the problem is proved via employing the theoretical results obtained in the previous section. The novelty of our approach comes from the fact that we consider two potential contact zones and the variational formulation allows us to determine simultaneously the displacement field and the Cauchy stress tensor.

Performing Platform Governance: Facebook and the Stage Management of Data Relations

Controversies surrounding social media platforms have provided opportunities for institutional reflexivity amongst users and regulators on how to understand and govern platforms. Amidst contestation, platform companies have continued to enact projects that draw upon existing modes of privatized governance. We investigate how social media companies have attempted to achieve closure by continuing to set the terms around platform governance. We investigate two projects implemented by Facebook (Meta)—authenticity regulation and privacy controls—in response to the Russian Interference and Cambridge Analytica controversies surrounding the 2016 U.S. Presidential Election. Drawing on Goffman’s metaphor of stage management, we analyze the techniques deployed by Facebook to reinforce a division between what is visible and invisible to the user experience. These platform governance projects propose to act upon front-stage data relations: information that users can see from other users—whether that is content that users can see from “bad actors”, or information that other users can see about oneself. At the same time, these projects relegate back-stage data relations—information flows between users constituted by recommendation and targeted advertising systems—to invisibility and inaction. As such, Facebook renders the user experience actionable for governance, while foreclosing governance of back-stage data relations central to the economic value of the platform. As social media companies continue to perform platform governance projects following controversies, our paper invites reflection on the politics of these projects. By destabilizing the boundaries drawn by platform companies, we open space for continuous reflexivity on how platforms should be understood and governed.

WELL-POSEDNESS OF A NONLINEAR INTERFACE PROBLEM DESCRIBED BY HEMIVARIATIONAL INEQUALITIES VIA BOUNDARY INTEGRAL OPERATORS

This paper is devoted to the well-posedness of a novel nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models nonmonotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which however lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of the paper, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results and, moreover, establish a stability result for extended real-valued HVIs with respect to the extended real-valued function as a parameter. Based on the latter general stability result, we provide various stability results for the interface problem, as well as the stability of a related bilateral obstacle interface problem with respect to the obstacles.

On optimal control in a nonlinear interface problem described by hemivariational inequalities

Abstract This article is devoted to the existence of optimal controls in various control problems associated with a novel nonlinear interface problem on an unbounded domain with non-monotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models non-monotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which, however, lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of this article, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results; moreover, we establish a stability result with respect to the extended real-valued function as a parameter. Based on the latter stability result, we prove the existence of optimal controls for four kinds of optimal control problems: distributed control on the bounded domain, boundary control, simultaneous boundary-distributed control governed by the interface problem, and control of the obstacle driven by a related bilateral obstacle interface problem.

A numerical range approach to Birkhoff–James orthogonality with applications

The main aim of this paper is to provide characterizations of Birkhoff–James orthogonality (BJ-orthogonality in short) in a number of families of Banach spaces in terms of the elements of significant subsets of the unit ball of their dual spaces, which makes the characterizations more applicable. The tool to do so is a fine study of the abstract numerical range and its relation with the BJ-orthogonality. Among other results, we provide a characterization of BJ-orthogonality for spaces of vector-valued bounded functions in terms of the domain set and the dual of the target space, which is applied to get results for spaces of vector-valued continuous functions, uniform algebras, Lipschitz maps, injective tensor products, bounded linear operators with respect to the operator norm and to the numerical radius, multilinear maps, and polynomials. Next, we study possible extensions of the well-known Bhatia–Šemrl theorem on BJ-orthogonality of matrices, showing results in spaces of vector-valued continuous functions, compact linear operators on reflexive spaces, and finite Blaschke products. Finally, we find applications of our results to the study of spear vectors and spear operators. We show that no smooth point of a Banach space can be BJ-orthogonal to a spear vector of Z. As a consequence, if X is a Banach space containing strongly exposed points and Y is a smooth Banach space with dimension at least two, then there are no spear operators from X to Y. Particularizing this result to the identity operator, we show that a smooth Banach space containing strongly exposed points has numerical index strictly smaller than one. These latter results partially solve some open problems.

Fixed points of $G$-monotone mappings in metric and modular spaces

Let $C$ be a bounded, closed and convex subset of a reflexive metric space with a digraph $G$ such that $G$-intervals along walks are closed and convex. In the main theorem we show that if $T\colon C\rightarrow C$ is a monotone $G$-nonexpansive mapping and there exists $c\in C$ such that $Tc\in [c,\rightarrow )_{G}$, then $T$ has a fixed point provided for each $a\in C$, $[a,a]_{G}$ has the fixed point property for nonexpansive mappings. In particular, it gives an essential generalization of the Dehaish-Khamsi theorem concerning partial orders in complete uniformly convex hyperbolic metric spaces. Some counterparts of this result for modular spaces, and for commutative families of mappings are given too.

The Wolff hull of a compact holomorphic self-map on an infinite dimensional ball

For large classes of (finite and) infinite dimensional complex Banach spaces Z, B its open unit ball and f:B→B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:B\\rightarrow B$$\\end{document} a compact holomorphic fixed-point free map, we introduce and define the Wolff hull, W(f), of f in ∂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial B$$\\end{document} and prove that W(f) is proximal to the images of all subsequential limits of the sequences of iterates (fn)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(f^n)_n$$\\end{document} of f. The Wolff hull generalises the concept of a Wolff point, where such a point can no longer be uniquely determined, and coincides with the Wolff point if Z is a Hilbert space. Recall that (fn)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(f^n)_n$$\\end{document} does not generally converge even in finite dimensions, compactness of f (i.e. f(B) is relatively compact) is necessary for convergence in the infinite dimensional Hilbert ball and all accumulation points Γ(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma (f)$$\\end{document} of (fn)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(f^n)_n$$\\end{document} map B into ∂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial B$$\\end{document} (for any topology finer than the topology of pointwise convergence on B). The target set of f is T(f)=⋃g∈Γ(f)g(B).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} T(f)=\\bigcup _{g \\in \\Gamma (f)} g(B). \\end{aligned}$$\\end{document}To locate T(f), we use a concept of closed convex holomorphic hull, Ch(x)⊂∂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Ch}}(x) \\subset \\partial B$$\\end{document} for each x∈∂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\in \\partial B$$\\end{document} and define a distinguished Wolff hull W(f). We show that the Wolff hull intersects all hulls from T(f), namely W(f)∩Ch(x)≠∅for allx∈T(f).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} W(f) \\cap {\ ext {Ch}}(x)\ e \\emptyset \\ \\ \\hbox {for all}\\ \\ x \\in T(f). \\end{aligned}$$\\end{document}If B is the Hilbert ball, W(f) is the Wolff point, and this is the usual Denjoy–Wolff result. Our results are for all reflexive Banach spaces having a homogeneous ball (or equivalently, for all finite rank JB∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$JB^*$$\\end{document}-triples). These include many well-known operator spaces, for example, L(H, K), where either H or K is finite dimensional.

The Dynamic Behavior of Conjugate Multipliers on Some Reflexive Banach Spaces of Analytic Functions

In this paper, we characterize the hypercyclic, mixing and chaotic conjugate multipliers on some reflexive spaces of analytic functions. Our results extend several well-known results in the existing literature. We present some nontrivial examples to show the validity of our results.

Investigation on continuous dependence and regularity solutions of functional integrodifferential equations

In this work, to show the existence and continuous dependence of integral solution for neutral integro-differential equation with finite delay situation. This result established by using integrated resolvent operator with support of Lipschitz continuous and uniqueness part using Banach contraction mapping. Also study the existence of strict solution on reflexive and general Banach spaces. In the last section, an example is provided related to this theory.

Multiple solutions to a quasilinear periodic boundary value problem with impulsive effects

The authors investigate the multiplicity of solutions to a quasilinear periodic boundary value problem with impulsive effects. They use variational methods and some critical points theorems for smooth functionals, due to Ricceri, that are defined on reflexive Banach spaces. They obtain the existence of at least three solutions to the problem. The applicability of the results is illustrated with an example.

Critical point approaches to nonlinear square root Laplacian equations

This work is devoted to the study of multiplicity results of solutions for a class of nonlinear equations involving the square root of the Laplacian. Indeed, we will use variational methods for smooth functionals, defined on reflexive Banach spaces, in order to achieve the existence of at least three solutions for the equations. Moreover, assuming that the nonlinear terms are nonnegative, we will prove that the solutions are nonnegative. Finally, by presenting an example, we will ensure the applicability of our results.

Weak solutions of a fractional orders quadratic functional integro-differential inclusion in a reflexive Banach space

Weak solutions of a fractional orders quadratic functional integro-differential inclusion in a reflexive Banach space

Inverse problems for evolutionary quasi-variational hemivariational inequalities with application to mixed boundary value problems

The aim of this paper is to examine an inverse problem of parameter identification in an evolutionary quasi-variational hemivariational inequality in infinite dimensional reflexive Banach spaces. First, the solvability and compactness of the solution set to the inequality are established by employing a fixed point argument and tools of non-linear analysis. Then, general existence and compactness results for the inverse problem have been proved. Finally, we illustrate the applicability of the results in the study of an identification problem for an initial-boundary value problem of parabolic type with mixed multivalued and non-monotone boundary conditions and a state constraint.

A Note on Stone Spaces

The goal of this paper is to characterize each of compact, totally disconnected, Stone relation spaces, and Stone reflexive spaces as well as examine the relationships between them. Finally, we investigate some properties of them and compare our results.

ON TWO-SIDED UNIDIRECTIONAL MEAN VALUE INEQUALITY IN A FRÉCHET SMOOTH SPACE

The paper is devoted to a new unidirectional mean value inequality for the Fréchet subdifferential of a continuous function. This mean value inequality finds an intermediate point and localizes its value both from above and from below; for this reason, the inequality is called two-sided. The inequality is considered for a continuous function defined on a Fréchet smooth space. This class of Banach spaces includes the case of a reflexive space and the case of a separable Asplund space. As some application of these inequalities, we give an upper estimate for the Fréchet subdifferential of the upper limit of continuous functions defined on a reflexive space.