Banach Space Of Continuous Functions Research Articles

Subharmonic additions to the Beurling–Malliavin Theorems. II. On the completeness radius

The Beurling–Malliavin Theorem on the multiplier, considered in a subharmonic framework in the first part of our work, already in its original classical version within the framework of entire functions of exponential type, allowed in the 1960s to completely solve the problem of the radius of completeness of exponential system in the form of a remarkable criterion and exclusively in geometric terms for the exponents of this exponential system without any additional restrictions on the relative position of these exponents. The exact formulations of the Beurling– Malliavin Theorem on the radius of completeness in the introduction are somewhat adapted as a problem of the possible minimum growth along the real axis R of subharmonic functions with given constraints on their Riesz distributions of masses. In this largely overview part of the paper, we discuss the Beurling–Malliavin Theorem on the radius of completeness, along with its somewhat more general subharmonic manifestations. Thus, our results from 2014-16 allow us to obtain significantly more subtle results with respect to the Beurling – Malliavin Theorem itself on the radius of completeness with a defect excess of no more than 1 or 2 for exponents in classical rigid Banach spaces of continuous functions on a segment of fixed length or functions with integrable module in the p-th degree on such segments.

Banach spaces of continuous functions without norming Markushevich bases

AbstractWe investigate the question whether a scattered compact topological space K such that has a norming Markushevich basis (M‐basis, for short) must be Eberlein. This question originates from the recent solution, due to Hájek, Todorčević and the authors, to an open problem from the 1990s, due to Godefroy. Our prime tool consists in proving that does not embed in a Banach space with a norming M‐basis, thereby generalising a result due to Alexandrov and Plichko. Subsequently, we give sufficient conditions on a compact K for not to embed in a Banach space with a norming M‐basis. Examples of such conditions are that K is a zero‐dimensional compact space with a P‐point, or a compact tree of height at least . In particular, this allows us to answer the said question in the case when K is a tree and to obtain a rather general result for Valdivia compacta. Finally, we give some structural results for scattered compact trees; in particular, we prove that scattered trees of height less than ω2 are Valdivia.

Existence and uniqueness of blow-up solution to a fully fractional thermostat model

In this article, we deal with a fully fractional thermostat model in the settings of the Riemann–Liouville fractional derivatives. The equivalences between the fractional differential equations and the corresponding Volterra–Fredholm integral equations are rigorously derived. By choosing an appropriate weighted Banach space of continuous functions, we employ two standard fixed-point theorems, Leray–Schauder alternative and Banach contraction principle, to establish the existence of blow-up solutions — the unbounded mathematical solutions in the operational interval. Furthermore, an implicit numerical scheme based on the right product rectangle rule is presented, which provides the numerical approximation of the obtained solution. Some examples are provided to validate our theoretical findings, along with numerical simulations of the solutions.

Differentiability of the transition semigroup of the stochastic Burgers-Huxley equation and application to optimal control

In this present paper we consider the transition semigroup {Pt}t≥0 related to the stochastic Burgers-Huxley equation which describes a prototype model for describing the interaction between reaction mechanisms, convection effects and diffusion transports. We are proving that it has a regularizing effect in the Banach space of continuous functions C(0,1), that is, the transition semigroup {Pt}t≥0 possesses strong Feller property. Some estimates on derivative of the transition semigroup related to stochastic Burgers-Huxley equation are established based on the exponential moment estimates of the solution of stochastic Burgers-Huxley equation and its first variation equation. Finally, certain application in Hamilton-Jacobi-Bellman equation arising from stochastic optimal control problem is provided to illustrate our results.

Functional autoregressive process with seasonality

We study the estimation of a seasonality perturbed by a continuous time process admitting a -valued autoregressive representation where is the Banach space of continuous functions on We provide the almost sure convergence, asymptotic normality and compact iterated logarithm law. Following Antoniadis A. (1982) we construct in the framework of functional autoregressive processes (non i.i.d. case), confidence balls for the seasonality in the space from compact iterated logarithm law. Then when seasonality belongs to a finite dimensional space (dimensional reduction), we study the seasonality estimation giving its asymptotic properties. Finally, we examine an estimator of the dimension of this space when it is unknown. Numerical simulations illustrate the asymptotic results of the estimators.

A generalization of Ascoli–Arzelá theorem inCnwith application in the existence of a solution for a class of higher-order boundary value problem

PurposeA generalization of Ascoli–Arzelá theorem in Banach spaces is established. Schauder's fixed point theorem is used to prove the existence of a solution for a boundary value problem of higher order. The authors’ results are obtained under, rather, general assumptions.Design/methodology/approachFirst, a generalization of Ascoli–Arzelá theorem in Banach spaces inCnis established. Second, this new generalization with Schauder's fixed point theorem to prove the existence of a solution for a boundary value problem of higher order is used. Finally, an illustrated example is given.FindingsThere is no funding.Originality/valueIn this work, a new generalization of Ascoli–Arzelá theorem in Banach spaces inCnis established. To the best of the authors’ knowledge, Ascoli–Arzelá theorem is given only in Banach spaces of continuous functions. In the second part, this new generalization with Schauder's fixed point theorem is used to prove the existence of a solution for a boundary value problem of higher order, where the derivatives appear in the non-linear terms.

ω-граничнi множини для iмпульсно-збуреного параболiчного рiвняння в просторi неперервних функцiй

We consider an important class of disturbed infinite-dimensional systems provided by evolutionary equations with impulsive effects. An impulsive dynamical system acting in the phase space H is a hybrid system described by an evolutionary equation together with impulsive disturbances which, in turn, are characterized by a closed set M ⊂ H (impulsive set) and a mapping I: M -> H (impulsive mapping). A solution of such a system, by definition, is a left-continuous function y satisfying evolutionary equation for all times t for which y(t) \notin M and jumping to a new state I(y(t)) at all times t for which y(t) reaches M. The main problem in the study of qualitative behavior of impulsive dynamical systems by classical methods is that solutions usually do not depend continuously on initial data. A number of examples illustrate already in the finite-dimensional case that new complex scenarios of possible limit behaviors of impulsive trajectories can occur, which are fundamentally impossible in the non-impulsive case. Results concerning attracting sets for infinite-dimensional impulsive systems have appeared only a few years ago. But all those results concern the limit behavior of impulsive semiflows in Hilbert space and the presence of scalar product played crucial role in those construction. In this paper we want to investigate ω-limit attracting sets for impulsive-disturbed parabolic system in the sup-norm of Banach space of continuous functions.

The form of locally defined operators in waterman spaces

Abstract A representation formula for locally defined operators acting between Banach spaces of continuous functions of bounded variation in the Waterman sense is presented. Moreover, the Nemytskij composition operators will be investigated and some consequences for locally bounded as well as uniformly bounded local operators will be given.

The chaotic Black–Scholes equation with time-dependent coefficients

In Emamirad et al. (in: Semigroups of Operators - Theory and Applications, Springer, 2014; and Proc. Amer. Math. Soc. 140:2043–2052, 2012), the Black–Scholes semigroup was studied in various Banach spaces of continuous functions regarding chaotic behavior and the null volatility limit. Here we let the volatility and the interest rates be continuous functions on the half line $$[0,\infty )$$ , and we consider the generalized Black–Scholes equation as a linear nonautonomous abstract Cauchy problem. It is shown that this problem is wellposed and an explicit formula for the corresponding strongly continuous evolution family is given. Furthermore, a hypercyclicity criterion for strongly continuous evolution families on separable Banach spaces is proved and applied to the Black–Scholes evolution family. Finally, the chaotic behavior is defined for strongly continuous evolution families and it is shown that the Black–Scholes evolution family is chaotic when the volatility and the interest rates are periodic functions.

Functional data analysis in the Banach space of continuous functions

Functional data analysis is typically conducted within the $L^{2}$-Hilbert space framework. There is by now a fully developed statistical toolbox allowing for the principled application of the functional data machinery to real-world problems, often based on dimension reduction techniques such as functional principal component analysis. At the same time, there have recently been a number of publications that sidestep dimension reduction steps and focus on a fully functional $L^{2}$-methodology. This paper goes one step further and develops data analysis methodology for functional time series in the space of all continuous functions. The work is motivated by the fact that objects with rather different shapes may still have a small $L^{2}$-distance and are therefore identified as similar when using a $L^{2}$-metric. However, in applications it is often desirable to use metrics reflecting the visualization of the curves in the statistical analysis. The methodological contributions are focused on developing two-sample and change-point tests as well as confidence bands, as these procedures appear to be conducive to the proposed setting. Particular interest is put on relevant differences; that is, on not trying to test for exact equality, but rather for prespecified deviations under the null hypothesis. The procedures are justified through large-sample theory. To ensure practicability, nonstandard bootstrap procedures are developed and investigated addressing particular features that arise in the problem of testing relevant hypotheses. The finite sample properties are explored through a simulation study and an application to annual temperature profiles.

On Banach spaces of continuous functions on finite products of separable compact lines

On Banach spaces of continuous functions on finite products of separable compact lines

Which multiplication operators are surjective isometries?

Let F be a Banach space of continuous functions over a connected locally compact space X. We present several sufficient conditions on F guaranteeing that the only multiplication operators on F that are surjective isometries are scalar multiples of the identity. The conditions are given via the properties of the inclusion operator from F into C(X), as well as in terms of geometry of F. An important tool in our investigation is the notion of Birkhoff Orthogonality.

On the second-order Fréchet derivatives of eigenvalues of Sturm–Liouville problems in potentials

The works of V.A. Vinokurov have shown that eigenvalues and normalized eigenfunctions of Sturm–Liouville problems are analytic in potentials, considered as mappings from the Lebesgue space to the space of real numbers and the Banach space of continuous functions respectively. Moreover, the first-order Frechet derivatives are known and play an important role in many problems. In this paper, we will find the second-order Frechet derivatives of eigenvalues in potentials, which are also proved to be negative definite quadratic forms for some cases.

Large separated sets of unit vectors in Banach spaces of continuous functions

The paper concerns the problem of whether a nonseparable $\mathcal {C}(K)$ space must contain a set of unit vectors whose cardinality equals the density of $\mathcal {C}(K)$, and such that the distances between any two distinct vectors are always greater

On surjections between Banach spaces of continuous functions on separable nonmetrizable compact lines

On surjections between Banach spaces of continuous functions on separable nonmetrizable compact lines

Mean Ergodic Multiplication Operators on Weighted Spaces of Continuous Functions

Multiplication operators on weighted Banach spaces and locally convex spaces of continuous functions have been thoroughly studied. In this note, we characterize when continuous multiplication operators on a weighted Banach space and on a weighted inductive limit of Banach spaces of continuous functions are power bounded, mean ergodic or uniformly mean ergodic. The behaviour of the operator on weighted inductive limits depends on the properties of the defining sequence of weights and it differs from the Banach space case.

Stability of ε-isometric embeddings into Banach spaces of continuous functions

Let X be a Banach space, T be a compact Hausdorff space and C(T) be the real Banach space of all continuous functions on T endowed with the supremum norm. We show that if there exists a standard ε-isometric embedding f:X→C(T), then there are nonempty closed subset S⊆T and a linearly isometric embedding g:X→span‾(f(X)|S)⊂C(S) defined as g(u)=limn→∞⁡f(2nu)|S2n for each u∈X satisfying that‖f(u)|S−g(u)‖≤4εfor all u∈X. Making use of this result and the well known simultaneous extension operator E:C(S)→C(T), we also prove that the existence of a standard ε-isometric embedding f:X→C(T) implies the existence of a linearly isometric embedding E∘g:X→span‾(E(f(X)|S))⊆C(T) whenever T is metrizable. These conclusions generalize several well-known results. For any compact Hausdorff space (resp. compact metric space) T, we further obtain that if g(X) is complemented in span‾(f(X)|S) (resp. E∘g(X) is complemented in span‾(E(f(X)|S))), then the standard ε-isometric embedding f:X→C(T) is stable.

Positive solutions of nonlinear delayed differential equations with impulses

The paper is concerned with the long-term behavior of solutions to scalar nonlinear functional delayed differential equations y ̇ ( t ) = − f ( t , y t ) , t ≥ t 0 . It is assumed that f : [ t 0 , ∞ ) × C ↦ R is a continuous mapping satisfying a local Lipschitz condition with respect to the second argument and C ≔ C ( [ − r , 0 ] , R ) , r > 0 is the Banach space of continuous functions. The problem is solved of the existence of positive solutions if the equation is subjected to impulses y ( t s + ) = b s y ( t s ) , s = 1 , 2 , … , where t 0 ≤ t 1 < t 2 < ⋯ and b s > 0 , s = 1 , 2 , … . A criterion for the existence of positive solutions on [ t 0 − r , ∞ ) is proved and their upper estimates are given. Relations to previous results are discussed as well.

English

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation (P) $$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$ where {ℒ(t): t ∈ R} is a family of linear operators from a Banach space E into itself and f: R × E → E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a 0, we let C([−d, 0],E) be the Banach space of continuous functions from [−d, 0] into E and f d : [a, b] × C([−d, 0],E) → E. Let $$\hat {\mathcal{L}}:[a,b] \to L(E)$$ be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τ t x(s) = x(t + s) for each s ∈ [−d, 0]. We prove that, under certain conditions, the differential equation with delay (Q) $$\dot x\left( t \right) = \hat {\mathcal{L}}\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right),ift \in \left[ {a,b} \right],$$ has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too.

On some geometric properties of Banach spaces of continuous functions on separable compact lines

We study properties of Banach spaces $C(L)$ of all continuous scalar (real or complex) functions on compact lines&nbsp;$L$. First we show that if $L$ is a separable compact line, then for every closed linear subspace $X$ of $C(L)$ with separable dual the