Fixed Points of Closed and Compact Composite Sequences of Operators and Projectors in a Class of Banach Spaces

Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces which can be also considered as Banach's spaces are discussed through the paper. The class of composite operators under study can include, in particular, sequences of projection operators under, in general, oblique projective operators. In this paper we are concerned with composite operators which include sequences of pairs of contractive operators involving, in general, oblique projection operators. The results are generalized to sequences of, in general, nonconstant bounded closed operators which can have bounded, closed, and compact limit operators, such that the relevant composite sequences are also compact operators. It is proven that in both cases, Banach contraction principle guarantees the existence of unique fixed points under contractive conditions.

On Best Proximity Point Theorems and Fixed Point Theorems for p-Cyclic Hybrid Self-Mappings in Banach Spaces

This paper relies on the study of fixed points and best proximity points of a class of so-called generalized point-dependent (K-Lambda)hybrid p-cyclic self-mappings relative to a Bregman distance Df, associated with a Gâteaux differentiable proper strictly convex function f in a smooth Banach space, where the real functions Lambda and K quantify the point-to-point hybrid and nonexpansive (or contractive) characteristics of the Bregman distance for points associated with the iterations through the cyclic self-mapping.Weak convergence results to weak cluster points are obtained for certain average sequences constructed with the iterates of the cyclic hybrid self-mappings.

On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite-dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.

Common Fixed Points of Generalized Rational Type Cocyclic Mappings in Multiplicative Metric Spaces

The aim of this paper is to present fixed point result of mappings satisfying a generalized rational contractive condition in the setup of multiplicative metric spaces. As an application, we obtain a common fixed point of a pair of weakly compatible mappings. Some common fixed point results of pair of rational contractive types mappings involved in cocyclic representation of a nonempty subset of a multiplicative metric space are also obtained. Some examples are presented to support the results proved herein. Our results generalize and extend various results in the existing literature.

Approximate Best Proximity Points of Cyclic Self-maps in Metric Spaces and Cyclic Asymptotic Regularity

3rd International Conference on Mathematical Modeling in Physical Sciences (IC-MSQUARE) Madrid, AUG 28-31, 2014 / editado por Vagenas, EC; Vlachos, DS; Bastos, C; Hofer, T; Kominis, Y; Kosmas, O; LeLay, G; DePadova, P; Rode, B; Suraud, E; Varga, K

Convergence Properties and Fixed Points of Two General Iterative Schemes with Composed Maps in Banach Spaces with Applications to Guaranteed Global Stability

This paper investigates the boundedness and convergence properties of two general iterative processes which involve sequences of self-mappings on either complete metric or Banach spaces. The sequences of self-mappings considered in the first iterative scheme are constructed by linear combinations of a set of self-mappings, each of them being a weighted version of a certain primary self-mapping on the same space. The sequences of self-mappings of the second iterative scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are also given to the important problem of global stability of a class of extended nonlinear polytopic-type parameterizations of certain dynamic systems.

An approach version of fuzzy metric spaces including an ad hoc fixed point theorem

In this paper, inspired by two very different, successful metric theories such us the real view-point of Lowen's approach spaces and the probabilistic field of Kramosil and Michalek's fuzzymetric spaces, we present a family of spaces, called fuzzy approach spaces, that are appropriate to handle, at the same time, both measure conceptions. To do that, we study the underlying metric interrelationships between the above mentioned theories, obtaining six postulates that allow us to consider such kind of spaces in a unique category. As a result, the natural way in which metric spaces can be embedded in both classes leads to a commutative categorical scheme. Each postulate is interpreted in the context of the study of the evolution of fuzzy systems. First properties of fuzzy approach spaces are introduced, including a topology. Finally, we describe a fixed point theorem in the setting of fuzzy approach spaces that can be particularized to the previous existing measure spaces.

Reinforcement learning with reference tracking control in continuous state spaces

The contribution described in this paper is an algorithm for learning nonlinear, reference tracking, control policies given no prior knowledge of the dynamical system and limited interaction with the system through the learning process. Concepts from the field of reinforcement learning, Bayesian statistics and classical control have been brought together in the formulation of this algorithm which can be viewed as a form of indirect self tuning regulator. On the task of reference tracking using a simulated inverted pendulum it was shown to yield generally improved performance on the best controller derived from the standard linear quadratic method using only 30 s of total interaction with the system. Finally, the algorithm was shown to work on the simulated double pendulum proving its ability to solve nontrivial control tasks. © 2011 IEEE.

Hope in the Most Unlikely Spaces: Thawra and the Contemporary Arabic Novel

In the early to mid-twentieth century, many novelists in the Arab world championed Arab nationalism in their literary reflections on the social and political struggles of their countries, depicting these struggles primarily in terms of spatial binaries that pitted the Arab world against the West, even as they imported Western literary models of progress and modernity into their own work. The intense experience of national awakening that infused their writing often placed these authors at a literary disadvantage, for in their literature, all too often the depth and diversity of Arabic cultures and the complexity of socio-political struggles across the Arab world were undermined by restrictive spatial discourses that tended to focus only on particular versions of Arab history and on a seemingly unifying national predicament. Between the Arab defeat of 1967 and the present day, however, an increasing number of Arab authors have turned to less restrictive forms of spatial discourse in search of a language that might offer alternative narratives of hope beyond the predictable, and seemingly thwarted, trajectories of nationalism. This study traces the ways in which contemporary Arab authors from Egypt and the Sudan have endeavoured to re-think and re-define the Arab identity in ever-changing spaces where elements of the local and the global, the traditional and the modern, interact both competitively and harmoniously. I examine the spatial language and the tropes used in three Arabic novels, viewing them through the lens of thawra (revolution) in both its socio-political and artistic manifestations. Linking the manifestations of thawra in each text to different scenes of revolution in the Arab world today, in Chapter Two, I consider how, at a stage when the Sudan of the sixties was both still dealing with colonial withdrawal and struggling to establish itself as a nation-state, the geographical and textual landscapes of Tayeb Salih‟s Season of Migration to the North depict the ongoing dilemma of the Sudanese identity. In Chapter Three, I examine Alaa iii al-Aswany‟s The Yacoubian Building in the context of a socially diseased and politically corrupt Egypt of the nineties: social, political, modern, historical, local, and global elements intertwine in a dizzyingly complex spatial network of associations that sheds light on the complicated reasons behind today‟s Egyptian thawra. In Chapter Four, the final chapter, Gamal al-Ghitani‟s approach to his Egypt in Pyramid Texts drifts far away from Salih‟s anguished Sudan and al-Aswany‟s chaotic Cairo to a realm where thawra manifests itself artistically in a sophisticated spatial language that challenges all forms of spatial hegemony and, consequently, old and new forms of social, political, and cultural oppression in the Arab world.

Gendered Spaces and Intimate Citizenship: The Case of Breastfeeding

This article situates breastfeeding politics in the context of intimate citizenship, where women’s capability to care in a range of social spaces is at stake. Drawing on the work of Lefebvre and Fenster, the article considers the extent to which recent breastfeeding promotion work by the Health Promotion Agency in Northern Ireland has sought to reconceive of social spaces in ways that have the potential to improve intimate citizenship for breastfeeding women.

Culture and the Irish Border:Spaces for Conflict Transformation

The Irish border has been described as a ‘natural’ cultural divide between the island's two dominant indigenous ethno-national communities. However, an examination of key resources of ethno-national group culture - religion, sport and language - provides evidence to challenge this representation. Moreover, in the post-1994 period of conflict transformation, evidence is also presented to support the proposition that the Irish border region has developed into a cultural space in which Irish nationalist and Ulster unionist ethno-national communities can explore cultural differences and commonalities through cross-border, cross-community contact and communication in small group encounters. This space underpins the reconfiguration of the border from barrier to political bridge between North and South. European Union (EU) Peace programmes for Ireland, beginning in 1995, provided the support for a cross-border approach to escaping the cage of ethno-national conflict in Northern Ireland. However, post-2004 EU enlargement signalled the beginning of the end for EU Peace funding and severe economic recession has undermined the expectation of British-Irish intergovernmental intervention to support cross-border partnerships and their work. Therefore, the outlook for the sustainability of this cross-border cultural space is gloomy with potentially deleterious consequences for the continued reconfiguration of the border from barrier to bridge.

Sensing Spaces:A Body-oriented Architectural Design Education

The contemporary dominance of visuality has turned our understanding of space into a mode of unidirectional experience that externalizes other sensual capacities of the body while perceiving the built environment. This affects not only architectural practice but also architectural education when an introduction to the concept of space is often challenging, especially for the students who have limited spatial and sensual training. Considering that an architectural work is not perceived as a series of retinal pictures but as a repeated multi-sensory experience, the problem definitions in the design studio need to be disengaged from the dominance of a ‘focused vision’ and be re-constructed in a holistic manner. A method to address this approach is to enable the students to refer to their own sensual experiences of the built environment as a part of their design processes. This paper focuses on a particular approach to the second year architectural design teaching which has been followed in the Department of Architecture at Izmir University of Economics for the last three years. The very first architectural project of the studio and the program, entitled ‘Sensing Spaces’, is conducted as a multi-staged design process including ‘sense games, analyses of organs and their interpretations into space’. The objectives of this four-week project are to explore the sense of space through the design of a three-dimensional assembly, to create an awareness of the significance of the senses in the design process and to experiment with re-interpreted forms of bodily parts. Hence, the students are encouraged to explore architectural space through their ‘tactile, olfactory, auditory, gustative and visual stimuli’. In this paper, based on a series of examples, architectural space is examined beyond its boundaries of structure, form and function, and spatial design is considered as an activity of re-constructing the built environment through the awareness of bodily senses.

Social Work Home Visits to Children and Families in the UK: A Foucauldian Perspective

The home visit is at the heart of social work practice with children and families; it is what children and families' social workers do more than any other single activity (except for recording), and it is through the home visit that assessments are made on a daily basis about risk, protection and welfare of children. And yet it is, more than any other activity, an example of what Pithouse has called an ‘invisible trade’: it happens behind closed doors, in the most secret and intimate spaces of family life. Drawing on conceptual tools associated with the work of Foucault, this article sets out to provide a critical, chronological review of research, policy and practice on home visiting. We aim to explain how and in what ways changing discourses have shaped the emergence, legitimacy, research and practice of the social work home visit to children and families at significant time periods and in a UK context. We end by highlighting the importance for the social work profession of engagement and critical reflection on the identified themes as part of their daily practice.

Wounded Spaces: When Planning Degraded Cairo’s Urban Memory

This paper examines the position of planning practices operated under precise guidelines for displaying modernity. Cultivating the spatial qualities of Cairo since the 1970s has unveiled centralised ideologies and systems of governance and economic incentives. I present a discussion of the wounds that result from the inadequate upgrading ventures in Cairo, which I argue, created scars as enduring evidence of unattainable planning methods and processes that undermined its locales. In this process, the paper focuses on the consequences of eviction rather than the planning methods in one of the city’s traditional districts. Empirical work is based on interdisciplinary research, public media reports and archival maps that document actions and procedures put in place to alter the visual, urban, and demographic characteristics of Cairo’s older neighbourhoods against a backdrop of decay to shift towards a global spectacular. The paper builds a conversation about the power and fate these spaces were subject to during hostile transformations that ended with their being disused. Their existence became associated with sores on the souls of its ex-inhabitants, as outward signs of inward scars showcasing a lack of equality and social justice in a context where it was much needed.

Besov spaces and the Laplacian on fractal H-sets

Nesta tese são estudados espaços de Besov de suavidade generalizada em espaços euclidianos, numa classe de fractais designados conjuntos-h e em estruturas abstractas designadas por espaços-h. Foram obtidas caracterizações e propriedades para estes espaços de funções. Em particular, no caso de espaços de Besov em espaços euclidianos, foram obtidas caracterizações por diferenças e por decomposições em átomos não suaves, foi provada uma propriedade de homogeneidade e foram estudados multiplicadores pontuais. Para espaços de Besov em conjuntos-h foi obtida uma caracterização por decomposições em átomos não suaves e foi construído um operador extensão. Com o recurso a cartas, os resultados obtidos para estes espaços de funções em fractais foram aplicados para definir e trabalhar com espaços de Besov de suavidade generalizada em estruturas abstractas. Nesta tese foi também estudado o laplaciano fractal, considerado a actuar em espaços de Besov de suavidade generalizada em domínios que contêm um conjunto-h fractal. Foram obtidos resultados no contexto de teoria espectral para este operador e foi estudado, à custa deste operador, um problema de Dirichlet fractal no contexto de conjuntos-h.