Showing papers on "Banach space published in 2014"
Book•
16 Feb 2014
TL;DR: In this paper, the authors present an overview of the history of Gaussian Processes and their application to Banach Space Theory, including Bernouilli Processes, Random Fourier Series and Trigonometric Sums, and the fundamental Conjectures.
Abstract: 0. Introduction.- 1. Philosophy and Overview of the Book.- 2. Gaussian Processes and the Generic Chaining.- 3. Random Fourier Series and Trigonometric Sums, I. - 4. Matching Theorems I.- 5. Bernouilli Processes.- 6. Trees and the Art of Lower Bounds.- 7. Random Fourier Series and Trigonometric Sums, II.- 8. Processes Related to Gaussian Processes.- 9. Theory and Practice of Empirical Processes.- 10. Partition Scheme for Families of Distances.- 11. Infinitely Divisible Processes.- 12. The Fundamental Conjectures.- 13. Convergence of Orthogonal Series Majorizing Measures.- 14. Matching Theorems, II: Shor's Matching Theorem. 15. The Ultimate Matching Theorem in Dimension => 3.- 16. Applications to Banach Space Theory.- 17. Appendix: What this Book is Really About.- 18. Appendix: Continuity.- References. Index.
216 citations
TL;DR: An extension of Darbo's fixed point theorem associated with measures of noncompactness is given, and some results on the existence of coupled fixed points for a class of condensing operators in Banach spaces are presented.
125 citations
01 Jan 2014
TL;DR: In this article, a review of standard topics from functional analysis such as quasinormed and normed linear spaces and closed, compact and Fredholm linear operators on Banach spaces is devoted to a review.
Abstract: This chapter is devoted to a review of standard topics from functional analysis such as quasinormed and normed linear spaces and closed, compact and Fredholm linear operators on Banach spaces. These topics form a necessary background for what follows. In Sects. 3.1–3.3 we study linear operators and functionals, quasinormed and normed linear spaces. In a normed linear space we consider continuous linear functionals as generalized coordinates of the space. In Sect. 3.4 we prove the Riesz–Markov representation theorem which describes an intimate relationship between Radon measures and non-negative linear functionals on the spaces of continuous functions. This fact constitutes an essential link between measure theory and functional analysis, providing a powerful tool for constructing Markov transition functions in Chap. 9.
121 citations
TL;DR: An existence result of optimal multi-control pairs governed by the presented system is proved and a new kind of Sobolev type appears in terms of two linear operators.
90 citations
TL;DR: By using fixed point theorems of concave operators in partial ordering Banach spaces, the existence and uniqueness of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations for any given parameter are established.
85 citations
TL;DR: In this article, a category-theoretic framework for universal homogeneous objects is developed, with some applications in the theory of Banach spaces, linear orderings, and in the topology of compact Hausdorff spaces.
78 citations
TL;DR: In this article, a sequential variant of the discrepancy principle is analyzed, where exact penalization must be excluded, and discrepancy principle must stop after a finite number of iterations, under two natural assumptions.
Abstract: The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a sequential variant of the discrepancy principle is analysed. In many cases, such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here, we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.
69 citations
Book•
15 Sep 2014
TL;DR: In this article, the authors study the problem of higher smoothness in separable real Banach spaces with respect to the Stone-Weierstrass theorem and its structural consequences.
Abstract: This book is about the subject of higher smoothness in separable real Banach spaces. It brings together several angles of view on polynomials, both in finite and infinite setting. Also a rather thorough and systematic view of the more recent results, and the authors work is given. The book revolves around two main broad questions: What is the best smoothness of a given Banach space, and its structural consequences? How large is a supply of smooth functions in the sense of approximating continuous functions in the uniform topology, i.e. how does the Stone-Weierstrass theorem generalize into infinite dimension where measure and compactness are not available? The subject of infinite dimensional real higher smoothness is treated here for the first time in full detail, therefore this book may also serve as a reference book.
68 citations
TL;DR: In this article, the authors prove the existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces, in terms of a Riemann-Liouville fractional derivative.
Abstract: We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange optimal control problem is considered, and existence of a multi-integral solution obtained. Main tools include fractional calculus, semigroup theory, fractional power of operators, a singular version of Gronwall's inequality, and Leray-Schauder fixed point theorem. An example illustrating the theory is given.
67 citations
TL;DR: In this paper, the authors investigated the concept of spatial distribution for data in infinite dimensional Banach spaces and proved some Glivenko-Cantelli and Donsker-type results for the empirical spatial distribution process in finite dimensional spaces.
Abstract: The spatial distribution has been widely used to develop various nonparametric procedures for finite dimensional multivariate data. In this paper, we investigate the concept of spatial distribution for data in infinite dimensional Banach spaces. Many technical difficulties are encountered in such spaces that are primarily due to the noncompactness of the closed unit ball. In this work, we prove some Glivenko-Cantelli and Donsker-type results for the empirical spatial distribution process in infinite dimensional spaces. The spatial quantiles in such spaces can be obtained by inverting the spatial distribution function. A Bahadur-type asymptotic linear representation and the associated weak convergence results for the sample spatial quantiles in infinite dimensional spaces are derived. A study of the asymptotic efficiency of the sample spatial median relative to the sample mean is carried out for some standard probability distributions in function spaces. The spatial distribution can be used to define the spatial depth in infinite dimensional Banach spaces, and we study the asymptotic properties of the empirical spatial depth in such spaces. We also demonstrate the spatial quantiles and the spatial depth using some real and simulated functional data.
62 citations
TL;DR: A new characterization of the uniform convexity of Banach spaces was obtained in this article, in the sense of the Bishop-Phelps-Bollobas theorem for bilinear forms on.
Abstract: A new characterization of the uniform convexity of Banach space is obtained in the sense of the Bishop–Phelps–Bollobas theorem. It is also proved that the couple of Banach spaces has the Bishop–Phelps–Bollobas property for every Banach space when is uniformly convex. As a corollary, we show that the Bishop–Phelps–Bollobas theorem holds for bilinear forms on .
TL;DR: The relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces was studied in this article. But it was not shown that the norm of an arbitrary Banach space is octagonal unless, and only if, every convex combination of w ⁎ -slices in the dual unit ball has diameter 2.
18 Feb 2014
TL;DR: In this article, the authors characterize the metric spaces whose free space has the bounded approximation property through a Lipschitz analogue of the local reflexivity principle, and show that there exist compact metric spaces with free spaces that fail the approximation property.
Abstract: We characterize the metric spaces whose free space has the bounded approximation property through a Lipschitz analogue of the local reflexivity principle. We show that there exist compact metric spaces whose free spaces fail the approximation property.
TL;DR: A general framework to find the mild solutions for impulsive fractional integro-differential equations with nonlocal condition is established, which will provide an effective way to deal with such problems.
TL;DR: In this article, the existence and multiplicity results for a class of second-order difference equations are established, and the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented.
Abstract: Abstract In this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach is based on variational methods and, in particular, on a local minimum theorem and its variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.
TL;DR: In this article, it was shown that any frequently hypercyclic operator T acting on a reflexive Banach space admits an invariant probability measure with full support, which may be required to vanish on the set of all periodic vectors for T; and if an operator admits an ergodic measure, then it has a comeager set of distributionally irregular vectors.
TL;DR: In this article, the authors characterize quasi-static rate-independent evolutions by means of their graph parame-trization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance.
Abstract: We characterize quasi-static rate-independent evolutions, by means of their graph parame- trization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals F of class C 1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals Fn and its Γ-limit F we provide, under suitable assumptions, a convergence result for the associated quasi-static evolutions. Finally, we apply this approach to a phase field model in brittle fracture.
Posted Content•
TL;DR: In this article, a Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional over closed, smooth Riemannian manifolds of arbitrary dimension was proved.
Abstract: In this monograph, we develop results on global existence and convergence of solutions to abstract gradient flows on Banach spaces for a potential function that obeys the Lojasiewicz-Simon gradient inequality. We prove a Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional over closed, smooth Riemannian manifolds of arbitrary dimension and apply the resulting framework to prove new results for the gradient flow equation for the Yang-Mills energy functional on a principal bundle, with compact Lie structure group, over a closed, smooth Riemannian manifolds, including the following. If the initial connection is close enough to a local minimum of the Yang-Mills energy functional, in a norm sense when the base manifold has arbitrary dimension or in an energy sense when the base manifold has dimension four, then the Yang-Mills gradient flow exists for all time and converges to a Yang-Mills connection. If the initial connection is allowed to have arbitrary energy but we restrict to the setting of a Hermitian vector bundle over a compact, complex, Hermitian (but not necessarily Kaehler) surface and the initial connection has curvature of type (1,1), then the Yang-Mills gradient flow exists for all time, though bubble singularities may (and in certain cases must) occur in the limit as time tends to infinity.
TL;DR: In this paper, a Poincare operator for nonlinear evolution equations with non-instantaneous impulses on Banach spaces is proposed and the existence of a global compact connected attractor is derived.
Abstract: In this paper, we consider periodic solutions for a class of nonlinear evolution equations with noninstantaneous impulses on Banach spaces. By constructing a Poincare operator, which is a composition of the maps and using the techniques of a priori estimate, we avoid assuming that periodic solution is bounded like in [1–4] and try to present new su cient conditions on the existence of periodic mild solutions for such problems by utilizing semigroup theory and Leray-Schauder’s xed point theorem. Furthermore, existence of a global compact connected attractor for the Poincare operator is derived.
TL;DR: In this article, the authors present a survey of some of the results which have been obtained in the last twelve years on the non-linear geometry of Banach spaces, focusing on the contribution of the late Nigel Kalton.
Abstract: This is a survey of some of the results which were obtained in the last twelve years on the non-linear geometry of Banach spaces. We focus on the contribution of the late Nigel Kalton.
TL;DR: In this paper, the statistical properties of a general class of two-dimensional hyperbolic systems with singularities were studied by constructing Banach spaces on which the associated transfer operators are quasi-compact.
Abstract: We study the statistical properties of a general class of two-dimensional hyperbolic systems with singularities by constructing Banach spaces on which the associated transfer operators are quasi-compact. When the map is mixing, the transfer operator has a spectral gap and many related statistical properties follow, such as exponential decay of correlations, the central limit theorem, the identification of Ruelle resonances, large deviation estimates and an almost-sure invariance principle. To demonstrate the utility of this approach, we give two applications to specific systems: dispersing billiards with corner points and the reduced maps for certain billiards with focusing boundaries.
TL;DR: In this article, a new algorithm for solving the multiple-sets split feasibility problem in Banach spaces is proposed, which does not need weak-to-weak continuity of the duality mapping.
Abstract: In this article, we propose a new algorithm for solving the multiple-sets split feasibility problem in Banach spaces. Under some mild conditions we establish the norm convergence of the proposed algorithm. One advantage of the proposed algorithm is that its norm convergence does not need the weak-to-weak continuity of the duality mapping.
TL;DR: This paper investigates the generalized Hyers-Ulam stability of the functional [email protected]?k"2,...,k"n=01fx"[email protected]?i=2n(-1)^k^"^ix"i-2^n^-^1f(x"1)-2+f(-x"i)=0 for integer values of n such that n>=2.
TL;DR: In this paper, the boundary value problem of determining the parameter of an elliptic equation u 00 (t) +Au(t) = f(t +p (0 6 t 6 T), u(0) = '; u(T) = ; u( ) = ; 0 < < T, with a positive operator A in an arbitrary Banach space E is studied.
Abstract: The boundary value problem of determining the parameter of an elliptic equation u 00 (t) +Au(t) = f(t) +p (0 6 t 6 T), u(0) = '; u(T) = ; u( ) = ; 0 < < T , with a positive operator A in an arbitrary Banach space E is studied. The exact estimates are obtained for the solution of this problem in Holder norms. Coercive stability estimates for the solution of boundary value problems for multi-dimensional elliptic equations are established.
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TL;DR: In this paper, the Bochner norm of γ(Omega,X) for functions with values in an arbitrary Banach space was shown to be equivalent to γ (Omega and X).
Abstract: Using notions from the geometry of Banach spaces we introduce square functions $\gamma(\Omega,X)$ for functions with values in an arbitrary Banach space $X$. We show that they have very convenient function space properties comparable to the Bochner norm of $L_2(\Omega,H)$ for a Hilbert space $H$. In particular all bounded operators $T$ on $H$ can be extended to $\gamma(\Omega,X)$ for all Banach spaces $X$. Our main applications are characterizations of the $H^{\infty}$--calculus that extend known results for $L_p$--spaces from \cite{CowlingDoustMcIntoshYagi}. With these square function estimates we show, e. g., that a $c_0$--group of operators $T_s$ on a Banach space with finite cotype has an $H^{\infty}$--calculus on a strip if and only if $e^{-a|s|}T_s$ is $R$--bounded for some $a > 0$. Similarly, a sectorial operator $A$ has an $H^{\infty}$--calculus on a sector if and only if $A$ has $R$--bounded imaginary powers. We also consider vector valued Paley--Littlewood $g$--functions on $UMD$--spaces.
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TL;DR: The point degree spectrum of a represented space is introduced as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees, and so on and creates a connection among various areas of mathematics including computability theory, descriptive set theory, infinite dimensional topology and Banach space theory.
Abstract: We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees, and so on. The notion of point degree spectrum creates a connection among various areas of mathematics including computability theory, descriptive set theory, infinite dimensional topology and Banach space theory. Through this new connection, for instance, we construct a family of continuum many infinite dimensional Cantor manifolds with property $C$ whose Borel structures at an arbitrary finite rank are mutually non-isomorphic. This provides new examples of Banach algebras of real valued Baire class two functions on metrizable compacta, and strengthen various theorems in infinite dimensional topology such as Pol's solution to Alexandrov's old problem.
01 Jan 2014
TL;DR: The present study has its origin in problems of optimal resource allocation, especially those related to the possibilities of a price mechanism, and focuses on the case where by " optimal" is meant " efficient" resource allocation.
Abstract: The present study has its origin in problems of optimal resource allocation, especially those related to the possibilities of a price mechanism. While for some purposes Pareto-optimality might be the more relevant concept, we have confined ourselves here to the case where by " optimal" is meant " efficient" resource allocation.
TL;DR: In this paper, the family of expanders constructed by finite quotients of a lattice in such a group does not admit a uniform embedding in any Banach space of type > 1.
Abstract: We extend Vincent Lafforgue's results to Sp4. As applications, the family of expanders constructed by finite quotients of a lattice in such a group does not admit a uniform embedding in any Banach space of type > 1, and any affine isometric action of such a group, or of any cocompact lattice in it, in a Banach space of type > 1 has a fixed point.
TL;DR: This paper establishes a set of sufficient conditions for the controllability of fractional semilinear integro-differential inclusions in Banach spaces via resolvent operators and uses Bohnenblust-Karlin's fixed point theorem to prove the main results.
TL;DR: A new class of impulsive differential equations is introduced, more suitable to characterize memory processes of the drugs in the bloodstream and the consequent absorption for the body and the study of existence and uniqueness theorems interesting.