Showing papers on "Banach space published in 2002"

Vector-Valued Laplace Transforms and Cauchy Problems

Abstract: Linear differential equations in Banach spaces are systematically treated with the help of Laplace transforms. The central tool is an “integrated version” of Widder’s theorem (characterizing Laplace transforms of bounded functions). It holds in any Banach space (whereas the vector-valued version of Widder’s theorem itself holds if and only if the Banach space has the Radon-Nikodým property). The Hille-Yosida theorem and other generation theorems are immediate consequences. The method presented here can be applied to operators whose domains are not dense.

Recent Developments in the Navier-Stokes Problem

Abstract: INTRODUCTION What is this Book About? SOME RESULTS OF REAL HARMONIC ANALYSIS Real Interpolation, Lorentz Spaces, and Sobolev Embedding Besov Spaces and Littlewood-Paley Decomposition Shift-Invariant Banach Spaces of Distributions and Related Besov Spaces Vector-Valued Integrals Complex Interpolation, Hardy Space, and Calderon-Zygmund Operators Vector-Valued Singular Integrals A Primer to Wavelets Wavelets and Functional Spaces The Space BMO A GENERAL FRAMEWORK FOR SHIFT-INVARIANT ESTIMATES FOR THE NAVIER-STOKES EQUATIONS Weak Solutions for the Navier-Stokes Equations Divergence-Free Vector Wavelets The Mollified Navier-Stokes Equations CLASSICAL EXISTENCE RESULTS FOR THE NAVIER-STOKES EQUATIONS The Leray Solutions for the Navier-Stokes Equations Kato's Mild Solutions for the Navier-Stokes Equations NEW APPROACHES OF MILD SOLUTIONS The Mild Solutions of Koch and Tataru: The Space BMO-1 Generalization of the Lp Theory: Navier-Stokes and Local Measures Further Results on Local Measures Regular Initial Values Besov Spaces of Negative Order Pointwise Multipliers of Negative Order Further Adapted Spaces for the Navier-Stokes Equations Cannone's Approach of Self-Similarity DECAY AND REGULARITY RESULTS FOR WEAK AND MILD SOLUTIONS Space-Analytic Solutions of the Navier-Stokes Equations Space Localization and Navier-Stokes Equations Time Decay for the Solutions to the Navier-Stokes Equations Uniqueness of Ld Solutions Further Results on Uniqueness of Mild Solutions Stability and Lyapunov Functionals LOCAL ENERGY INEQUALITIES FOR THE NAVIER-STOKES EQUATIONS ON R3 The Caffarelli, Kohn, and Nirenberg Regularity Criterion On the Dimension of the Set of Singular Points Local Existence (in Time) of Suitable Locally Square Integrable Weak Solutions Global Existence of Suitable Locally Square Integrable Weak Solutions Leray's Conjecture on Self-Similar Singularities CONCLUSION Singular Initial Values REFERENCES BIBLIOGRAPHY INDEX NOMINUM INDEX RERUM

Strong Convergence of a Proximal-Type Algorithm in a Banach Space

Abstract: In this paper, we study strong convergence of the proximal point algorithm It is known that the proximal point algorithm converges weakly to a solution of a maximal monotone operator, but it fails to converge strongly Then, in [Math Program, 87 (2000), pp 189--202], Solodov and Svaiter introduced the new proximal-type algorithm to generate a strongly convergent sequence and established a convergence property for it in Hilbert spaces Our purpose is to extend Solodov and Svaiter's result to more general Banach spaces Using this, we consider the problem of finding a minimizer of a convex function

Another control condition in an iterative method for nonexpansive mappings

Abstract: Reich [4] Shioj, i and Takahashi [6 essentiall] y extended Lions and respectively' ,Wittman's result tso the framework of uniformly smooth Banach spaces. Reich [5] alsoextended Wittman's resul to thte class of Banach spaces whic arhe uniformly smoothand have a weakly sequentially continuous duality map. Moreover the contro, l sequence(a

Subexponential decay of correlations

Abstract: We describe a method for proving subexponential lower bounds for correlations functions, and apply it to study decay of correlations for maps with countable Markov partitions. One result is that LS Young’s upper estimates [Y2] are optimal in many situations. Our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on Banach spaces, which we apply to the iterates of the transfer operator.

Semismooth Newton Methods for Operator Equations in Function Spaces

Abstract: We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes nonlinear complementarity problem (NCP)-function-based reformulations of infinite-dimensional nonlinear complementarity problems and thus covers a very comprehensive class of applications. Our results generalize semismoothness and $\alpha$-order semismoothness from finite-dimensional spaces to a Banach space setting. For this purpose, a new infinite-dimensional generalized differential is used that is motivated by Qi's finite-dimensional C-subdifferential [Research Report AMR96/5, School of Mathematics, University of New South Wales, Australia, 1996]. We apply these semismoothness results to develop a Newton-like method for nonsmooth operator equations and prove its local q-superlinear convergence to regular solutions. If the underlying operator is $\alpha$-order semismooth, convergence of q-order $1+\alpha$ is proved. We also establish the semismoothness of composite operators and develop corresponding chain rules. The developed theory is accompanied by illustrative examples and by applications to NCPs and a constrained optimal control problem.

Controllability of nonlinear systems in Banach spaces: a survey

Abstract: This paper presents a survey on research using fixed-point theorems and semigroup theory to study the controllability of nonlinear systems and functional integrodifferential systems in Banach spaces. Also discussed is the use of this technique in K-controllability and boundary controllability problems for nonlinear systems and integro-differential systems in abstract spaces.

ON THE HYERS-ULAM STABILITY OF THE BANACH SPACE-VALUED DIFFERENTIAL EQUATION y'=λy

Abstract: Let I be an open interval and X a complex Banach space. Let a non-zero complex number with Re . If is a strongly differentiable map from I to X with , then we show that the distance between and the set of all solutions to the differential equation y'=y is at most $\varepsilon/Re\lambda$.

Classical solutions of the periodic Camassa—Holm equation

Abstract: We study the periodic Cauchy problem for the Camassa—Holm equation and prove that it is locally well-posed in the space of continuously differentiable functions on the circle. The approach we use consists in rewriting the equation and deriving suitable estimates which permit application of o.d.e. techniques in Banach spaces. We also describe results in fractional Sobolev H s spaces and in Appendices provide a direct well-posedness proof for arbitrary real s > 3/2 based on commutator estimates of Kato and Ponce as well as include a derivation of the equation on the diffeomorphism group of the circle together with related curvature computations.

Banach space representations and Iwasawa theory

Abstract: We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringoK[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗oK[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL2(ℤp).

Methods in Nonlinear Integral Equations

Abstract: Preface. Notation. Overview. I: Fixed Point Methods. 1. Compactness in Metric Spaces. 2. Completely Continuous Operators on Banach Spaces. 3. Continuous Solutions of Integral Equations via Schauder's Theorem. 4. The Leray-Schauder Principle and Applications. 5. Existence Theory in LP Spaces. References: Part I. II: Variational Methods. 6. Positive Self-Adjoint Operators in Hilbert Spaces. 7. The Frechet Derivative and Critical Points of Extremum. 8. The Mountain Pass Theorem and Critical Points of Saddle Type. 9. Nontrivial Solutions of Abstract Hammerstein Equations. References Part II. III: Iterative Methods. 10. The Discrete Continuation Principle. 11. Monotone Iterative Methods. 12. Quadratically Convergent Methods. References: Part III. Index.

Additive results for the generalized Drazin inverse

Abstract: Additive perturbation results for the generalized Drazin inverse of Banach space operators are presented. Precisely, if Ad denotes the generalized Drazin inverse of a bounded linear operator A on an arbitrary complex Banach space, then in some special cases (A + B)d is computed in terms of Ad and Bd. Thus, recent results of Hartwig, Wang and Wei (Linear Algebra Appl. 322 (2001), 207–217) are extended to infinite dimensional settings with simplified proofs.

On error bounds for lower semicontinuous functions

Abstract: We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents). For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f. In a reflexive Banach space X, we further obtain several sufficient and necessary conditions for the existence of error bounds in terms of the lower Dini derivative of f.

Plane with A∞-Weighted Metric not Bilipschitz Embeddable to Rn

Abstract: A planar set $G \subset {\bb R}^2$ is constructed that is bilipschitz equivalent to ( $G, d^z$ ), where ( $G, d$ ) is not bilipschitz embeddable to any uniformly convex Banach space. Here, $z \in (0, 1)$ and $d^z$ denotes the $z$ th power of the metric $d$ . This proves the existence of a strong $A_{\infty}$ weight in ${\bb R}^2$ , such that the corresponding deformed geometry admits no bilipschitz mappings to any uniformly convex Banach space. Such a weight cannot be comparable to the Jacobian of a quasiconformal self-mapping of ${\bb R}^2$ .

Iterative methods for nonlinear operator equations in banach spaces

Abstract: Contents: Introduction and preliminaries Accretive and monotone operator theory Iterative solutions of non-linear equations for strongly accretive and strongly pseudocontractive mappings Iterative solutions of non-linear equations for a class of non self-mappings Iterative solutions of non-linear equations for strongly quasiaccretive and hemicontractive mappings Steepest descent approximations for accretive type mapping equations General approximations to accretive type mapping equations in Banach spaces Iterative solutions of non-linear equations for multi-values accretive and pseudocrontractive type mappings Stability results for the iterative sequences Iterative approximations of asymptotically pseudocontractive mappings Iterative approximations of solutions to variational inclusions in Banach spaces.

A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces

Abstract: When the set of closed subspaces of C(∆), where ∆ is the Cantor set, is equipped with the standard Effros–Borel structure, the graph of the basic relations between Banach spaces (isomorphism, being isomorphic to a subspace, quotient, direct sum. . . ) is analytic non-Borel. Many natural families of Banach spaces (such as reflexive spaces, spaces not containing `1(ω), . . .) are coanalytic non-Borel. Some natural ranks (rank of embedding, Szlenk indices) are shown to be coanalytic ranks. Applications are given to universality questions. Analogous results are shown for basic sequences modulo equivalence. 0. Introduction. Classifying Banach spaces is notoriously difficult, and it is natural to conjecture that the obstruction to such a classification lies in the topological complexity of the relevant relations, and in particular of the isomorphism equivalence relation. To support this conjecture, one needs of course a natural and usable frame in which such topological notions can be handled. The purpose of the present work is to provide such a frame. The collection of separable Banach spaces is not a set, and we first need a proper parametrization of this collection. We choose to consider it as the set of all closed subspaces of the space C(∆) of continuous functions on the Cantor set. It is indeed well known that every separable Banach space is isometric to a subspace of C(∆). This choice could be considered as arbitrary; however we show that natural but different choices lead to the same levels of complexity. To investigate the topological complexity of natural families of Banach spaces, we use the theory of analytic sets, introduced by Suslin and Lusin. The basic results of this theory are presented e.g. in [K-L1], [K] or [Z]. There is a strong interplay between analytic sets and classical analysis, for which 2000 Mathematics Subject Classification: Primary 46B20.

Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in banach spaces

Abstract: The purpose of this paper is to study the weak and strong convergence of implicit iteration process to a common fixed point for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Refs. [1], 5-7, 10-11.

An infinite Ramsey theorem and some Banach-space dichotomies

Abstract: A problem of Banach asks whether every infinite-dimensional Banach space which is isomorphic to all its infinite-dimensional subspaces must be isomorphic to a separable Hilbert space. In this paper we prove a result of a Ramsey-theoretic nature which implies an interesting dichotomy for subspaces of Banach spaces. Combined with a result of Komorowski and TomczakJaegermann, this gives a positive answer to BanachXs problem. We then generalize the Ramsey-theoretic result and deduce a further dichotomy for Banach spaces with an unconditional basis.

Almost fréchet differentiability of lipschitz mappings between infinite-dimensional banach spaces

Abstract: We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every -Frechet differentiability, and that every Lipschitz mapping from an asymptotically uniformly smooth space to a finite-dimensional space has such points. The latter result improves, with a simpler proof, an earlier result of the second and third authors. We also survey some of the known results on the notions of asymptotic smoothness and convexity, prove some new properties, and present some new proofs of existing results.2000 Mathematical Subject Classification: 46G05, 46T20.