BANACH SPACES

Given a finite covering by closed convex sets of BX , the unit ball of an infinite-dimensional Banach space, we investigate whether there is a set of the covering that contains balls of radius close to 1 and (a) arbitrarily high finite dimension or (b) infinite dimension. In case (a) the answer is affirmative, but for the case (b) we just get radius close to 1/2 and finite codimension under much more restrictive hypotheses.

This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are especially interested in discretization schemes based on frames. The central objective is to derive an adaptive frame algorithm which is guaranteed to converge for a wide range of cases. As a core ingredient we use the concept of Gelfand frames which induces equivalences between smoothness norms and weighted sequence norms of frame coefficients. It turns out that this Gelfand characteristic of frames is closely related to their localization properties. We also give constructive examples of Gelfand wavelet frames on bounded domains. Finally, an application to the efficient adaptive computation of canonical dual frames is presented.

Let A be a Banach algebra over R or C with center Z(A). In this paper, we show that, if a non-injective continuous derivation of A satisfies some local differential identities, then A must be commutative. We give several applications, and we provide examples to show that some hypotheses of our theorems are necessary.

A new measure of noncompactness on Banach spaces is defined from the Hausdorff measure of noncompactness, giving a quantitative version of a classical result by R. S. Phillips [Trans. Am. Math. Soc., 48, 516–541 (1940; Zbl 0025.34202; JFM 66.0554.01)]. From the main result, classical results are obtained now as corollaries and we have an application to interpolation theory of Banach spaces.

The main goal of this paper is the study of quasi s-numbers of multilinear operators among Banach spaces. The relationships among multilinear variants of approximation, Kolmogorov and Gelfand numbers of operators and their generalized linear adjoint are shown. In the multilinear case, analogous theorems which are well-known in the linear case, are stated and proved. The estimates of measures of non-compactness of multilinear operators in terms of measures of the adjoint operators are also proved.

In this paper, we consider the optimal robust disturbance attenuation problem (ORDAP) for multi-input multioutput (MIMO) uncertain plants. Duality theory is used to show existence of optimal feedback laws. Next a key is the dual space of C?n::n [l, 21, and vise-versa since Gnxn and q ; k n are finite dimensional and therefore reflexive. If X denotes a finite dimensional complex

In this article, we give an explicit solution to the nonstandard H" problem which includes the Optimal Robust Disturbance Attenuation Problem (ORDAP) and the Two-Disc Problem. Our solution depends on an interplay between function theory, Banach space duality and operator theory. The solution depends on an operator which is analogous to the Sarason o p erator for the standard H m problem. This operator allows to quantify the optimal performance of feedback systems and provides an explicit formula for the optimal controller. The nonstandard H" problem is perhaps the simplest control problem which deals with significant plant uncertainty. Surprisingly, relatively little work has been done on it despite the large number of papers in the H" control literature.

This work is a survey of results and problems connected with selected properties of bilinear mappings between function spaces. Section 1 is devoted to the lack of the counterpart of the Banach openness principle for bilinear surjections between Banach spaces. Some counterexamples and positive results are discussed. Then we deal with variants of openness for multiplication in spaces C(K), for a compact K, and in other Banach algebras and function spaces. In Section 2, several phenomena of dichotomies for operators of multiplication and convolution in spaces of integrable or continuous functions are presented. Some natural properties of these operators hold either always or the sets of objects having the given property are small. The smallness is described by meagerness and porosity. Let us introduce some basic notation. The ball with center x and radius r in a given metric space X is denoted by B X (x, r) (if X is fixed, index X will be omitted). The interior of a set A ⊂ X will be written as int(A).

espanolSe desarrolla una generalizacion del metodo de punto proximal clasico y el metodo de punto proximal con distancias de Bregman bajo condiciones de convexidad. Partiendo de una sucesion arbitraria de funciones de Bregman convergente puntualmente, el metodo propuesto permite generalizar los casos clasicos que han sido desarrollados para una funcion Bregman fija, considerando propiedades que regulan el comportamiento de la sucesion de distancias de Bregman. Como consecuencia, se obtiene un metodo que converge al minimizador de la funcion objetivo. EnglishA generalization of the classical proximal point method and the method of proximal point with Bregman distances is developed under conditions of convexity. Starting from an arbitrary punctually convergent sucession of Bregman functions, our method allows both the generalization to the classic cases that have been developed for a fixed Bregman function and the addition of properties that regulate the behavior of the succession of ...

We study the existence, uniqueness and asymptotic behavior, as well as the stability of a special kind of traveling wave solutions for competitive PDE systems involving intrinsic growth, competition, crowding effects and diffusion. The traveling waves are exclusive in the sense that as the variable goes to positive or negative infinity, different species are close to extinction or carrying capacity. We perform an appropriate affine transformation of the traveling wave equations into monotone form and construct appropriate upper and lower solutions. By this means, we reduce the existence proof to application of well-known theory about monotone traveling wave systems (cf.

Analisamos equações integro-diferenciais do tipo Barbashin quando os multiplicadores c = c(t, s) e núcleos k = k(t, s, µ) são estacionários. Consideramos os casos analisamos o problema quando o multiplicador cé limitado e quando nãoé limitado. A existência e unicidade de soluções são deduzidas através de teoremas de ponto-fixo.

Apresentaremos neste artigo a demonstração de uma desigualdade do tipo Wirtinger-uma desigualdade que envolve integrais de funções do R n que estima a norma de uma função u através da norma de seu gradiente ∇u. Iniciaremos com um pouco da história da própria desigualdade de Wirtinger e em seguida partiremos para a demonstração a qual nos propusemos inicialmente e, para tal, utilizaremos alguns importantes conceitos, corolários e teoremas de análise funcional e de medida e integração. Vale ressaltar aqui que a desigualdade apresentada nãoé um caso particular da desigualdade de Poincaré.

Duality principles in Gabor theory such as the Ron-Shen duality principle and the Wexler-Raz biorthogonality relations play a fundamental role for analyzing Gabor systems. In this article we present a general approach to derive duality principles in abstract frame theory. For each sequence in a separable Hilbert space we define a corresponding sequence dependent only on two orthonormal bases. Then we characterize exactly properties of the first sequence in terms of the associated one, which yields duality relations for the abstract frame setting. In the last part we apply our results to Gabor systems.

The purpose of this paper is to introduce a broader definition of Riemann integrability and to compare it with the traditional Riemann integral. We will examine two classic pathological examples in Integration Theory: the original and modified Dirichlet functions. Although these functions are not integrable in the traditional Riemann sense, we will show that the generalized integral can evaluate them, yielding a valid result.

The behavior of bilinear operators acting on interpolation of Banach spaces for the ρ method in relation to the compactness is analyzed. Similar results of Lions-Peetre, Hayakawa and Person's compactness theorems are obtained for the bilinear case and the ρ method.

This paper is devoted to the study of relationships between solutions of Stampacchia and Minty vector variational-like inequalities, weak and strong Pareto solutions of vector optimization problems and vector critical points in Banach spaces under pseudo-invexity and pseudo-monotonicity hypotheses. We have extended the results given by Gang and Liu (2008) [22] to Banach spaces and the relationships obtained for weak efficient points in Santos et al. (2008) [21] are completed and enabled to relate vector critical points, weak efficient points, solutions of the Minty and Stampacchia weak vector variationallike inequalities problems and solutions of perturbed vector variational-like inequalities problems.

Every Köthe echelon Fréchet space X that is Montel and not isomorphic to a countable product of copies of the scalar eld admits a power bounded continuous linear operator T such that I − T does not have closed range, but the sequence of arithmetic means of the iterates of T converge to 0 uniformly on the bounded sets in X. On the other hand, if X is a Fréchet space which does not have a quotient isomorphic to a nuclear Köthe echelon space with a continuous norm, then the sequence of arithmetic means of the iterates of any continuous linear operator T (for which (1/n)T n converges to 0 on the bounded sets) converges uniformly on the bounded subsets of X, i.e., T is uniformly mean ergodic, if and only if the range of I − T is closed. This result extends a theorem due to Lin for such operators on Banach spaces. The connection of Browder'equality for power bounded operators on Fréchet spaces to their uniform mean ergodicity is exposed. An analysis of the mean ergodic properties of the classical Cesàro operator on Banach sequence spaces is also given.

In this paper, we consider the difference equation on an arbitrary Banach space (X, II. IL)9 A (q,Az,) + fn(zn) = 0, where {qn} is a positive sequence and j,, is X-valued. We shall give conditions so that for a given I E X, there exists a solution of this equation asymptotically equal to I.

We show that the space of bounded linear operators between spaces of continuous functions on compact Hausdorff topological spaces has the Bishop-Phelps-Bollobás property. A similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and Hausdorff topological space into a uniformly convex space, and for the class of compact operators from a Banach space into a predual of an L 1-space.

In this paper, we introduce a partial order on a cone metric space and prove a Caristi-type theorem. Furthermore, we prove fixed point theorems for single-valued nondecreasing and weakly increasing mappings, and multi-valued mappings on an ordered cone metric space.

In this article, we find the solution of a nonlinear fractional integro-differential equations with impulsive and integral conditions via an efficient method. The method of solution in this article involves the concept of fixed point theory in Banach spaces.

We introduce a new class of nonlinear mappings, the class of asymp- totically �-hemicontractive mappings in the intermediate sense and approximate the unique common �xed point of a family of three of these mappings in Banach spaces. Our results improve and generalize the results of Xue and Fan [Zhiqun Xue, Ruiqin Fan, Some comments on Noor's iterations in Banach spaces, Applied Mathematics and Computation 206 (2008) 12-15] which in turn is a correction of the results of Ra�q [Arif Ra�q, Modified Noor iterations for nonlinear equations in Banach spaces, Applied Mathematics and Computation 182 (2006) 589-595

Let X be a Banach space and L(X) be the Banach algebra of bounded operators on X. In this note we prove that if we have a compact subset K of a commutative sub-algebra of L(X), and given ε > 0, then it is possible to define a new norm in X, equivalent to its given norm, in such a way that inside a neighborhood Uε of this compact set in the subalgebra, the norms of all the operators differ from their spectral radius in less than ε. If X is a Hilbert space then it is possible to define this new norm as an Hilbertian norm. When one wants to estimate the norms of functions of bounded operators the properties of norms on L(X) usually are very helpful. But imposing assumptions on the spectral radius of an operator is better than imposing on its norm, because the first is intrinsic and the second one is not. But for certain sets of bounded operators one can change the norm of the space to an equivalent norm in such a way that the norms of the bounded operators are as near as we wish to their spectral radius. This fact has been widely used by many authors, including C. Pugh classical paper [1], for example. In our previous papers Rodrigues-Sola Morales [2] and [3] we also used this fact, and in [2] we proved that if we have a finite family of bounded linear operators T n , such that T n commutes with T m for every m, n, then it is possible to define a new norm on the Banach space X, equivalent to the initial norm,

The purpose of this paper is to review the results obtained by the authors on linearization of dynamical systems in infinite dimensional Banach spaces, especially in the C 1 case, and also to present some open problems that we believe that are still important for the understanding of the theory.

This paper is devoted to modeling and analyzing the behavior of a platoon that is formed of semiautomated vehicles and behaves like a virtual train in which the first vehicle is in the leading position while the other members of the train track just the preceding one. It is supposed that the dynamic models of the carts are only approximately known but their loads and the nature of the coupling (connection) between the carts and their lodas are completely unknown by the controller. It also is assumed that various local tracking strategies can be applied between the members as "adaptive" and "non-adaptive" "distance and relative velocity tracking", and "acceleration tracking". It is also assumed that the drives of the carts have limited driving forces that is modeled in the simulations by the use of smooth sigmoid functions. The carts are approximately modeled as rigid bodies while their loads are represented by masses connected to the appropriate carts by elastic springs having viscous friction. The aim is to guarantee the possible smoothest motion for the last element of the train that may be followed by various participants of the road communication whose driving comfort has to be kept in mind. This task corresponds to the control of an imprecisely known strongly coupled nonlinear system also having unmodeled coupled internal degrees of freedom. It is shown via computer simulation that the novel, geometrically interpreted adaptive control that in certain cases can be a simple alternative of Lyapunov's 2 nd Method can successfully solve this complicated task in decentralized manner, and can also reduce the jogging of the loads of the carts.

In this paper, we study the M-iteration process in hyperbolic spaces and prove some strong and -convergence theorems of this iteration process for generalized nonexpansive mappings. Moreover, we establish the weak w 2 -stability and data dependence theorems for a class of contractive-type mappings by using M-iteration process. The results presented here extend and improve some recent results announced in the current literature.

Let {e iλnt } n∈Z be an exponential Schauder Basis for L 2 (0, 1), for λ n ∈ R, and let {r n (t)} n∈Z be its dual Schauder Basis. Let A be a non-empty subset of the integers containing exactly M elements. We prove that for α > 0 the weighted system {t α • r n (t)} n∈Z\A is exact in the space L 2 (0, 1), that is, it is complete and minimal in L 2 (0, 1), if and only if M − 1 2 ≤ α < M + 1 2. We also show that such a system is not a Riesz Basis for L 2 (0, 1). In particular, the weighted trigonometric system {t α • e 2πint } n∈Z\A is exact in L 2 (0, 1), if and only if α ∈ [M − 1 2 , M + 1 2), but it is not a Schauder Basis for L 2 (0, 1).

This paper presents an integration of Nye's extended Einstein field equations with the King-Oyewole Unified Geometric Wave Theory (KO-UGWT) in Banach space formalism. We develop a comprehensive mathematical framework that combines quantum corrections to general relativity with advanced concepts from KO-UGWT, including computational density and higher-dimensional constructs. The resulting unified theory offers novel insights into quantum gravity and cosmology, predicting fine-structure constant variations, CMB non-Gaussianities, and modifications to gravitational wave propagation. Our framework provides a set of testable predictions spanning various scales of physical phenomena, from laboratory quantum experiments to cosmological observations.

This paper presents a comprehensive framework integrating the King-Oyewole Unified Geometric Wave Theory (KO-UGWT) and Enhanced Grand Unified Theory (EGUT) to address fundamental questions in quantum mechanics, cosmology, and gravity. By synthesizing concepts from Banach space formalism, Hilbert space fragmentation, holographic principles, and the Generalized Uncertainty Principle, we develop a unified approach to model quantum state normalization, cosmic expansion, and dark energy. Our framework extends quantum state representation using Banach spaces, incorporates computational density concepts, and utilizes tensor mathematics to describe field interactions and spacetime curvature. We explore the implications of this integrated theory for quantum correlations, black hole thermodynamics, and the nature of spacetime. The paper offers potential resolutions to the cosmological constant problem, proposes a dynamic dark energy model, and predicts specific quantum gravitational effects. Furthermore, we outline experimental setups and observational tests to validate our theoretical predictions, ranging from laboratory experiments with ultracold atoms to cosmic observations of high-energy phenomena.

This paper presents a unified theoretical framework that integrates the Unified Geometric Wave Theory (UGWT) with advanced concepts in quantum gravity, computational complexity, and cosmology. By synthesizing UGWT's wave-centric approach with extended quantum state normalization in Banach spaces and computational density in cosmic structures, we develop a comprehensive model that bridges microscopic quantum phenomena with macroscopic cosmological processes. Our framework addresses fundamental questions in physics, including the nature of quantum gravity, the origin of cosmic structure, and the evolution of the universe. We present rigorous mathematical formulations that unify quantum field theories, gravitational physics, and cosmological models, incorporating concepts such as infinity groupoids and emergent topology. Key contributions include a generalized wave equation in Banach spaces, an information-theoretic formulation of computational density, and a set of quantum gravity field equations. We propose a dynamic dark energy model arising from nested universe structures and offer potential resolutions to long-standing issues such as the black hole information paradox and cosmological singularities. Furthermore, we propose experimental tests and observational consequences of this unified framework, spanning from high-energy physics to cosmological observations. This interdisciplinary approach provides new insights into the fundamental nature of reality, the emergence of classical spacetime from quantum processes, and the role of information in the universe's structure and evolution.

This paper explores the possibility of retrieving information from black holes by leveraging quantum entanglement properties. By extending previous work on extended quantum state normalization and cosmological models, we hypothesize that information entangled within black holes can be accessed through their interconnected nested structures. We synthesize theoretical models and recent research to provide a comprehensive framework for these phenomena, potentially revolutionizing quantum communication and computing across cosmic scales. Our approach integrates concepts from Banach space formalism, Hilbert space fragmentation, and the ruliad to propose novel methods for quantum information processing using cosmic-scale quantum effects.

The integration of quantum mechanics and cosmology has led to novel theoretical advancements, particularly in understanding the dynamic interplay between black holes and the universe's expansion. This manuscript explores the concept of nested universes within black holes, using advanced mathematical frameworks such as Banach spaces and Hilbert space fragmentation. By treating quantum state magnitudes as continuous variables and using density as a measure of entanglement, we propose a unified model that accounts for the complex interactions within and between nested universes. Our approach builds upon our previous work on extended quantum state normalization [1] and computational density in black holes [2], offering a novel perspective on the structure of the universe and the fundamental laws governing it.

This paper presents a novel quantum computing paradigm that integrates Banach space formalism, Hilbert space fragmentation, and n-dimensional spheres to enhance quantum state representation and manipulation. By extending quantum state normalization beyond traditional constraints, this approach provides a flexible and robust framework for advanced quantum computing applications, including quantum simulations, error correction, and communication protocols. We discuss the theoretical foundations, mathematical formalism, and potential implications of this integrated framework, demonstrating its potential to address key challenges in quantum computing and pave the way for more powerful and versatile quantum technologies.

Dr. Femi Oyewole explores the integration of Banach space formalism with the entropy and volume of black holes, relating the concept of computational density to entanglement density. By leveraging recent advancements in mathematical physics and quantum mechanics, this paper provides a comprehensive framework that enhances quantum computing and communication. It demonstrates how this integration offers new insights into quantum state representation, quantum error correction, and the development of advanced quantum algorithms. The approach bridges the gap between quantum mechanics and computational physics, with significant implications for the future of quantum information processing in extreme gravitational environments.

This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).

Dr. Femi Oyewole presents a novel framework that integrates Belief Propagation with Quantum Messages (BPQM) with Banach space formalism, Hilbert space fragmentation, and n-dimensional spheres to enhance quantum state representation and manipulation. This approach provides a flexible and robust foundation for advanced quantum computing applications, including quantum simulations, error correction, and communication protocols. The paper discusses the theoretical foundations, mathematical formalism, and potential implications of this integrated framework, demonstrating its potential to address key challenges in quantum computing and pave the way for more powerful and versatile quantum technologies.

This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).

Our previous work introduced a novel mathematical framework for quantum computing, incorporating five continuous variables as measures of probability: position within the sphere, magnitude, phase, volume, and density, as opposed to the traditionally used two continuous variables (position and phase based only on surface area). This paper further develops these findings and proposes an integration of these methodologies to support a quantum computing paradigm where the state magnitude ‘r’ is a continuous variable normalized based on the volume of a sphere. Additionally, we explore the use of density as a measure of entanglement in parallel with its role as a continuous variable. This integration could potentially offer new insights into state-dependent physical quanti- ties and the stabilization of quantum states, paving the way for more robust and versatile quantum computing systems.

Recent advancements in quantum mechanics have explored extending traditional Hilbert space formalism to more generalized frameworks. This paper proposes an integration of Banach space formalism and Hilbert space fragmentation to support a quantum computing paradigm where the state magnitude is a continuous variable. We introduce a novel normalization scheme based on n-dimensional spheres, offering new insights into state-dependent physical quantities and quantum state stabilization. This approach paves the way for more robust and versatile quantum computing systems, potentially addressing key challenges in quantum state manipulation and error correction.

Con la ayuda de métodos topológicos se demuestra existencia y estabilidad de soluciones periódicas para ciertas ecuaciones diferenciales escalares de segundo orden con singularidades en la variable de estado. En el Capítulo 1 hacemos una breve descripción del estado del arte de las ecuaciones diferenciales con singularidades. En el Capítulo 2 presentamos las herramientas topológicas que usaremos en esta memoria: métodos de sub-super soluciones, teoría de grado y algunas propiedades del índice topológico. En el Capítulo 3 nos ocupamos de la existencia y estabilidad de soluciones periódicas de ecuaciones no lineales del tipo Liénard con una o dos singularidades débiles. El método de la demostración se basa en la construcción de una región positivamente invariante del flujo combinado con propiedades del índice topológico. En el Capítulo 4, usando cotas a priori, grado de Leray-Schauder e índice topológico, nosotros damos resultados de existencia, multiplicidad y estabilidad de soluciones periódicas de un modelo masa-resorte no autónomo que idealiza la acción electrostática de un sistema micro-electro-mecánico (MEMS). Una bifurcación silla-nodo se revela, en forma análoga con en el caso autónomo. Finalmente, en el Capítulo 5, usando grado de Leray-Schauder y cotas a priori, se obtienen condiciones suficientes para la existencia de soluciones periódicas para familias de ecuaciones del tipo Lazer-Solimini con retraso dependiente del estado.

We show that the space of bounded linear operators between spaces of continuous functions on compact Hausdorff topological spaces has the Bishop-Phelps-Bollobás property. A similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and Hausdorff topological space into a uniformly convex space, and for the class of compact operators from a Banach space into a predual of an L 1-space.

In the present paper, we study the J-iterative scheme of Bhutia and Tiwary (J. Linear Topol. Algebra, 8(4), (2019), 237-250) in Kohlenbach hyperbolic space. We prove the weak w 2-stability and data dependence theorems of this iterative scheme for contraction mappings. We also give some-convergence and strong convergence theorems for generalized α-nonexpansive mappings and finite families of total asymptotically nonexpansive mappings using J-iterative scheme. The results presented here can be viewed as a generalization of several well-known results in CAT(0) space and uniformly convex Banach space.