Arbitrary Banach Space Research Articles

Randomized approximation of summable sequences — adaptive and non-adaptive

We prove lower bounds for the randomized approximation of the embedding ℓ1m↪ℓ∞m based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix N∈Rn×m. These lower bounds reflect the increasing difficulty of the problem for m→∞, namely, a term logm in the complexity n. This result implies that non-compact operators between arbitrary Banach spaces are not approximable using non-adaptive Monte Carlo methods. We also compare these lower bounds for non-adaptive methods with upper bounds based on adaptive, randomized methods for recovery for which the complexity n only exhibits a (loglogm)-dependence. In doing so we give an example of linear problems where the error for adaptive vs. non-adaptive Monte Carlo methods shows a gap of order n1/2(logn)−1/2.

A multiparametric family of solutions to the Volterra linear integral equation of the first kind

We study the Volterra integral equation of the first kind with an integral operator of order n, a singularity and a sufficiently smooth kernel in a certain Banach space with weight. It reduces to an integro-differential equation with two terms on the left-hand side. The first term corresponds to an equation for which an explicitly multiparameter family of solutions is constructed. For the second term, we obtain an equation with an operator whose norm in an arbitrary Banach space is arbitrarily small near zero. Such splitting of the integral operator allows one to construct a particular and general solutions to the integro-differential equation in the corresponding Banach space in the form of convergent series. Thus, under certain restrictions on the operator pencil corresponding to a given integral operator, a multi-parameter family of solutions is being constructed for the original integral equation.

Exact calculus for s-Holder subgradients in Banach spaces

We give in this paper some useful calculus results related to the s-H$\ddot{o}$lder subdifferential of extended-real valued functions defined on arbitrary real Banach spaces.

A resolvent approach for the stability of the critical or essential spectra to perturbed C0-semigroups on Banach spaces with application in biological transport

We give sufficient conditions in terms of the resolvents assuring the stability of the critical or essential spectra to perturbed C0-semigroups on an arbitrary given Banach space. Our theoretical result applies in Lp-biological transport, 1≤p<+∞. Our results improve and complete, significatively, well known earlier ones.

On the Existence and Uniqueness of Fuzzy Mild Solution of Fractional Evolution Equations

In this paper, the nonlocal Cauchy problem is discussed for the fuzzy fractional evolution equations in an arbitrary Banach space for order q ∈ (1, 2) and the criteria on the existence and uniqueness of mild fuzzy solutions are obtained by using Schauder’s fixed point theorem. An example to illustrate the applications of main results is also given.

On Whitney-type extension theorems on Banach spaces for C1,ω, C1,+, [formula omitted], and [formula omitted]-smooth functions

Our paper is a complement to a recent article by D. Azagra and C. Mudarra (2021, [2]). We show how older results on semiconvex functions with modulus ω easily imply extension theorems for C1,ω-smooth functions on super-reflexive Banach spaces which are versions of some theorems of Azagra and Mudarra. We present also some new interesting consequences which are not mentioned in their article, in particular extensions of C1,ω-smooth functions from open quasiconvex sets. They proved also an extension theorem for CB1,+-smooth functions (i.e., functions with uniformly continuous derivative on each bounded set) on Hilbert spaces. Our version of this theorem and new extension results for C1,+ and Cloc1,+-smooth functions (i.e., functions with uniformly, resp. locally uniformly continuous derivative), all of which are proved on arbitrary super-reflexive Banach spaces, are further main contributions of our paper. Some of our proofs use main ideas of the article by D. Azagra and C. Mudarra, but all are formally completely independent on their article.

An extension of Brøndsted-Rockafellar's theorem with applications

In 1965, Brøndsted and Rockafellar [6] established that the domain of Fenchel subdifferential of a closed proper and convex function is dense in its domain. Our purpose is to show that the theorem is valid if instead, one considers functions more general than convex, these functions are called paraconvex. As an application, we extend Attouch's Theorem as well as a result of Combari and Thibault [7] concerning the graph convergence of subdifferentials of a sequence of convex functions to the graph convergence of appropriate subdifferentials of a sequence of paraconvex functions in the setting of arbitrary Banach space.

On the Study of Semilinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative

This work aims to study the existing results of mild solutions for a semi-linear Atangana-Baleanu-Caputo fractional differential equation with order $ 0 &amp;lt; \theta &amp;lt; 1 $ in an arbitrary Banach space. We rely on some arguments to present the mild solution to our problem in terms of an $ \theta $-resolvent family. Then we study the existence of this mild solution by using Krasnoselskii's fixed point theorem. Finally, we give an example to prove our results.

On simultaneous similarity of families of commuting operators

Characterization of simultaneous similarity for commuting m m - tuples of operators is an open problem even in finite-dimensional spaces; known as “A wild problem in linear algebra”. In this paper we offer a criterion for simultaneous similarity of m m -tuples of k k -cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m m -tuples of commuting square matrices.

On stability of almost surjective functional equations of uniformly convex Banach spaces

Let Y be a uniformly convex space with power type p, and let (G,+) be an abelian group, delta ,varepsilon geq 0, 0< r<1. We first show a stability result for approximate isometries from an arbitrary Banach space into Y. This is a generalization of Dolinar’ results for (delta ,r)-isometries of Hilbert spaces and L_{p} (1< p<infty ) spaces. As a result, we prove that if a standard mapping F:Grightarrow Y satisfies d(u,F(G))leq delta |u|^{r} for every uin Y and |∥F(x)−F(y)∥−∥F(x−y)∥|≤ε,x,y∈G,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\bigl\\vert \\bigl\\Vert F(x)-F(y) \\bigr\\Vert - \\bigl\\Vert F(x-y) \\bigr\\Vert \\bigr\\vert \\leq \\varepsilon , \\quad x,y \\in G, $$\\end{document} then there is an additive operator A:Grightarrow Y such that ∥F(x)−Ax∥=o(∥F(x)∥)as ∥F(x)∥→∞.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\bigl\\Vert F(x)-Ax \\bigr\\Vert =o\\bigl( \\bigl\\Vert F(x) \\bigr\\Vert \\bigr) \\quad \ ext{as } \\bigl\\Vert F(x) \\bigr\\Vert \\rightarrow \\infty . $$\\end{document}

Weak precompactness in projective tensor products

We give a sufficient condition for a pair of Banach spaces (X,Y) to have the following property: whenever W1⊆X and W2⊆Y are sets such that {x⊗y:x∈W1,y∈W2} is weakly precompact in the projective tensor product X⊗̂πY, then either W1 or W2 is relatively norm compact. For instance, such a property holds for the pair (ℓp,ℓq) if 1<p,q<∞ satisfy 1/p+1/q≥1. Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ℓ1 into X⊗̂πY for arbitrary Banach spaces X and Y, in connection with the compactness of all operators from X to Y∗.

Phase-isometries on the unit sphere of CL-spaces

A mapping f:SX→SY between the unit spheres of two real Banach spaces X and Y is said to be a phase-isometry if it satisfies{‖f(x)+f(y)‖,‖f(x)−f(y)‖}={‖x+y‖,‖x−y‖} for all x,y∈SX. When f is surjective, X is a real CL-space and Y is an arbitrary real Banach space, we establish in this paper that there exists a phase-function ε:SX→{−1,1} such that ε⋅f is an isometry which is the restriction of a linear isometry from X to Y.

Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order Dzhrbashyan–Nersesyan derivatives, and an unknown element. The inverse problem is given by an equation, special initial value conditions for lower Dzhrbashyan–Nersesyan derivatives, and an overdetermination condition, which is defined by a linear continuous operator. Applying the fixed-point method for contraction mapping a theorem on the existence of a local unique solution is proved under the condition of local Lipschitz continuity of the nonlinear mapping. Analogous nonlocal results were obtained for the case of the nonlocally Lipschitz continuous nonlinear operator in the equation. The obtained results for the problem in arbitrary Banach spaces were used for the research of nonlinear inverse problems with time-dependent unknown coefficients at lower-order Dzhrbashyan–Nersesyan time-fractional derivatives for integro-differential equations and for a linearized system of dynamics of fractional Kelvin–Voigt viscoelastic media.

On the property (C) of Corson and other sequential properties of Banach spaces

A well-known result of R. Pol states that a Banach space X has property (C) of Corson if and only if every point in the weak*-closure of any convex set C⊆BX⁎ is actually in the weak*-closure of a countable subset of C. Nevertheless, it is an open problem whether this is in turn equivalent to the countable tightness of BX⁎ with respect to the weak*-topology. Frankiewicz, Plebanek and Ryll-Nardzewski provided an affirmative answer under MA+¬CH for the class of C(K)-spaces. In this article we provide a partial extension of this latter result by showing that under the Proper Forcing Axiom (PFA) the following conditions are equivalent for an arbitrary Banach space X:(1)X has property E′;(2)X has weak*-sequential dual ball;(3)X has property (C) of Corson;(4)(BX⁎,w⁎) has countable tightness. This provides a partial extension of a former result of Arhangel'skii. In addition, we show that every Banach space with property E′ has weak*-convex block compact dual ball.

Abstract almost periodicity for group actions on uniform topological spaces

AbstractWe present a unified theory for the almost periodicity of functions with values in an arbitrary Banach space, measures and distributions via almost periodic elements for the action of a locally compact abelian group on a uniform topological space. We discuss the relation between Bohr- and Bochner-type almost periodicity, and similar conditions, and how the equivalence among such conditions relates to properties of the group action and the uniformity. We complete the paper by demonstrating how various examples considered earlier all fit in our framework.

Daugavet property of Banach algebras of holomorphic functions and norm-attaining holomorphic functions

We show that the duals of Banach algebras of scalar-valued bounded holomorphic functions on the open unit ball BE of a Banach space E lack weak⁎-strongly exposed points. Consequently, we obtain that some Banach algebras of holomorphic functions on an arbitrary Banach space have the Daugavet property which extends the observation of P. Wojtaszczyk [56]. Moreover, we present a new denseness result by proving that the set of norm-attaining vector-valued holomorphic functions on the open unit ball of a dual Banach space is dense provided that its predual space has the metric π-property. Besides, we obtain several equivalent statements for the Banach space of vector-valued homogeneous polynomials to be reflexive, which improves the result of J. Mujica [47], J. A. Jaramillo and L. A. Moraes [39]. As a byproduct, we generalize some results on polynomial reflexivity due to J. Farmer [35].

Abstract Univariate Neural Network Approximation Using a q-Deformed and λ-Parametrized Hyperbolic Tangent Activation Function

In this work, we perform univariate approximation with rates, basic and fractional, of continuous functions that take values into an arbitrary Banach space with domain on a closed interval or all reals, by quasi-interpolation neural network operators. These approximations are achieved by deriving Jackson-type inequalities via the first modulus of continuity of the on hand function or its abstract integer derivative or Caputo fractional derivatives. Our operators are expressed via a density function based on a q-deformed and λ-parameterized hyperbolic tangent activation sigmoid function. The convergences are pointwise and uniform. The associated feed-forward neural networks are with one hidden layer.

Inertial Iteration Scheme for Approximating Fixed Points of Lipschitz Pseudocontractive Maps in Arbitrary Real Banach Spaces

"We study a perturbed inertial Krasnoselskii-Mann-type algorithm and prove that the algorithm is an approximate fixed point sequence for Lipschitz pseudocontractive maps in arbitrary real Banach spaces. Strong convergence results are then established for our inertial iteration scheme for approximation of fixed points of Lipschitz pseudocontractive maps and solutions of certain important accretive-type operator equations in certain real Banach spaces. Implementation of our algorithm is illustrated using numerical examples in both finite and infinite dimensional Banach spaces. Our results improve rate of convergence and extend several related recent results. "

Invariant manifolds for difference equations with generalized trichotomies

On an arbitrary Banach space, assuming that a linear nonautonomous difference equation \linebreak $x_{m+1} = A_m x_m$ admits a very general type of trichotomy, we establish conditions for the existence of global Lipschitz invariant center manifolds of the perturbed equation $x_{m+1} = A_m x_m + f_m(x_m)$. Our results not only improve results already existing in the literature, but also include new cases.

Arithmetic Progressions and Chaos in Linear Dynamics

We characterize chaotic linear operators on reflexive Banach spaces in terms of the existence of long arithmetic progressions in the sets of return times. We also show that this characterization does not hold for arbitrary Banach spaces. To achieve this, we study $$\mathcal {F}$$ -hypercyclicity for a family of subsets of the natural numbers associated to the existence of arbitrarily long arithmetic progressions.