The cofinal property of the Reflexive Indecomposable Banach spaces
Argyros, Spiros A.; Raikoftsalis, Theocharis
2010-01-01
It is shown that every separable reflexive Banach space is a quotient of a reflexive Hereditarily Indecomposable space, which yields that every separable reflexive Banach is isomorphic to a subspace of a reflexive Indecomposable space. Furthermore, every separable reflexive Banach space is a quotient of a reflexive complementably $\\ell_p$ saturated space with $1
Reflexive Operator Algebras on Banach Spaces
Merlevède, Florence; Peligrad, Costel; Peligrad, Magda
2012-01-01
In this paper we study the reflexivity of a unital strongly closed algebra of operators with complemented invariant subspace lattice on a Banach space. We prove that if such an algebra contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property, then it is reflexive, i.e. it contains every operator that leaves invariant every closed subspace in the invariant subspace lattice of the algebra. In particular, such algebras coincide with their...
First and Second Order Convex Sweeping Processes in Reflexive Smooth Banach Spaces
In this paper we establish new characterizations of the normal cone of closed convex sets in reflexive smooth Banach spaces and then we use those results to prove the existence of solutions for first order convex sweeping processes and their variants in reflexive smooth Banach spaces. The case of second order convex sweeping processes is also studied. (author)
Let E be a real reflexive Banach space with uniformly Gateaux differentiable norm. Let K be a nonempty closed convex subset of E. Suppose that every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive mappings. Let T1, T2, ..., TN be a family of nonexpansive self-mappings of K, with F := intersectioni=1N Fix(Ti) ≠ 0, F = Fix(TNTN-1... T1) = Fix(T1TN ... T2) = ... Fix(TN-1TN-2... T1TN). Let { λn} be a sequence in (0, 1) satisfying the following conditions: C1 : lim λn 0; C2 : Σ λn = ∞ . For a fixed δ element of (0, 1), define Sn : K → K by Snx := (1 - δ )x + δTnx for all x element of K where Tn = Tn mod N. For arbitrary fixed u, x0 element of K, let B := { x element of K : TNTN-1... T1x γx+(1- γ)u, for some γ > 1} be bounded and let the sequence {xn} be defined iteratively by xn+1 λn+1u + (1 - λn+1 )Sn+1xn, for n ≥ 0. Assume that lim n→∞ vertical bar vertical bar Tnxn - Tn+1 xn vertical bar vertical bar = 0. Then, {xn} converges strongly to a common fixed point of the family T1, T2, ..., TN. Convergence theorem is also proved for non-self maps. (author)
Casazza, Peter G.; Lammers, Mark C.
1999-01-01
We show that the classification problem for genus $n$ Banach spaces can be reduced to the unconditionally primary case and that the critical case there is $n=2$. It is further shown that a genus $n$ Banach space is unconditionally primary if and only if it contains a complemented subspace of genus $(n-1)$. We begin the process of classifying the genus 2 spaces by showing they have a strong decomposition property.
Multipliers of Banach valued weighted function spaces
Serap Öztop
2000-10-01
Full Text Available We generalize Banach valued spaces to Banach valued weighted function spaces and study the multipliers space of these spaces. We also show the relationship between multipliers and tensor product of Banach valued weighted function spaces.
Separably injective Banach spaces
Avilés, Antonio; Castillo, Jesús M F; González, Manuel; Moreno, Yolanda
2016-01-01
This monograph contains a detailed exposition of the up-to-date theory of separably injective spaces: new and old results are put into perspective with concrete examples (such as l∞/c0 and C(K) spaces, where K is a finite height compact space or an F-space, ultrapowers of L∞ spaces and spaces of universal disposition). It is no exaggeration to say that the theory of separably injective Banach spaces is strikingly different from that of injective spaces. For instance, separably injective Banach spaces are not necessarily isometric to, or complemented subspaces of, spaces of continuous functions on a compact space. Moreover, in contrast to the scarcity of examples and general results concerning injective spaces, we know of many different types of separably injective spaces and there is a rich theory around them. The monograph is completed with a preparatory chapter on injective spaces, a chapter on higher cardinal versions of separable injectivity and a lively discussion of open problems and further lines o...
$\\alpha$-minimal Banach spaces
Rosendal, Christian
2011-01-01
A Banach space with a Schauder basis is said to be $\\alpha$-minimal for some countable ordinal $\\alpha$ if, for any two block subspaces, the Bourgain embeddability index of one into the other is at least $\\alpha$. We prove a dichotomy that characterises when a Banach space has an $\\alpha$-minimal subspace, which contributes to the ongoing project, initiated by W. T. Gowers, of classifying separable Banach spaces by identifying characteristic subspaces.
Albiac, Fernando
2016-01-01
This text provides the reader with the necessary technical tools and background to reach the frontiers of research without the introduction of too many extraneous concepts. Detailed and accessible proofs are included, as are a variety of exercises and problems. The two new chapters in this second edition are devoted to two topics of much current interest amongst functional analysts: Greedy approximation with respect to bases in Banach spaces and nonlinear geometry of Banach spaces. This new material is intended to present these two directions of research for their intrinsic importance within Banach space theory, and to motivate graduate students interested in learning more about them. This textbook assumes only a basic knowledge of functional analysis, giving the reader a self-contained overview of the ideas and techniques in the development of modern Banach space theory. Special emphasis is placed on the study of the classical Lebesgue spaces Lp (and their sequence space analogues) and spaces of continuous f...
Factorization of weakly compact operators between Banach spaces and Fréchet or (LB)-spaces
Bonet Solves, José Antonio; Wright, J. D. Maitland
2012-01-01
In this note we show that weakly compact operators from a Banach space X into a complete (LB)-space E need not factorize through a reflexive Banach space. If E is a Fréchet space, then weakly compact operators from a Banach space X into E factorize through a reflexive Banach space. The factorization of operators from a Fréchet or a complete (LB)-space into a Banach space mapping bounded sets into relatively weakly compact sets is also investigated.
A pythagorean approach in Banach spaces
Gao Ji
2006-01-01
Let be a Banach space and let be the unit sphere of . Parameters , , , and , where and are introduced and studied. The values of these parameters in the spaces and function spaces are estimated. Among the other results, we proved that a Banach space with , or is uniform nonsquare; and a Banach space with , or has uniform normal structure.
Extremely strict ideals in Banach spaces
T S S R K RAO
2016-08-01
Motivated by the notion of an ideal introduced by Godefroy {\\it et al.} ({\\it Studia Math.} {\\bf 104} (1993) 13–59), in this article, we introduce and study the notion of an extremely strict ideal. For a Poulsen simplex $K$, we show that the space of affine continuous functions on $K$ is an extremely strict ideal in the space of continuous functions on $K$. For injective tensor product spaces, we prove a cancelation theorem for extremely strict ideals. We also exhibit non-reflexive Banach spaces which are not strict ideals in their fourth dual.
Fusion Frames and -Frames in Banach Spaces
Amir Khosravi; Behrooz Khosravi
2011-05-01
Fusion frames and -frames in Hilbert spaces are generalizations of frames, and frames were extended to Banach spaces. In this article we introduce fusion frames, -frames, Banach -frames in Banach spaces and we show that they share many useful properties with their corresponding notions in Hilbert spaces. We also show that -frames, fusion frames and Banach -frames are stable under small perturbations and invertible operators.
Tokarev, Eugene
2002-01-01
In the article is introduced a new class of Banach spaces that are called sub B-convex. Namely, a Banach space X is said to be B -convex if it may be represented as a direct sum l_1+ W, where W is B-convex. It will be shown that any separable sub B-convex Banach space X may be almost isometrically embedded in a separable Banach space G(X) of the same cotype as X, which has a series of properties. Namely, G(X) is an approximate envelope, i.e. any separable Banach space which is finitely repres...
Characterizing R-duality in Banach spaces
Christensen, Ole; Xiao, Xiang Chun; Zhu, Yu Can
2013-01-01
R-duals of certain sequences in Hilbert spaces were introduced by Casazza, Kutyniok and Lammers in 2004 and later generalized to Banach spaces by Xiao and Zhu. In this paper we provide some characterizations of R-dual sequences in Banach spaces.......R-duals of certain sequences in Hilbert spaces were introduced by Casazza, Kutyniok and Lammers in 2004 and later generalized to Banach spaces by Xiao and Zhu. In this paper we provide some characterizations of R-dual sequences in Banach spaces....
Existence of zeros for operators taking their values in the dual of a Banach space
Ricceri Biagio
2004-01-01
Full Text Available Using continuous selections, we establish some existence results about the zeros of weakly continuous operators from a paracompact topological space into the dual of a reflexive real Banach space.
Isometries on Banach spaces function spaces
Fleming, Richard J
2002-01-01
Fundamental to the study of any mathematical structure is an understanding of its symmetries. In the class of Banach spaces, this leads naturally to a study of isometries-the linear transformations that preserve distances. In his foundational treatise, Banach showed that every linear isometry on the space of continuous functions on a compact metric space must transform a continuous function x into a continuous function y satisfying y(t) = h(t)x(p(t)), where p is a homeomorphism and |h| is identically one.Isometries on Banach Spaces: Function Spaces is the first of two planned volumes that survey investigations of Banach-space isometries. This volume emphasizes the characterization of isometries and focuses on establishing the type of explicit, canonical form given above in a variety of settings. After an introductory discussion of isometries in general, four chapters are devoted to describing the isometries on classical function spaces. The final chapter explores isometries on Banach algebras.This treatment p...
Regularization methods in Banach spaces
Schuster, Thomas; Hofmann, Bernd; Kazimierski, Kamil S
2012-01-01
Regularization methods aimed at finding stable approximate solutions are a necessary tool to tackle inverse and ill-posed problems. Usually the mathematical model of an inverse problem consists of an operator equation of the first kind and often the associated forward operator acts between Hilbert spaces. However, for numerous problems the reasons for using a Hilbert space setting seem to be based rather on conventions than on an approprimate and realistic model choice, so often a Banach space setting would be closer to reality. Furthermore, sparsity constraints using general Lp-norms or the B
Coefficient Quantization for Frames in Banach Spaces
Casazza, P. G.; Dilworth, S. J.; Odell, E.; Schlumprecht, Th.; Zsak, Andras
2007-01-01
Let $(e_i)$ be a fundamental system of a Banach space. We consider the problem of approximating linear combinations of elements of this system by linear combinations using quantized coefficients. We will concentrate on systems which are possibly redundant. Our model for this situation will be frames in Banach spaces.
Banach spaces of analytic functions
Hoffman, Kenneth
2007-01-01
A classic of pure mathematics, this advanced graduate-level text explores the intersection of functional analysis and analytic function theory. Close in spirit to abstract harmonic analysis, it is confined to Banach spaces of analytic functions in the unit disc.The author devotes the first four chapters to proofs of classical theorems on boundary values and boundary integral representations of analytic functions in the unit disc, including generalizations to Dirichlet algebras. The fifth chapter contains the factorization theory of Hp functions, a discussion of some partial extensions of the f
Simultaneous approximation in scales of Banach spaces
The problem of verifying optimal approximation simultaneously in different norms in a Banach scale is reduced to verification of optimal approximation in the highest order norm. The basic tool used is the Banach space interpolation method developed by Lions and Peetre. Applications are given to several problems arising in the theory of finite element methods
Bloch spaces on bounded symmetric domains in complex Banach spaces
DENG; Fangwen
2006-01-01
We give a definition of Bloch space on bounded symmetric domains in arbitrary complex Banach space and prove such function space is a Banach space. The properties such as boundedness, compactness and closed range of composition operators on such Bloch space are studied.
On Λ-Type Duality of Frames in Banach Spaces
Renu Chugh
2013-11-01
Full Text Available Frames are redundant system which are useful in the reconstruction of certain classes of spaces. The dual of a frame (Hilbert always exists and can be obtained in a natural way. Due to the presence of three Banach spaces in the definition of retro Banach frames (or Banach frames duality of frames in Banach spaces is not similar to frames for Hilbert spaces. In this paper we introduce the notion of Λ-type duality of retro Banach frames. This can be generalized to Banach frames in Banach spaces. Necessary and sufficient conditions for the existence of the dual of retro Banach frames are obtained. A special class of retro Banach frames which always admit a dual frame is discussed.
A Gaussian Average Property for Banach Spaces
Casazza, Peter G.; Nielsen, Niels Jorgen
1996-01-01
In this paper we investigate a Gaussian average property of Banach spaces. This property is weaker than the Gordon Lewis property but closely related to this and other unconditional structures. It is also shown that this property implies that certain Hilbert space valued operators defined on subspaces of the given space can be extended.
On Banach spaces without the approximation property
Reinov, Oleg I.
2002-01-01
A. Szankowski's example is used to construct a Banach space similar to that of "An example of an asymptotically Hilbertian space which fails the approximation property", P.G. Casazza, C.L. Garc\\'{\\i}a, W.B. Johnson [math.FA/0006134 ()].
Boundary Controllability of Integrodifferential Systems in Banach Spaces
K Balachandran; E R Anandhi
2001-02-01
Sufficient conditions for boundary controllability of integrodifferential systems in Banach spaces are established. The results are obtained by using the strongly continuous semigroup theory and the Banach contraction principle. Examples are provided to illustrate the theory.
The Topological Degree in Ordered Banach Spaces
Adrian DUMA; Ileana DUMA
2008-01-01
This paper is devoted to the applications of classical topological degrees to nonlinear problems involving various classes of operators acting between ordered Banach spaces. In this frame-work, the Leray-Schauder, Browder-Petryshyn, and Amann-Weiss degree theories are considered, and several existence results are obtained. The non-Archimedean case is also discussed.
Wang, Min
2016-04-01
This paper aims to establish the Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces. For this purpose, we firstly prove a very general existence result for generalized mixed variational inequalities, provided that the mapping involved has the so-called mixed variational inequality property and satisfies a rather weak coercivity condition. Finally, we establish the Tikhonov regularization method for generalized mixed variational inequalities. Our findings extended the results for the generalized variational inequality problem (for short, GVIP(F, K)) in R^n spaces (He in Abstr Appl Anal, 2012) to the generalized mixed variational inequality problem (for short, GMVIP(F,φ , K) ) in reflexive Banach spaces. On the other hand, we generalized the corresponding results for the generalized mixed variational inequality problem (for short, GMVIP(F,φ ,K) ) in R^n spaces (Fu and He in J Sichuan Norm Univ (Nat Sci) 37:12-17, 2014) to reflexive Banach spaces.
Noncompact Perturbation of Sweeping Process with Delay in Banach Spaces
A. G. Ibrahim; Aladsani, F.
2013-01-01
We have proven an existence theorem concerning the existence of solutions for a functional evolution inclusion governed by sweeping process with closed convex sets depending on time and state and with a noncompact nonconvex perturbation in Banach spaces. This work extends some recent existence theorems concerning sweeping processes from Hilbert spaces setting to Banach spaces setting. Moreover, it improves some recent existence results for sweeping processes in Banach spaces.
Viscosity Approximation Method for Infinitely Many Asymptotically Nonexpansive Maps in Banach Spaces
Ruo Feng RAO
2011-01-01
In the framework of reflexive Banach spaces satisfying a weakly continuous duality map,the author uses the viscosity approximation method to obtain the strong convergence theorem for iterations with infinitely many asymptotically nonexpansive mappings.The main results obtained in this paper improve and extend some recent results.
The General Iterative Methods for Asymptotically Nonexpansive Semigroups in Banach Spaces
Rabian Wangkeeree
2012-01-01
Full Text Available We introduce the general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups which is a unique solution of some variational inequalities. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. The main result extends various results existing in the current literature.
Boundedness of biorthogonal systems in Banach spaces
Hájek, Petr Pavel; Montesinos, V.
2010-01-01
Roč. 177, č. 1 (2010), s. 145-154. ISSN 0021-2172 R&D Projects: GA ČR GA201/07/0394 Institutional research plan: CEZ:AV0Z10190503 Keywords : M-basis * Banach spaces Subject RIV: BA - General Mathematics Impact factor: 0.630, year: 2010 http://link.springer.com/article/10.1007%2Fs11856-010-0041-x
Characterization of Banach valued BMO functions and UMD Banach spaces by using Bessel convolutions
Betancor, Jorge J; Rodríguez-Mesa, Lourdes
2011-01-01
In this paper we consider the space $BMO_o(\\mathbb{R},X)$ of bounded mean oscillations and odd functions on $\\mathbb{R}$ taking values in a UMD Banach space $X$. The functions in $BMO_o(\\mathbb{R},X)$ are characterized by Carleson type conditions involving Bessel convolutions and $\\gamma$-radonifying norms. Also we prove that the UMD Banach spaces are the unique Banach spaces for which certain $\\gamma$-radonifying Carleson inequalities for Bessel-Poisson integrals of $BMO_o(\\mathbb{R},X)$ functions hold.
Malliavin calculus and decoupling inequalities in Banach spaces
Maas, Jan
2008-01-01
We develop a theory of Malliavin calculus for Banach space valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Ito isometry to Banach spaces. In the white noise case we obtain two sided L^p-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Ito chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the ...
Introduction to Banach spaces and algebras
Allan, Graham
2010-01-01
Banach spaces and algebras are a key topic of pure mathematics. Graham Allan's careful and detailed introductory account will prove essential reading for anyone wishing to specialise in functional analysis and is aimed at final year undergraduates or masters level students. Based on the author's lectures to fourth year students at Cambridge University, the book assumes knowledge typical of first degrees in mathematics, including metric spaces, analytic topology, and complexanalysis. However, readers are not expected to be familiar with the Lebesgue theory of measure and integration.The text be
Unbounded operators on Banach spaces over the quaternion field
Ludkovsky, S. V.
2004-01-01
Resolvents of quasi-linear operators and operator algebras in Banach spaces over the quaternion field are investigated. Spectral theory of unbounded nonlinear operators in quaternion Banach spaces is studied. Strongly continuous semigroups of quaternion quasi-linear operators are investigated and the polar decomposition theorem for them is proved.
The heart of the Banach spaces
Wegner, Sven-Ake
2016-01-01
Let an exact category in the sense of Quillen be given. Assume that in this category every morphism has a kernel and that every kernel is an inflation. In their famous 1982 paper, Beilinson, Bernstein and Deligne consider in this situation a t-structure on the derived category and remark that its heart can be described as a category of formal quotients. They further point out, that the category of Banach spaces is an example, and that here a similar category of formal quotients was studied by...
BANACH CONTRACTION PRINCIPLE ON CONE RECTANGULAR METRIC SPACES
Ismat Beg
2009-08-01
Full Text Available We introduce the notion of cone rectangular metric space and prove {sc Banach} contraction mapping principle in cone rectangular metric space setting. Our result extends recent known results.
On Shrinking and Boundedly Complete Schauder Frames of Banach spaces
Liu, Rui
2009-01-01
This paper studies Schauder frames in Banach spaces, a concept which is a natural generalization of frames in Hilbert spaces and Schauder bases in Banach spaces. The associated minimal and maximal spaces are introduced, as are shrinking and boundedly complete Schauder frames. Our main results extend the classical duality theorems on bases to the situation of Schauder frames. In particular, we will generalize James' results on shrinking and boundedly complete bases to frames. Secondly we will ...
Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces
Jong Soo Jung
2008-01-01
Abstract Let be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of has the fixed point property for nonexpansive mappings. Let be a nonempty closed convex subset of , a contractive mapping (or a weakly contractive mapping), and nonexpansive mapping with the fixed point set . Let be generated by a new composite iterative scheme: , , . It is proved that converges strongly to a point in , which is a s...
Shenghua Wang
2013-01-01
Full Text Available We first introduce the concept of Bregman asymptotically quasinonexpansive mappings and prove that the fixed point set of this kind of mappings is closed and convex. Then we construct an iterative scheme to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of Bregman asymptotically quasinonexpansive mappings in reflexive Banach spaces and prove strong convergence theorems. Our results extend the recent ones of some others.
Open problems in the geometry and analysis of Banach spaces
Guirao, Antonio J; Zizler, Václav
2016-01-01
This is a collection of some easily-formulated problems that remain open in the study of the geometry and analysis of Banach spaces. Assuming the reader has a working familiarity with the basic results of Banach space theory, the authors focus on concepts of basic linear geometry, convexity, approximation, optimization, differentiability, renormings, weak compact generating, Schauder bases and biorthogonal systems, fixed points, topology and nonlinear geometry. The main purpose of this work is to help convince young researchers in Functional Analysis that the theory of Banach spaces is a fertile field of research, full of interesting open problems. Inside the Banach space area, the text should help expose young researchers to the depth and breadth of the work that remains, and to provide the perspective necessary to choose a direction for further study. Some of the problems presented herein are longstanding open problems, some are recent, some are more important and some are only "local" problems. Some would ...
On the uniqueness of minimal projections in Banach spaces
Ewa Szlachtowska; Dominik Mielczarek
2012-01-01
Let \\(X\\) be a uniformly convex Banach space with a continuous semi-inner product. We investigate the relation of orthogonality in \\(X\\) and generalized projections acting on \\(X\\). We prove uniqueness of orthogonal and co-orthogonal projections.
Some Questions Arising from the Homogeneous Banach Space Problem
Casazza, Peter G.
1992-01-01
We review the current state of the homogeneous Banach space problem. We then formulate several questions which arise naturally from this problem, some of which seem to be fundamental but new. We give many examples defining the bounds on the problem. We end with a simple construction showing that every infinite dimensional Banach space contains a subspace on which weak properties have become stable (under passing to further subspaces). Implications of this construction are considered.
Construction of generalized atomic decompositions in Banach spaces
Raj Kumar
2014-06-01
Full Text Available G-atomic decompositions for Banach spaces with respect to a model space of sequences have been introduced and studied as a generalization of atomic decompositions. Examples and counter example have been provided to show its existence. It has been proved that an associated Banach space for G-atomic decomposition always has a complemented subspace. The notion of a representation system is introduced and exhibits its relation with G-atomic decomposition. Also It has been observed that G-atomic decompositions are exactly compressions of Schauder decompositions for a larger Banach space. We give a characterization for finite G-atomic decomposition in terms of finite-dimensional expansion of identity. Keywords: complemented coefficient spaces, finite-dimensional expansion of identity, G-atomic decomposition, representation system.
Metric embeddings bilipschitz and coarse embeddings into Banach spaces
Ostrovskii, Mikhail I
2013-01-01
Embeddings of discrete metric spaces into Banach spaces recently became an important tool in computer science and topology. The book will help readers to enter and to work in this very rapidly developing area having many important connections with different parts of mathematics and computer science. The purpose of the book is to present some of the most important techniques and results, mostly on bilipschitz and coarse embeddings. The topics include embeddability of locally finite metric spaces into Banach spaces is finitely determined, constructions of embeddings, distortion in terms of Poinc
Atomic decompositions of Banach-space-valued martingales
刘培德; 侯友良
1999-01-01
Several theorems for atomic decompositions of Banach-space-valued martingales are proved. As their applications, the relationship among some martingale spaces such as H_α(X) and _ρ(?)_α in the case 0<α≤1 are studied. It is shown that there is a close connection between the results and the smoothness and convexity of the value spaces.
Banach Spaces with respect to Operator-Valued Norms
Kim, Yun-Su
2008-01-01
We introduce the notions of L(H)-valued norms and Banach spaces with respect to L(H)-valued norms. In particular, we introduce Hilbert spaces with respect to L(H)-valued inner products. In addition, we provide several fundamental examples of Hilbert spaces with respect to L(H)-valued inner products.
Additive maps preserving the reduced minimum modulus of Banach space operators
Bourhim, Abdellatif
2009-01-01
Let ${\\mathcal B}(X)$ be the algebra of all bounded linear operators on an infinite dimensional complex Banach space $X$. We prove that an additive surjective map $\\phi$ on ${\\mathcal B}(X)$ preserves the reduced minimum modulus if and only if either there are bijective isometries $U:X\\to X$ and $V:X\\to X$ both linear or both conjugate linear such that $\\phi(T)=UTV$ for all $T\\in{\\mathcal B}(X)$, or $X$ is reflexive and there are bijective isometries $U:X^*\\to X$ and $V:X\\to X^*$ both linear or both conjugate linear such that $\\phi(T)=UT^*V$ for all $T\\in{\\mathcal B}(X)$. As immediate consequences of the ingredients used in the proof of this result, we get the complete description of surjective additive maps preserving the minimum, the surjectivity and the maximum moduli of Banach space operators.
A GLIDING HUMP PROPERTY AND BANACH-MACKEY SPACES
CHARLES SWARTZ
2001-08-01
Full Text Available We consider the Banach-Mackey property for pairs of vector spaces E and E' which are in duality. Let A be an algebra of sets and assume that P is an additive map from A into the projection operators on E. We define a continuous gliding hump property for the map P and show that pairs with this gliding hump property and anoter measure theoretic property are Banach-Mackey pairs, i. e., weakly bounded subsets of E are strongly bounded. Examples of vector valued function spaces, such as the space of Pettis integrable functions, which satisfy these conditions are given
(s, μ)-similar operators in the Banach spaces
The theory of the operator ideals formed by means of S function is developed. The problem of the construction of the operator acting from one Banach space to another whose S numbers are near to the given ones, is solved. Several conditions, sufficient for that any wholly continuous operator in the Gilbert space were transferred to the given pair of the Banach spaces without distorting too much the values of its S-numbers, are given. All the considered operators are assumed to be linear and continuous ones
Banach's space: Lviv and the Scottish cafe
D.Henderson
2004-01-01
Full Text Available In a sense Stefan Banach is the patron saint of this workshop. He was one of the most prominent former residents of Lviv. His ``hangout'', the Scottish Cafe, is shown in the street scene that is the cover of the book of abstracts.
On well posedness of best simultaneous approximation problems in Banach spaces
LI; Chong
2001-01-01
［1］Pinkus, A., Uniqueness in vector-valued approximation, J. Approx. Theory, 1993, 73: 17—92.［2］Li, C., Watson, G. A., On best simultaneous approximation, J. Approx. Theory, 1997, 91: 332—348.［3］Tanimoto, S., A characterization of best simultaneous approximations, J. Approx. Theory, 1989, 59: 359—361.［4］Watson, G. A., A characterization of best simultaneous approximations, J. Approx. Theory, 1993, 75: 175—182.［5］Li, C., Watson, G. A., On best simultaneous approximation of a finite set of functions, Computer Math. Applic., 1999, 37: 1—9.［6］Stechkin, S. B., Approximation properties of sets in normed linear spaces, Rev. Roumaine Math. Pures and Appl. (in Russian), 1963, 8: 5—18.［7］Borwein, J. M., Fitzpatrick, S., Existence of nearest points in Banach spaces, Can. J. Math., 1989, 41: 702—720.［8］De Blasi, F. S., Myjak, J., On a generalized best approximation problem, J. Approx. Theory, 1998, 94: 54—72.［9］Georgiev, P. G., The strong Ekeland variational principle, the strong drop theorem and applications, J. Math. Anal. Appl., 1988, 131: 1—21.［10］Lau, K. S., Almost Chebyshev subsets in reflexive Banach spaces, Indiana Univ. Math. J., 1978, 27: 791—795.［11］Li, C., On mutually nearest and mutually furthest points in reflexive Banach spaces, J. Approx. Theory, 2000, 103: 1—17.［12］Li, C., On well posed generalized best approximation problems, J. Approx. Thory, 2000, 107: 96—108.［13］Li, C., Almost K-Chebyshev sets, Acta. Math. Sin. (in Chinese), 1990, 33: 251—259.［14］De Blasi, F. S., Myjak, J., Papini, P. L., On mutually nearest and mutually furthest points of sets in Banach spaces, J. Approx. Theory, 1992, 70: 142—155.［15］Dontchev, A., Zolezzi, T., Well Posed Optimization Problems, Lecture Notes in Math., Vol. 1543, New York: Springer-Verlag, 1993.［16］Li, C., Wang, X. H., Almost Chebyshev set with respect to bounded subsets, Science in China, Ser. A, 1997, 40: 375—383.［17］Ni, R
Riesz Isomorphisms of Tensor Products of Order Unit Banach Spaces
T S S R K Rao
2009-06-01
In this paper we formulate and prove an order unit Banach space version of a Banach–Stone theorem type theorem for Riesz isomorphisms of the space of vector-valued continuous functions. Similar results were obtained recently for the case of lattice-valued continuous functions in [5] and [6].
Banach spaces and descriptive set theory selected topics
Dodos, Pandelis
2010-01-01
This volume deals with problems in the structure theory of separable infinite-dimensional Banach spaces, with a central focus on universality problems. This topic goes back to the beginnings of the field and appears in Banach's classical monograph. The novelty of the approach lies in the fact that the answers to a number of basic questions are based on techniques from Descriptive Set Theory. Although the book is oriented on proofs of several structural theorems, in the main text readers will also find a detailed exposition of numerous “intermediate” results which are interesting in their own right and have proven to be useful in other areas of Functional Analysis. Moreover, several well-known results in the geometry of Banach spaces are presented from a modern perspective.
The reconstruction property in Banach spaces and a perturbation theorem
Casazza, P.G.; Christensen, Ole
2008-01-01
Perturbation theory is a fundamental tool in Banach space theory. However, the applications of the classical results are limited by the fact that they force the perturbed sequence to be equivalent to the given sequence. We will develop a more general perturbation theory that does not force equiva...
On some impulsive fractional differential equations in Banach spaces
JinRong Wang
2010-01-01
Full Text Available This paper deals with some impulsive fractional differential equations in Banach spaces. Utilizing the Leray-Schauder fixed point theorem and the impulsive nonlinear singular version of the Gronwall inequality, the existence of \\(PC\\-mild solutions for some fractional differential equations with impulses are obtained under some easily checked conditions. At last, an example is given for demonstration.
On nonlocal problems for fractional differential equations in Banach spaces
XiWang Dong
2011-01-01
Full Text Available In this paper, we study the existence and uniqueness of solutions to the nonlocal problems for the fractional differential equation in Banach spaces. New sufficient conditions for the existence and uniqueness of solutions are established by means of fractional calculus and fixed point method under some suitable conditions. Two examples are given to illustrate the results.
Improved bounds in the metric cotype inequality for Banach spaces
Giladi, Ohad; Naor, Assaf
2010-01-01
It is shown that if (X, ||.||_X) is a Banach space with Rademacher cotype q then for every integer n there exists an even integer m X we have \\sum_{j=1}^n \\Avg_x [ ||f(x+ (m/2) e_j)-f(x) ||_X^q ] n^{(1/2)+(1/q)}.
Smooth approximations of norms in separable Banach spaces
Hájek, Petr Pavel; Talponen, J.
2014-01-01
Roč. 65, č. 3 (2014), s. 957-969. ISSN 0033-5606 R&D Projects: GA ČR(CZ) GAP201/11/0345 Institutional support: RVO:67985840 Keywords : Banach space * approximation Subject RIV: BA - General Mathematics Impact factor: 0.640, year: 2014 http://qjmath.oxfordjournals.org/content/65/3/957
Multipliers of pg-Bessel sequences in Banach spaces
Abdollahpour, M. R.; Najati, A.; Gavruta, P.
2015-01-01
In this paper, we introduce (p,q)g-Bessel multipliers in Banach spaces and we show that under some conditions a (p,q)g-Bessel multiplier is invertible. Also, we show the continuous dependency of (p,q)g-Bessel multipliers on their parameters.
Banach空间中的Xd Bessel列%Sequences of Xd Bessel for a Banach space
王亚丽; 曹怀信; 张巧卫
2011-01-01
研究了Banach空间X中的Xd Bessel列、Xd框架、Xd独立框架、Xd紧框架与Xd Riesz基.证明了当Xd为BK-空间时,(BXXd,‖·‖)是数域F上的Banach空间；当Xd是BK-空间且X自反时,通过定义算子Tf,建立了空间BXXd与算子空间B(X*,Xd)之间的等距同构,为利用算子论的方法研究Xd Bessel列提供了必要的理论依据.最后,给出了Banach空间X中Xd Bessel列的等价刻画并证明了独立的Xd框架与Xd Riesz基是一致的.%Xd Bessel sequences, Xd frames, Xd independent frames, Xd tight frames and Xd Riesz basis for a Banach space X ate introduced and discussed. It is proved that (Bxxd || ? || ) is a Banach space when Xd is a BK-space. By de fining an operator Tf, an isometric isomorphism from Bxxd to B(X* ,Xd) is established when Xd is a BK-space and X is reflexive, which provides a necessary theoretical basis for studying Xd Bessel sequences by the operator theory. Finally, the equivalent characterizations of Xd Bessel sequences for a Banach space X are given. Also, it is proved that independ ent Xd frames and Xd Riesz bases for a Banach space X are the same.
Quantitative coarse embeddings of quasi-Banach spaces into a Hilbert space
Kraus, Michal
2015-01-01
We study how well a quasi-Banach space can be coarsely embedded into a Hilbert space. Given any quasi-Banach space X which coarsely embeds into a Hilbert space, we compute its Hilbert space compression exponent. We also show that the Hilbert space compression exponent of X is equal to the supremum of the amounts of snowflakings of X which admit a bi-Lipschitz embedding into a Hilbert space.
Jeribi, Aref
2015-01-01
Uncover the Useful Interactions of Fixed Point Theory with Topological StructuresNonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications is the first book to tackle the topological fixed point theory for block operator matrices with nonlinear entries in Banach spaces and Banach algebras. The book provides researchers and graduate students with a unified survey of the fundamental principles of fixed point theory in Banach spaces and algebras. The authors present several exten
Consistency of Regularized Learning Schemes in Banach Spaces
Combettes, Patrick L.; Salzo, Saverio; Villa, Silvia
2014-01-01
This paper proposes a unified framework for the investigation of learning theory in Banach spaces of features via regularized empirical risk minimization. The main result establishes the consistency of such learning schemes under general conditions on the loss function, the geometry of the feature space, the regularization function, and the regularization parameters. The focus is placed on Tikhonov-like regularization with totally convex functions. This broad class of regularizers provides a ...
Integrals and Banach spaces for finite order distributions
Talvila, Erik
2011-01-01
Let $\\Bc$ denote the real-valued functions continuous on the extended real line and vanishing at $-\\infty$. Let $\\Br$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\\infty$. Define $\\acn$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in $\\Bc$. Similarly with $\\arn$ from $\\Br$. A type of integral is defined on distributions in $\\acn$ and $\\arn$. The multipliers are iterated integrals of functions of bounded variation. For each $n\\in\\N$, the spaces $\\acn$ and $\\arn$ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to $\\Bc$ and $\\Br$, respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space $\\ac^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The space $\\ar^1$ contains all finite signed Borel measure...
Some problems on ordinary differential equations in Banach spaces
Hájek, Petr Pavel; Vivi, P.
2010-01-01
Roč. 104, č. 2 (2010), s. 245-255. ISSN 1578-7303 R&D Projects: GA AV ČR IAA100190801; GA ČR GA201/07/0394 Institutional research plan: CEZ:AV0Z10190503 Keywords : Banach space * ODE * Peano's theorem Subject RIV: BA - General Mathematics Impact factor: 0.400, year: 2010 http://link.springer.com/article/10.5052%2FRACSAM.2010.16
Polynomial algebras and smooth functions in Banach spaces
D'Alessandro, Stefania; Hájek, Petr Pavel
2014-01-01
Roč. 266, č. 3 (2014), s. 1627-1646. ISSN 0022-1236 R&D Projects: GA ČR(CZ) GAP201/11/0345; GA MŠk(CZ) 7AMB12FR003 Institutional support: RVO:67985840 Keywords : polynomials in Banach space Subject RIV: BA - General Mathematics Impact factor: 1.322, year: 2014 http://www.sciencedirect.com/science/article/pii/S0022123613004588
Weak compactness and sigma-Asplund generated Banach spaces
Fabian, Marián; Montesinos, V.; Zizler, Václav
2007-01-01
Roč. 181, č. 2 (2007), s. 125-152. ISSN 0039-3223 R&D Projects: GA AV ČR IAA1019301; GA AV ČR(CZ) IAA100190610 Institutional research plan: CEZ:AV0Z10190503 Keywords : epsilon-Asplund set * epsilon-weakly compact set * weakly compactly generated Banach space Subject RIV: BA - General Mathematics Impact factor: 0.568, year: 2007
Some Properties of the Injective Tensor Product of Banach Spaces
Xiao Ping XUE; Yong Jin LI; Qing Ying BU
2007-01-01
Let X and Y be Banach spaces such that X has an unconditional basis. Then X?Y, the injective tensor product of X and Y, has the Radon-Nikodym property (respectively, the analytic Radon-Nikodym property, the near Radon-Nikodym property, non-containment of a copy of co, weakly sequential completeness) if and only if both X and Y have the same property and each continuous linear operator from the predual of X to Y is compact.
Tools for Malliavin calculus in UMD Banach spaces
Pronk, Matthijs
2012-01-01
In this paper we study the Malliavin derivatives and Skorohod integrals for processes taking values in an infinite dimensional space. Such results are motivated by their applications to SPDEs and in particular financial mathematics. Vector-valued Malliavin theory in Banach space E is naturally restricted to spaces E which have the so-called UMD property, which arises in harmonic analysis and stochastic integration theory. We provide several new results and tools for the Malliavin derivatives and Skorohod integrals in an infinite dimensional setting. In particular, we prove weak characterizations, a chain rule for Lipschitz functions, a sufficient condition for pathwise continuity and an Ito formula for non-adapted processes.
Greedy Algorithms for Reduced Bases in Banach Spaces
DeVore, Ronald
2013-02-26
Given a Banach space X and one of its compact sets F, we consider the problem of finding a good n-dimensional space X n⊂X which can be used to approximate the elements of F. The best possible error we can achieve for such an approximation is given by the Kolmogorov width dn(F)X. However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in Buffa et al. (Modél. Math. Anal. Numér. 46:595-603, 2012) in the case X=H is a Hilbert space. The results of Buffa et al. (Modél. Math. Anal. Numér. 46:595-603, 2012) were significantly improved upon in Binev et al. (SIAM J. Math. Anal. 43:1457-1472, 2011). The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces. © 2013 Springer Science+Business Media New York.
Some Banach spaces of Dirichlet series
Bailleul, Maxime; Lefèvre, Pascal
2013-01-01
The Hardy spaces of Dirichlet series denoted by ${\\cal H}^p$ ($p\\ge1$) have been studied in [12] when p = 2 and in [3] for the general case. In this paper we study some Lp-generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces denoted ${\\cal A}^p$ and ${\\cal B}^p$. We recover classical properties of spaces of analytic functions: boundedness of point evaluation, embeddings between these spaces and "Littlewood-Paley" formulas when p = 2. We also show that the...
An approximation problem of the finite rank operator in Banach spaces
王玉文; 潘少荣
2003-01-01
By the method of geometry of Banach spaces, we have proven that a bounded linear operatorin Banach space is a compact linear one iff it can be uniformly approximated by a sequence of the finite rankbounded homogeneous operators, which reveals the essence of the counter example given by Enflo.
Adrian PETRU(S)EL; Jen-Chih YAO
2009-01-01
In this paper, we prove a strong convergence theorem for resolvents of accretive operators in a Banach space by the viscosity approximation method with a generalized contraction mapping. The proximal point algorithm in a Banach space is also considered. The results extend some very recent theorems of W. Takahashi.
On sets minimizing their weighted length in uniformly convex separable Banach spaces
De Pauw, Thierry; Lemenant, Antoine; Millot, Vincent
2013-01-01
We study existence and partial regularity relative to the weighted Steiner problem in Banach spaces. We show $C^1$ regularity almost everywhere for almost minimizing sets in uniformly rotund Banach spaces whose modulus of uniform convexity verifies a Dini growth condition.
On moduli of convexity in Banach spaces
Reif Jiří
2005-01-01
Let be a normed linear space, an element of norm one, and and the local modulus of convexity of . We denote by the greatest such that for each closed linear subspace of the quotient mapping maps the open -neighbourhood of in onto a set containing the open -neighbourhood of in . It is known that . We prove that there is no universal constant such that , however, such a constant exists within the class of Hilbert spaces . If is a Hilbert space with , then .
Stability of interconnected dynamical systems described on Banach spaces
Rasmussen, R. D.; Michel, A. N.
1976-01-01
New stability results for a large class of interconnected dynamical systems (also called composite systems or large scale systems) described on Banach spaces are established. In the present approach, the objective is always the same: to analyze large scale systems in terms of their lower order and simpler subsystems and in terms of their interconnecting structure. The present results provide a systematic procedure of analyzing hybrid dynamical systems (i.e., systems that are described by a mixture of different types of equations). To demonstrate the method of analysis advanced, two specific examples are considered.
On nonautonomous second-order differential equations on Banach space
Nguyen Thanh Lan
2001-04-01
Full Text Available We show the existence and uniqueness of classical solutions of the nonautonomous second-order equation: uÃ¢Â€Â³(t=A(tuÃ¢Â€Â²(t+B(tu(t+f(t, 0Ã¢Â‰Â¤tÃ¢Â‰Â¤T; u(0=x0, uÃ¢Â€Â²(0=x1 on a Banach space by means of operator matrix method and apply to Volterra integrodifferential equations.
Iterative solutions of nonlinear equations in smooth Banach spaces
Let E be a smooth Banach space over the real field, φ not= K is contained in E closed convex and bounded, T:K → K uniformly continuous and strongly pseudo-contractive. It is proved that the Ishikawa iteration process converges strongly to the unique fixed point of T. Applications of this result to the operator equations Au=f or u+Au=f where A is a strongly accretive mapping of E into itself and under various continuity assumptions on A are also given. (author). 41 refs
Geometry and Gâteaux smoothness in separable Banach spaces
Hájek, Petr Pavel; Montesinos, V.; Zizler, Václav
2012-01-01
Roč. 6, č. 2 (2012), s. 201-232. ISSN 1846-3886 R&D Projects: GA ČR(CZ) GAP201/11/0345; GA AV ČR IAA100190901 Institutional research plan: CEZ:AV0Z10190503 Keywords : Gâteaux differentiable norms * extreme points * Radon-Nikodým property Subject RIV: BA - General Math ematics Impact factor: 0.529, year: 2012 http://oam.ele- math .com/06-15/Geometry-and-Gateaux-smoothness-in-separable-Banach-spaces
Linearization of class C1 for contractions on Banach spaces
Rodrigues, Hildebrando M.; Solà-Morales, J.
In this work we prove a C1-linearization result for contraction diffeomorphisms, near a fixed point, valid in infinite-dimensional Banach spaces. As an intermediate step, we prove a specific result of existence of invariant manifolds, which can be interesting by itself and that was needed on the proof of our main theorem. Our results essentially generalize some classical results by P. Hartman in finite dimensions, and a result of Mora-Sola-Morales in the infinite-dimensional case. It is shown that the result can be applied to some abstract systems of semilinear damped wave equations.
Near Convexity, Near Smoothness and Approximative Compactness of Half Spaces in Banach Spaces
Zi Hou ZHANG; Yu ZHOU; Chun Yan LIU
2016-01-01
The authors discuss the dual relation of nearly very convexity and property WS.By two kinds of near convexities and two kinds of near smoothness,the authors prove a series of characterizations such that every half-space in Banach space X and every weak* half-space in the dual space X* are approximatively weakly compact and approximatively compact.They show a sufficient condition such that a Banach space X is a Asplund space.Using upper semi-continuity of duality mapping,the authors also give two characterizations of property WS and property S.
On moduli of convexity in Banach spaces
Reif Jiří
2005-01-01
Full Text Available Let be a normed linear space, an element of norm one, and and the local modulus of convexity of . We denote by the greatest such that for each closed linear subspace of the quotient mapping maps the open -neighbourhood of in onto a set containing the open -neighbourhood of in . It is known that . We prove that there is no universal constant such that , however, such a constant exists within the class of Hilbert spaces . If is a Hilbert space with , then .
FRAME MULTIRESOLUTION ANALYSIS AND INFINITE TREES IN BANACH SPACES ON LOCALLY COMPACT ABELIAN GROUPS
S. S. Panday
2004-01-01
We extend the concept of frame multiresolution analysis to a locally compact abelian group and use it to define certain weighted Banach spaces and the spaces of their antifunctionals. We define analysis and synthesis operators on these spaces and establish the continuity of their composition. Also, we prove a general result to characterize infinite trees in the above Banach spaces of antifunctionals. This paper paves the way for the study of corresponding problems associated with some other types of Banach spaces on locally compact abelian groups including modulation spaces.
Smoothness in Banach spaces. Selected problems
Fabian, Marián; Montesinos, V.; Zizler, Václav
2006-01-01
Roč. 100, č. 2 (2006), s. 101-125. ISSN 1578-7303 R&D Projects: GA ČR(CZ) GA201/04/0090; GA AV ČR(CZ) IAA100190610 Institutional research plan: CEZ:AV0Z10190503 Keywords : smooth norm * renorming * weakly compactly generated space Subject RIV: BA - General Mathematics
QUASI-LOCAL CONJUGACY THEOREMS IN BANACH SPACES
ZHANG WEIRONG; MA JIPU
2005-01-01
Let f: U(xo)() E → F be a C1 map and f'(x0) be the Frechet derivative of f at x0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties: f'(x0) is double splitting and R(f'(x)) ∩ N(T0+) = {0} near x0. However,in infinite dimensional Banach spaces, f'(x0) is not always double splitting (i.e., the generalized inverse of f'(x0) does not always exist), but its bounded outer inverse of f'(x0) always exists.Only using the C1 map f and the outer inverse T0# of f'(x0), the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if x0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.
Homomorphisms and functional calculus on algebras on entire functions on Banach spaces
H. M. Pryimak
2015-07-01
Full Text Available The paper is devoted to study homomorphisms of algebras of entire functionson Banach spaces to a commutative Banach algebra. In particular, it is proposed amethod to construct homomorphisms vanishing on homogeneouspolynomials of degree less or equal that a fixed number $n.$
Stochastic integration in Banach spaces theory and applications
Mandrekar, Vidyadhar
2015-01-01
Considering Poisson random measures as the driving sources for stochastic (partial) differential equations allows us to incorporate jumps and to model sudden, unexpected phenomena. By using such equations the present book introduces a new method for modeling the states of complex systems perturbed by random sources over time, such as interest rates in financial markets or temperature distributions in a specific region. It studies properties of the solutions of the stochastic equations, observing the long-term behavior and the sensitivity of the solutions to changes in the initial data. The authors consider an integration theory of measurable and adapted processes in appropriate Banach spaces as well as the non-Gaussian case, whereas most of the literature only focuses on predictable settings in Hilbert spaces. The book is intended for graduate students and researchers in stochastic (partial) differential equations, mathematical finance and non-linear filtering and assumes a knowledge of the required integrati...
Yan Tang
2013-01-01
Full Text Available Suppose that C is a nonempty closed convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. A viscosity iterative process is constructed in this paper. A strong convergence theorem is proved for a common element of the set of fixed points of a finite family of pseudocontractive mappings and the set of solutions of a finite family of monotone mappings. And the common element is the unique solution of certain variational inequality. The results presented in this paper extend most of the results that have been proposed for this class of nonlinear mappings.
Porous sets for mutually nearest points in Banach spaces
Chong Li
2008-01-01
Full Text Available Let \\(\\mathfrak{B}(X\\ denote the family of all nonempty closed bounded subsets of a real Banach space \\(X\\, endowed with the Hausdorff metric. For \\(E, F \\in \\mathfrak{B}(X\\ we set \\(\\lambda_{EF} = \\inf \\{\\|z - x\\| : x \\in E, z \\in F \\}\\. Let \\(\\mathfrak{D}\\ denote the closure (under the maximum distance of the set of all \\((E, F \\in \\mathfrak{B}(X \\times \\mathfrak{B}(X\\ such that \\(\\lambda_{EF} \\gt 0\\. It is proved that the set of all \\((E, F \\in \\mathfrak{D}\\ for which the minimization problem \\(\\min_{x \\in E, z\\in F}\\|x - z\\|\\ fails to be well posed in a \\(\\sigma\\-porous subset of \\(\\mathfrak{D}\\.
Approximative compactness and continuity of metric projector in Banach spaces and applications
HUDZIK; Henryk; KOWALEWSKI; Wojciech; WISLA; Marek
2008-01-01
First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way.It is known that if C is a semi-Chebyshev closed and approximately compact set in a Banach space X,then the metric projectorπC from X onto C is continuous.Under the assumption that X is midpoint locally uniformly rotund,we prove that the approximative compactness of C is also necessary for the continuity of the projectorπC by the method of geometry of Banach spaces.Using this general result we find some necessary and sufficient conditions for T to have a continuous Moore-Penrose metric generalized inverse T~+,where T is a bounded linear operator from an approximative compact and a rotund Banach space X into a midpoint locally uniformly rotund Banach space Y.
On Property () in Banach Lattices, Calderón–Lozanowskiĭ and Orlicz–Lorentz Spaces
Paweł Kolwicz
2001-08-01
The geometry of Calderón–Lozanowskiĭ spaces, which are strongly connected with the interpolation theory, was essentially developing during the last few years (see [4, 9, 10, 12, 13, 17]). On the other hand many authors investigated property () in Banach spaces (see [7, 19, 20, 21, 25, 26]). The first aim of this paper is to study property () in Banach function lattices. Namely a criterion for property () in Banach function lattice is presented. In particular we get that in Banach function lattice property () implies uniform monotonicity. Moreover, property () in generalized Calderón–Lozanowskiĭ function spaces is studied. Finally, it is shown that in Orlicz–Lorentz function spaces property () and uniform convexity coincide.
In this work, we establish a sufficient condition for the controllability of the first-order impulsive neutral functional differential inclusions with infinite delay in Banach spaces. The results are obtained by using the Dhage's fixed point theorem.
张石生
2001-01-01
The purpose of this paper is to study the existence and approximation problem of solutions for a class of variational inclusions with accretive mappings in Banach spaces. The results extend and improve some recent results.
Approximating zero points of accretive operators with compact domains in general Banach spaces
Miyake Hiromichi
2005-01-01
Full Text Available We prove strong convergence theorems of Mann's type and Halpern's type for resolvents of accretive operators with compact domains and apply these results to find fixed points of nonexpansive mappings in Banach spaces.
Exponential dichotomies for linear discrete-time systems in Banach spaces
Ioan-Lucian Popa
2012-04-01
Full Text Available In this paper we investigate some dichotomy concepts for linear difference equations in Banach spaces. Characterizations of these concepts are given. Some illustrating examples clarify the relations between these concepts.
Geometrical and Topological Properties of Bumps and Starlike Bodies in Banach Spaces
Azagra Rueda, Daniel; Jiménez Sevilla, María del Mar
2002-01-01
While the topological and geometrical properties of convex bodies in Banach spaces are quite well understood (including their topological and smooth classification), much less is known about the structure of starlike bodies. Starlike bodies are important objects in nonlinear functional analysis as they appear as level sets of $n$-homogeneous polynomials on Banach spaces. Significant progress in the study of starlike bodies has been done in the last years by the efforts of Manuel Cepedello, Ro...
Strong convergence theorems for uniformly L-Lipschitzian mappings in Banach spaces
Let E be a real reflexive Banach space with uniform Gateaux differentiable norm, K be a nonempty bounded closed and convex subset of E , T : K → K be a uniformly L-Lipschitzian mapping such that F (T) := {x element of K : Tx = x} ≠ 0, u element of K be fixed and let {αn}n≥0 and {γn}n≥0 subset of (0, 1) be such that limn→∞ αn = 0 = limn→ ∞ γn and limn→ ∞(βn - 1)/ αn = 0, where βn Σj=0n λj and λj = 1 + αjγjL. Let Sn := (1 - αnγn)I + αnγnTn. It is proved that there exists some integer N0 > 1, such that for each n ≥ N0, there exists unique xn element of K such that xn = αnu+(1 -αn) 1/ (n + 1) Σj=0n Sjxn. If φ : E → R is defined by φ (y) := LIMn vertical bar vertical bar xn -y vertical bar vertical bar2 for all y element of E here LIM denotes a Banach limit, vertical bar vertical bar xn - Txn vertical bar vertical bar → 0 as n → ∞ and Kmin intersection F (T) ≠ 0, where Kmin := {x element of E : φ (x) = min (u element of K) φ (u) }, then it is proved that {xn} converges strongly to a fixed point of T. As an application, it is proved that the iterative process, z0 element of K, zn+1 alpha#nu + (1 - αn) 1/ (n + 1) Σj=0n Sjzn , n ≥ 0, under suitable conditions on the iteration parameters, converges strongly to a fixed point of T. (author)
On quasi-convex mappings of order α in the unit ball of a complex Banach space
LIU Taishun; XU Qinghua
2006-01-01
In this paper, a class of biholomorphic mappings named quasi-convex mapping of order α in the unit ball of a complex Banach space is introduced. When the Banach space is confined to Cn, we obtain the relation between this class of mappings and the convex mappings.Furthermore, the growth and covering theorems of this class of mappings are given on the unit ball of a complex Banach space Ⅹ. Finally, we get the second order terms coefficient estimations of the homogeneous expansion of quasi-convex mapping of order α defined on the polydisc in Cn and on the unit ball in a complex Banach space, respectively.
Robert F.Allen
2014-01-01
We study the bounded and the compact weighted composition operators from the Bloch space into the weighted Banach spaces of holomorphic functions on bounded homogeneous domains, with particular attention to the unit polydisk. For bounded homogeneous domains, we characterize the bounded weighted composition operators and determine the operator norm. In addition, we provide sufficient condi-tions for compactness. For the unit polydisk, we completely characterize the compact weighted composition operators, as well as provide ”computable” estimates on the operator norm.
Isometric uniqueness of a complementably universal Banach space for Schauder decompositions
Garbulińska, Joanna
2014-01-01
We present an isometric version of the complementably universal Banach space $\\mathbb{P}$ with a Schauder decomposition. The space $\\mathbb{P}$ is isomorphic to Pe{\\l}czy\\'nski's space with a universal basis as well as to Kadec' complementably universal space with the bounded approximation property.
On well posedness of best simultaneous approximation problems in Banach spaces
无
2001-01-01
The well posedness of best simultaneous approximation problems is considered. We establish the generic results on the well posedness of the best simultaneous approximation problems for any closed weakly compact nonempty subset in a strictly convex Kadec Banach space. Further, we prove that the set of all points in E(G) such that the best simultaneous approximation problems are not well posed is a σ-porous set in E(G) when X is a uniformly convex Banach space. In addition, we also investigate the generic property of the ambiguous loci of the best simultaneous approximation.
Real-Analytic Negligibility of Points and Subspaces in Banach Spaces, with Applications
Azagra Rueda, Daniel; Dobrowolski, Tadeusz
2002-01-01
We prove that every infinite-dimensional Banach space X having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to X \\ {0}. More generally, if X is an infinite-dimensional Banach space and F is a closed subspace of X such that there is a real-analytic seminorm on X whose set of zeros is F, and X / F is infinite-dimensional, then X and X \\ F are real-analytic diffeomorphic. As an application we show the existence of real-analytic free actions of the circle and t...
Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications
Azagra Rueda, Daniel; Dobrowolski, Tadeusz
1998-01-01
This article deals with smooth removability of compact sets in infinite-dimensional Banach spaces. The main result states that ifX is an infinite-dimensional Banach space which has a not necessarily equivalent Cp-smooth norm and K is a compact subset of X, then X and X r K are Cp diffeomorphic. The proof relies on the construction of a “deleting path” through a nontrivial refinement of Bessaga’s incomplete-norm technique. However, norms are not at present available and the construction req...
On Best Approximations from RS-sets in Complex Banach Spaces
Chong LI
2005-01-01
The concept of an RS-set in a complex Banach Space is introduced and the problem of best approximation from an RS-set in a complex space is investigated. Results consisting of characterizations, uniqueness and strong uniqueness are established.
Integrals with values in Banach spaces and locally convex spaces
Mikusinski, Piotr
2014-01-01
The purpose of this article is to present the construction and basic properties of the general Bochner integral. The approach presented here is based on the ideas from the book The Bochner Integral by J. Mikusinski where the integral is presented for functions defined on $\\mathbb{R}^N$. In this article we present a more general and simplified construction of the Bochner integral on abstract measure spaces. An extension of the construction to functions with values in a locally convex space is ...
Murthy, S; Arunkumar, M; V. Govindan
2015-01-01
In this paper, the authors introduce and investigate the general solution and generalized Ulam-Hyers stability of a generalized n-type additive-quadratic functional equation. g(x + 2y; u + 2v) + g(x 2y; u 2v) = 4[g(x + y; u + v) + g(x y; u v)] 6g(x; u) + g(2y; 2v) + g(2y;2v) 4g(y; v) 4g(y;v) Where is a positive integer with , in Banach Space and Banach Algebras using direct and fixed point methods.
The generalized hardy operator with kernel and variable integral limits in banach function spaces
Lang J; Gogatishvill A
1999-01-01
Let we have an integral operator where and are nondecreasing functions, and are non-negative and finite functions, and is nondecreasing in , nonincreasing in and for and . We show that the integral operator where and are Banach functions spaces with -condition is bounded if and only if . Where and
Bound and periodic solutions of the Riccati equation in Banach space
A. Ya. Dorogovtsev
1995-01-01
Full Text Available An abstract, nonlinear, differential equation in Banach space is considered. Conditions are presented for the existence of bounded solutions of this equation with a bounded right side, and also for the existence of stationary (periodic solutions of this equation with a stationary (periodic process in the right side.
Known results and open problems on C1 linearization in Banach spaces
Munhoz Rodrigues, Hildebrando; Solà-Morales Rubió, Joan
2012-01-01
The purpose of this paper is to review the results obtained by the authors on linearization of dynamical systems in infinite dimen- sional 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.
Common Fixed Point Theorems in Non-normal Cone Metric Spaces with Banach Algebras
HUANG Hua-ping; XU Shao-yuan; LIU Qiu-hua; MING Wei
2016-01-01
In this paper, we obtain a class of common fixed point theorems for generalized Lipschitz mappings in cone metric spaces with Banach algebras without the assumption of normality of cones. The results greatly generalize some results in the literature. Moreover, we give an example to support the main assertions.
Gu Feng
2006-01-01
Full Text Available The purpose of this paper is to study the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Chang and Cho (2003, Xu and Ori (2001, and Zhou and Chang (2002.
POSITIVE SOLUTIONS TO SEMI-LINEAR SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS IN BANACH SPACE
无
2008-01-01
In this paper,we study the existence of positive periodic solution to some second- order semi-linear differential equation in Banach space.By the fixed point index theory, we prove that the semi-linear differential equation has two positive periodic solutions.
曾六川
2003-01-01
A new class of almost asymptotically nonexpansive type mappings in Banach spaces is introduced, which includes a number of known classes of nonlinear Lipschitzian mappings and non-Lipschitzian mappings in Banach spaces as special cases; for example,the known classes of nonexpansive mappings, asymptotically nonexpansive mappings and asymptotically nonexpansive type mappings. The convergence problem of modified Ishikawa iterative sequences with errors for approximating fixed points of almost asymptotically nonexpansive type mappings is considered. Not only S. S. Chang' s inequality but also H.K. Xu' s one for the norms of Banach spaces are applied to make the error estimate between the exact fixed point and the approximate one. Moreover, Zhang Shi-sheng ' s method (Applied Mathematics and Mechanics ( English Edition ), 2001,22 (1) :25 - 34) for making the convergence analysis of modified Ishikawa iterative sequences with errors is extended to the case of almost asymptotically nonexpansive type mappings. The new convergence criteria of modified Ishikawa iterative sequences with errors for finding fixed points of almost asymptotically nonexpansive type mappings in uniformly convex Banach spaces are presented. Also, the new convergence criteria of modified Mann iterative sequences with errors for this class of mappings are immediately obtained from these criteria. The above results unify, improve and generalize Zhang Shi-sheng's ones on approximating fixed points of asymptotically nonexpansive type mappings by the modified Ishikawa and Mann iterative sequences with errors.
Convergence rates for an iteratively regularized Newton–Landweber iteration in Banach space
In this paper, we provide convergence and convergence rate results for a Newton-type method with a modified version of Landweber iteration as an inner iteration in a Banach space setting. Numerical experiments illustrate the performance of the method. (paper)
Non-expansive Mappings and Iterative Methods in Uniformly Convex Banach Spaces
Hai Yun ZHOU
2004-01-01
In this article, we will investigate the properties of iterative sequence for non-expansive mappings and present several strong and weak convergence results of successive approximations to fixed points of non-expansive mappings in uniformly convex Banach spaces. The results presented in this article generalize and improve various ones concerned with constructive techniques for the fixed points of non-expansive mappings.
Wan-sheng WANG; Shou-fu LI; Run-sheng YANG
2012-01-01
A series of contractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations (NFDEs) in Banach spaces are obtained,which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations (FDEs),neutral delay differential equations (NDDEs) and NFDEs of other types which appear in practice.
Fixed point iterations for quasi-contractive maps in uniformly smooth Banach spaces
Two well-known fixed point iteration methods are applied to approximate fixed points of quasi-contractive maps in real uniformly smooth Banach spaces. While our theorems generalize important known results, our method is of independent interest. (author). 25 refs
Approximate Fixed Point Theorems in Banach Spaces with Applications in Game Theory
2002-01-01
In this paper some new approximate fixed point theorems for multifunctions in Banach spaces are presented and a method is developed indicating how to use approximate fixed point theorems in proving the existence of approximate Nash equilibria for non-cooperative games.
Asymptotic behaviour of the solutions of Schroedinger equation with impulse effect in a Banach space
The present paper studies the asymptotic behaviour of the solutions of linear homogeneous differential Schroedinger equation with impulse effect in a Banach space and finds a dependence between their asymptotic behaviour and the spectrum of the linear Hamiltonian operator. 6 refs
A RANDOM FIXED POINT ITERATION FOR THREE RANDOM OPERATORS ON UNIFORMLY CONVEX BANACH SPACES
Binayak S. Choudhury
2003-01-01
In the present paper we introduce a random iteration scheme for three random operators defined on a closed and convex subset of a uniformly convex Banach space and prove its convergence to a common fixed point of three random operators. The result is also an extension of a known theorem in the corresponding non-random case.
On common fixed points of compatible mappings in metric and Banach spaces
M. S. Khan
1988-06-01
Full Text Available We prove a number of results concerning the existence of common fixed points of a family of maps satisfying certain contractive conditions in metric and Banach spaces. Results dealing with the stucture of the set of common fixed points of such maps are also given. Our work is an improvement upon the previously known results.
Strong Convergence Theorems for Strict Pseudocontractions in Uniformly Convex Banach Spaces
Lin Wei-Wei
2010-01-01
Full Text Available The viscosity approximation methods are employed to establish strong convergence theorems of the modified Mann iteration scheme to -strict pseudocontractions in -uniformly convex Banach spaces with a uniformly Gâteaux differentiable norm. The main result improves and extends many nice results existing in the current literature.
Some G-M-type Banach spaces and K-groups of operator algebras on them
ZHONG Huaijie; CHEN Dongxiao; CHEN Jianlan
2004-01-01
By providing several new varieties of G-M-type Banachspaces according to decomposable and compoundable properties, this paper discusses the operator structures of thesespaces and the K-theory of the algebra of the operators on these G-M-type Banach spaces throughcalculation of the K-groups of the operator ideals contained in the class of Riesz operators.
Psi-exponential dichotomy for linear differential equations in a Banach space
Atanaska Georgieva
2013-07-01
Full Text Available In this article we extend the concept psi-exponential and psi-ordinary dichotomies for homogeneous linear differential equations in a Banach space. With these two concepts we prove the existence of psi-bounded solutions of the appropriate inhomogeneous equation. A roughness of the psi-dichotomy is also considered.
Nonuniform Exponential Stability and Instability of Evolution Operators in Banach Space
Mihaela Tomescu; Andrea Minda
2006-01-01
In this paper is presenting a parallel between nonuniform exponential stability and nonuniform exponential instability of evolution operators in Banach spaces, beginning to present the concept of the evolution operator with nonuniform exponential decay, respectively growth, next with the concept of the nonuniform stability, respectively instability, nonuniform exponential stability, respectively instability, nonuniform integrable stability, respectively instability and...
Nonuniform Exponential Stability and Instability of Evolution Operators in Banach Space
Mihaela Tomescu
2006-10-01
Full Text Available In this paper is presenting a parallel between nonuniform exponential stability and nonuniform exponential instability of evolution operators in Banach spaces, beginning to present the concept of the evolution operator with nonuniform exponential decay, respectively growth, next with the concept of the nonuniform stability, respectively instability, nonuniform exponential stability, respectively instability, nonuniform integrable stability, respectively instability and relationship between this concepts.
Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces
2009-02-01
Full Text Available We study the influence of computational errors on the convergence to compact sets of orbits of nonexpansive mappings in Banach and metric spaces. We first establish a convergence theorem assuming that the computational errors are summable and then provide examples which show that the summability of errors is necessary for convergence.
ASYMPTOTICALLY ISOMETRIC COPIES OF lp (1≤p＜∞) AND c0 IN BANACH SPACES
Chen Dongyang
2006-01-01
Let X be a Banach space. If there exists a quotient space of X which is asymptotically isometric to l1, then X contains complemented asymptotically isometric copies of l1. Every infinite dimensional closed subspace of l1 contains a complemented subspace of l1 which is asymptotically isometric to l1. Let X be a separable Banach space such that X* contains asymptotically isometric copies of lp (1 ＜ p ＜∞). Then there exists a quotient space of X which is asymptotically isometric to lq (1/p+1/q=1). Complementedasymptotically isometric copies of c0 in K(X, Y) and W(X, Y) are discussed. Let X be a Gelfand-Phillips space. If X contains asymptotically isometric copies of c0, it has to contain complemented asymptotically isometric copies of c0.
N. L. Braha
2005-05-01
Full Text Available In this paper we will give a characterization of 1-absolutely summing operators using μ-approximate l_1 sequences. Exactly if (x_n _1^∞ is μ-approximate l_1 , basic and normalized sequence in Banach space X then every bounded linear operator T from X into Banach space Y is 1-absolutely summing if and only if Y is isomorphic to Hilbert space.
Boulbeba Abdelmoumen; Aref Jeribi; Maher Mnif
2012-01-01
In this paper,we define new measures called respectively graph measure of noncompactness and graph measure of weak noncompactness.Moreover,we apply the obtained results to discuss the incidence of some perturbation results realized in [2] on the behavior of essential spectra of such closed densely defined linear operators on Banach spaces.These results are exploited to investigate the essential spectra of a multidimensional neutron transport operator on L1 spaces.
Domains of uniqueness for $C_0$-semigroups on the dual of a Banach space
Lemle, Ludovic Dan
2008-01-01
Let $({\\cal X},\\|\\:.\\:\\|)$ be a Banach space. In general, for a $C_0$-semigroup \\semi on $({\\cal X},\\|\\:.\\:\\|)$, its adjoint semigroup \\semia is no longer strongly continuous on the dual space $({\\cal X}^{*},\\|\\:.\\:\\|^{*})$. Consider on ${\\cal X}^{*}$ the topology of uniform convergence on compact subsets of $({\\cal X},\\|\\:.\\:\\|)$ denoted by ${\\cal C}({\\cal X}^{*},{\\cal X})$, for which the usual semigroups in literature becomes $C_0$-semigroups.\\\\ The main purpose of this paper is to prove th...
M. B. Ghaemi; H. Majani; Cho, Y J; Eshaghi Gordji, M.
2011-01-01
Using the fixed point method, we investigate the stability of the systems of quadratic-cubic and additive-quadratic-cubic functional equations with constant coefficients form r-divisible groups into Ŝerstnev probabilistic Banach spaces.
Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces
This paper treats the embedding of finite-dimensional subsets of a Banach space B into finite-dimensional Euclidean spaces. When the Hausdorff dimension of X − X is finite, dH(X − X) k are injective on X. The proof motivates the definition of the 'dual thickness exponent', which is the key to proving that a prevalent set of such linear maps have Hölder continuous inverse when the box-counting dimension of X is finite and k > 2dB(X). A related argument shows that if the Assouad dimension of X − X is finite and k > dA(X − X), a prevalent set of such maps are bi-Lipschitz with logarithmic corrections. This provides a new result for compact homogeneous metric spaces via the Kuratowksi embedding of (X, d) into L∞(X)
Inexact Newton–Landweber iteration for solving nonlinear inverse problems in Banach spaces
By making use of duality mappings, we formulate an inexact Newton–Landweber iteration method for solving nonlinear inverse problems in Banach spaces. The method consists of two components: an outer Newton iteration and an inner scheme providing the increments by applying the Landweber iteration in Banach spaces to the local linearized equations. It has the advantage of reducing computational work by computing more cheap steps in each inner scheme. We first prove a convergence result for the exact data case. When the data are given approximately, we terminate the method by a discrepancy principle and obtain a weak convergence result. Finally, we test the method by reporting some numerical simulations concerning the sparsity recovery and the noisy data containing outliers. (paper)
Cauchy-Rassias Stability of Cauchy-Jensen Additive Mappings in Banach Spaces
Choonkil BAAK
2006-01-01
Let X, Y be vector spaces. It is shown that if a mapping f: X → Y satisfiesf(x+y/2+z)+f(x-y/2+z)=f(x)+2f(z), (0.1)f(x+y/2+z)-f(x-y/2+z)= f(y), (0.2)or2f(x+y/2+z)=f(x)+f(y)+2f(z) (0.3)for all x, y, z ∈ X, then the mapping f: X → Y is Cauchy additive.Furthermore, we prove the Cauchy-Rassias stability of the functional equations (0.1), (0.2) and (0.3) in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebra.
Common Fixed-Point Problem for a Family Multivalued Mapping in Banach Space
Zhanfei Zuo
2011-01-01
Full Text Available It is our purpose in this paper to prove two convergents of viscosity approximation scheme to a common fixed point of a family of multivalued nonexpansive mappings in Banach spaces. Moreover, it is the unique solution in to a certain variational inequality, where ∶=∩∞=0( stands for the common fixed-point set of the family of multivalued nonexpansive mapping {}.
Mann Type Implicit Iteration Approximation for Multivalued Mappings in Banach Spaces
Huimin He
2010-01-01
Full Text Available Let K be a nonempty compact convex subset of a uniformly convex Banach space E and let T be a multivalued nonexpansive mapping. For the implicit iterates x0∈K, xn=αnxn-1+(1-αnyn, yn∈Txn, n≥1. We proved that {xn} converges strongly to a fixed point of T under some suitable conditions. Our results extended corresponding ones and revised a gap in the work of Panyanak (2007.
Tomonari Suzuki
2006-01-01
We prove Browder's type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let be a bounded closed convex subset of a uniformly smooth Banach space . Let be an infinite family of commuting nonexpansive mappings on . Let and be sequences in satisfying for . Fix and define a sequence in by for . Then converges strongly to , where is the unique sunny nonexpansive retraction from onto .
Approximating Common Fixed Points of Lipschitzian Semigroup in Smooth Banach Spaces
Saeidi Shahram
2008-01-01
Abstract Let be a left amenable semigroup, let be a representation of as Lipschitzian mappings from a nonempty compact convex subset of a smooth Banach space into with a uniform Lipschitzian condition, let be a strongly left regular sequence of means defined on an -stable subspace of , let be a contraction on , and let , , and be sequences in (0, 1) such that , for all . Let , for all . Then, under suitable hypotheses on the constants, we show that converges stron...
Hyers-Ulam stability of linear second-order differential equations in complex Banach spaces
Yongjin Li
2013-08-01
Full Text Available We prove the Hyers-Ulam stability of linear second-order differential equations in complex Banach spaces. That is, if y is an approximate solution of the differential equation $y''+ alpha y'(t +eta y = 0$ or $y''+ alpha y'(t +eta y = f(t$, then there exists an exact solution of the differential equation near to y.
Nonlinear Evolution Governed by Accretive Operators in Banach Spaces: Error Control and Applications
Nochetto, Ricardo H; Savare, Giuseppe
2001-01-01
Nonlinear evolution equations governed by $m$-accretive operators in Banach spaces are discretized via the backward or forward Euler methods with variable stepsize. Computable a posteriori error estimates are derived in terms of the discrete solution and data, and shown to converge with optimal order $O(sqrttau)$. Applications to scalar conservation laws and degenerate parabolic equations (with or without hysteresis) in $L^1$, as well as to Hamilton-Jacobi equations in $C^...
MIXED MONOTONE ITERATIVE TECHNIQUES FOR SEMILINEAR EVOLUTION EQUATIONS IN BANACH SPACES
王良龙; 王志成
2004-01-01
This paper is concerned with initial value problems for semilinear evolution equations in Banach spaces. The abstract iterative schemes are constructed by combining the theory of semigroups of linear operators and the method of mixed monotone iterations. Some existence results on minimal and maximal (quasi)solutions are established for abstract semilinear evolution equations with mixed monotone or mixed quasimonotone nonlinear terms. To illustrate the main results, applications to ordinary differential equations and partial differential equations are also given.
Fixed point iterations for strictly hemi-contractive maps in uniformly smooth Banach spaces
It is proved that the Mann iteration process converges strongly to the fixed point of a strictly hemi-contractive map in real uniformly smooth Banach spaces. The class of strictly hemi-contractive maps includes all strictly pseudo-contractive maps with nonempty fixed point sets. A related result deals with the Ishikawa iteration scheme when the mapping is Lipschitzian and strictly hemi-contractive. Our theorems generalize important known results. (author). 29 refs
Suzuki Tomonari
2006-01-01
Full Text Available We prove Browder's type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let be a bounded closed convex subset of a uniformly smooth Banach space . Let be an infinite family of commuting nonexpansive mappings on . Let and be sequences in satisfying for . Fix and define a sequence in by for . Then converges strongly to , where is the unique sunny nonexpansive retraction from onto .
On some Banach space constants arising in nonlinear fixed point and eigenvalue theory
Erzakova Nina A
2004-01-01
Full Text Available As is well known, in any infinite-dimensional Banach space one may find fixed point free self-maps of the unit ball, retractions of the unit ball onto its boundary, contractions of the unit sphere, and nonzero maps without positive eigenvalues and normalized eigenvectors. In this paper, we give upper and lower estimates, or even explicit formulas, for the minimal Lipschitz constant and measure of noncompactness of such maps.
On some Banach space constants arising in nonlinear fixed point and eigenvalue theory
Martin Väth
2004-12-01
Full Text Available As is well known, in any infinite-dimensional Banach space one may find fixed point free self-maps of the unit ball, retractions of the unit ball onto its boundary, contractions of the unit sphere, and nonzero maps without positive eigenvalues and normalized eigenvectors. In this paper, we give upper and lower estimates, or even explicit formulas, for the minimal Lipschitz constant and measure of noncompactness of such maps.
Hierarchy of Hamilton equations on Banach Lie-Poisson spaces related to restricted Grassmannian
Golinski, Tomasz
2009-01-01
Using the Magri method one defines an involutive family of Hamiltonians on Banach Lie-Poisson space iR+UL_res^1 (which contains the restricted Grassmannian as a symplectic leaf) and on its complexification C+L_res^1. The hierarchy of Hamilton equations given by these Hamiltonians is investigated. The operator equations of Ricatti-type are included in this hierarchy. For a few particular cases one gives the explicit solutions.
Some s-numbers of an integral operator of Hardy type in Banach function spaces
Edmunds, D.; Gogatishvili, Amiran; Kopaliani, T.; Samashvili, N.
2016-01-01
Roč. 207, July (2016), s. 79-97. ISSN 0021-9045 R&D Projects: GA ČR GA13-14743S Institutional support: RVO:67985840 Keywords : Hardy type operators * Banach function spaces * s-numbers * compact linear operators Subject RIV: BA - General Mathematics Impact factor: 0.951, year: 2014 http://www. science direct.com/ science /article/pii/S0021904516000265
A reinterpretation of set differential equations as differential equations in a Banach space
Rasmussen, Martin; Rieger, Janosch; Webster, Kevin
2015-01-01
Set differential equations are usually formulated in terms of the Hukuhara differential, which implies heavy restrictions for the nature of a solution. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of $\\R^d$ with their support functions. Using this representation, we demonstrate how existence and uniqueness results can be applied to set differential equations. We provide a simple example,...
LI Shoufu
2005-01-01
A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.
Weak and Strong Convergence Theorems for Nonexpansive Mappings in Banach Spaces
Su Yongfu
2008-01-01
Full Text Available Abstract The purpose of this paper is to introduce two implicit iteration schemes for approximating fixed points of nonexpansive mapping and a finite family of nonexpansive mappings , respectively, in Banach spaces and to prove weak and strong convergence theorems. The results presented in this paper improve and extend the corresponding ones of H.-K. Xu and R. Ori, 2001, Z. Opial, 1967, and others.
Yun-zhi Zou
2012-01-01
Full Text Available A new class of generalized dynamical systems involving generalized f-projection operators is introduced and studied in Banach spaces. By using the fixed-point theorem due to Nadler, the equilibrium points set of this class of generalized global dynamical systems is proved to be nonempty and closed under some suitable conditions. Moreover, the solutions set of the systems with set-valued perturbation is showed to be continuous with respect to the initial value.
Hytonen, Tuomas; van Neerven, Jan; Portal, Pierre
2007-01-01
We study conical square function estimates for Banach-valued functions, and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-McIntosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator (A) with certain off-diagonal bounds, such that (A) always has a bounded (H^{\\infty})-functional calculus on these spaces. This provides a new way of proving functional ...
On Linear Isometries of Banach Lattices in $\\mathcal{C}_0()$-Spaces
José M Isidro
2009-11-01
Consider the space $\\mathcal{C}_0()$ endowed with a Banach lattice-norm $\\|\\cdot p\\|$ that is not assumed to be the usual spectral norm $\\|\\cdot p\\|_∞$ of the supremum over . A recent extension of the classical Banach-Stone theorem establishes that each surjective linear isometry of the Banach lattice $(\\mathcal{C}_0(),\\|\\cdot p\\|)$ induces a partition of into a family of finite subsets $S\\subset$ along with a bijection $T:→$ which preserves cardinality, and a family $[u(S):S\\in]$ of surjective linear maps $u(S):\\mathcal{C}(T(S))→\\mathcal{C}(S)$ of the finite-dimensional *-algebras $\\mathcal{C}(S)$ such that $$(U f)|_{T(S)}=u(S)(f|_S) \\quad \\forall f\\in\\mathcal{C}_0() \\quad \\forall S\\in.$$ Here we endow the space of finite sets $S\\subset$ with a topology for which the bijection and the map are continuous, thus completing the analogy with the classical result.
Casazza, Peter G.; Christensen, Ole
1995-01-01
We prove that a Hilbert space frame $\\fti$ contains a Riesz basis if every subfamily $\\ftj , J \\subseteq I ,$ is a frame for its closed span. Secondly we give a new characterization of Banach spaces which do not have any subspace isomorphic to $c_0$. This result immediately leads to an improvement of a recent theorem of Holub concerning frames consisting of a Riesz basis plus finitely many elements.
Fixed Point Theorems for Suzuki Generalized Nonexpansive Multivalued Mappings in Banach Spaces
Abkar A
2010-01-01
Full Text Available In the first part of this paper, we prove the existence of common fixed points for a commuting pair consisting of a single-valued and a multivalued mapping both satisfying the Suzuki condition in a uniformly convex Banach space. In this way, we generalize the result of Dhompongsa et al. (2006. In the second part of this paper, we prove a fixed point theorem for upper semicontinuous mappings satisfying the Suzuki condition in strictly spaces; our result generalizes a recent result of Domínguez-Benavides et al. (2009.
Convergence rates for the iteratively regularized Gauss–Newton method in Banach spaces
In this paper we consider the iteratively regularized Gauss–Newton method (IRGNM) in a Banach space setting and prove optimal convergence rates under approximate source conditions. These are related to the classical concept of source conditions that is available only in Hilbert space. We provide results in the framework of general index functions, which include, e.g. Hölder and logarithmic rates. Concerning the regularization parameters in each Newton step as well as the stopping index, we provide both a priori and a posteriori strategies, the latter being based on the discrepancy principle
Delta-semidefinite and delta-convex quadratic forms in Banach spaces
Kalton, N.; Konyagin, S. V.; Vesely, L.
2006-01-01
A continuous quadratic form ("quadratic form", in short) on a Banach space $X$ is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear operator $T\\colon X\\to X^*$ factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if $T$ is a UMD-operator. It follows, for instance, that each quadratic form on an infinite-dimensional $...
Existence results for a class of parabolic evolution equations in Banach spaces
WangJing; XueXingmei
2003-01-01
We discuss the existence results of the parabolic evolution equation d(x(t) + g(t,x(t)))/dt + A(t)x(t) =f( t ,x(t)) in Banach spaces, where A (t) generates an evolution system and functions f, g are continuous. We get the theorem of existence of a mild solution, the theorem of existence and uniqueness of a mild solution and the theorem of existence and uniqueness of an S-classieal (semi-classical) solution. We extend the cases when g(t) = 0 or A(t) = A.
Some compactness tests in Banach spaces by Cesaro means of Fourier coefficients
Öztürk, Seda
2015-09-01
Let H be a complex Banach space, T be the topological group of the unit circle with respect to the Euclidean topology, α be a strongly continuous isometric linear representation of T in H, {Fkα}k ∈ℤ be the family of Fourier coefficients with respect to α, and {σkα}k ∈ℤ be Cesaro means of the family {Fkα}k ∈ℤ . In this work, we give some compactness tests for closed subsets of H.
Oscillation and the mean ergodic theorem for uniformly convex Banach spaces
Avigad, Jeremy; Rute, Jason
2012-01-01
Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n x. We prove the following variational inequality in the case where T is power bounded from above and below: for any increasing sequence (t_k)_{k in N} of natural numbers we have sum_k || A_{t_{k+1}} x - A_{t_k} x ||^p
ANTI-PERIODIC SOLUTIONS FOR FIRST AND SECOND ORDER NONLINEAR EVOLUTION EQUATIONS IN BANACH SPACES
WEI Wei; XIANG Xiaoling
2004-01-01
In this paper, a new existence theorem of anti-periodic solutions for a class ofstrongly nonlinear evolution equations in Banach spaces is presentedThe equations con-tain nonlinear monotone operators and a nonmonotone perturbationMoreover, throughan appropriate transformation, the existence of anti-periodic solutions for a class of second-order nonlinear evolution equations is verifiedOur abstract results are illustrated by anexample from quasi-linear partial differential equations with time anti-periodic conditionsand an example from quasi-linear anti-periodic hyperbolic differential equations.
Approximation of fixed points of Lipschitz pseudo-contractive mapping in Banach spaces
Let K be a subset of a real Banach space X. A mapping T:K → X is called pseudo-contractive if the inequality ||x-y|| ≤ ||(1+r)(x-y)-r(Tx-Ty)|| holds for all x,y in K and r > 0. Fixed points of Lipschitz pseudo-contractive maps are approximated under appropriate conditions, by an iteration process of the type introduced by W.R. Mann. This gives an affirmative answer to the problem stated by T.L. Hicks and J.R. Rubicek (J. Math. Anal. Appl. 59 (1977) 504). (author). 28 refs
Existence of Solutions to Nonlinear Impulsive Volterra Integral Equations in Banach Spaces
CHEN Fangqi; TIAN Ruilan
2005-01-01
In this paper, the existence of solutions is studied for nonlinear impulsive Volterra integral equations with infinite moments of impulse effect on the half line R+ in Banach spaces.By the use of a new comparison result and recurrence method, the new existence theorems are achieved under a weaker compactness-type condition, which generalize and improve the related results for this class of equations with finite moments of impulse effect on finite interval and infinite moments of impulse effect on infinite interval.
Basic Properties of Banach Space Xμ%Banach空间Xμ的基本性质
郑列
2005-01-01
研究了Banach空间Xμ={c=(ci)i≥1:‖c‖μ=def ∞∑ i=1 iμ|Ci|∞},μ≥1.讨论了它的范数,紧性以及强收敛和弱收敛之间的关系.%Banach space Xμ= {c= (ci)i≥1: ‖c‖μ=def ∞∑ i=1 iμ|ci|∞},μ≥1. is studied. Our results relate to norm,compactness and the relation between strong convergence and weak * convergence.
Convergence theorems for a class of nonlinear maps in uniformly smooth Banach spaces
Let K be a nonempty closed and convex subset of a real uniformly smooth Banach space, E, with modulus of smoothness of power type q>1. Let T be a mapping of K into itself, T is an element of C (in the notion of Browder and Petryshyn; and Rhoades). It is proved that the Mann iteration process, under suitable conditions, converges strongly to the unique fixed point of T. If K is also bounded, then the Ishikawa iteration process converges to the fixed point of T. While our theorems generalize important known results, our method is also of independent interest. (author). 14 refs
Hanke-Raus heuristic rule for variational regularization in Banach spaces
Jin, Qinian
2016-08-01
We generalize the heuristic parameter choice rule of Hanke-Raus for quadratic regularization to general variational regularization for solving linear as well as nonlinear ill-posed inverse problems in Banach spaces. Under source conditions formulated as variational inequalities, we obtain a posteriori error estimates in term of the Bregman distance. By imposing certain conditions on the random noise, we establish four convergence results; one relies on the source conditions and the other three do not depend on any source conditions. Numerical results are presented to illustrate the performance.
On the regularity of mild solutions to complete higher order differential equations on Banach spaces
Nezam Iraniparast
2015-09-01
Full Text Available For the complete higher order differential equation u(n(t=Σk=0n-1Aku(k(t+f(t, t∈ R (* on a Banach space E, we give a new definition of mild solutions of (*. We then characterize the regular admissibility of a translation invariant subspace al M of BUC(R, E with respect to (* in terms of solvability of the operator equation Σj=0n-1AjXal Dj-Xal Dn = C. As application, almost periodicity of mild solutions of (* is proved.
Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces
Udomene Aniefiok
2006-01-01
Full Text Available Suppose is a nonempty closed convex subset of a real Banach space . Let be two asymptotically quasi-nonexpansive maps with sequences such that and , and . Suppose is generated iteratively by where and are real sequences in . It is proved that (a converges strongly to some if and only if ; (b if is uniformly convex and if either or is compact, then converges strongly to some . Furthermore, if is uniformly convex, either or is compact and is generated by , where , are bounded, are real sequences in such that and , are summable; it is established that the sequence (with error member terms converges strongly to some .
C0-semigroups of linear operators on some ultrametric Banach spaces
Toka Diagana
2006-01-01
Full Text Available C0-semigroups of linear operators play a crucial role in the solvability of evolution equations in the classical context. This paper is concerned with a brief conceptualization of C0-semigroups on (ultrametric free Banach spaces E. In contrast with the classical setting, the parameter of a given C0-semigroup belongs to a clopen ball Ωr of the ground field K. As an illustration, we will discuss the solvability of some homogeneous p-adic differential equations.
A model of CT dose profiles in Banach space; with applications to CT dosimetry
Weir, Victor J.
2016-07-01
In this paper the scatter component of computed tomography dose profiles is modeled using the solution to a nonlinear ordinary differential equation. This scatter function is summed with a modeled primary function of approximate trapezoidal shape. The primary dose profile is modeled to include the analytic continuation of the Heaviside step function. A mathematical theory is developed in a Banach space. The modeled function is used to accurately fit data from a 256-slice GE Revolution scanner. A 60 cm long body phantom is assembled and used for data collection with both a pencil chamber and a Farmer-type chamber.
Local subdifferentials and multivariational inequalities in Banach and Frechet spaces
Pavlo O. Kasyanov; Valery S. Mel'nik; Anna M. Piccirillo
2008-01-01
Some functional-topological concepts of subdifferential and locally subdifferential maps in Frechet spaces are established. Multivariational inequalities with an operator of the pseudo-monotone type, connected with subdifferential maps, are considered.
On growth and covering theorems of quasi-convex mappings in the unit ball of a complex Banach space
ZHANG; Wenjun(张文俊); LIU; Taishun(刘太顺)
2002-01-01
A class of biholomorphic mappings named "quasi-convex mapping" is introduced in the unitball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class ofstarlike mappings and contains the class of convex mappings properly, and it has the same growth and coveringtheorems as the convex mappings. Furthermore, when the Banach space is confined to Cn, the "quasi-convexmapping" is exactly the "quasi-convex mapping of type A" introduced by K. A. Roper and T. J. Suffridge.
Isomorphisms on weighed Banach spaces of harmonic and holomorphic functions
Jorda Mora, Enrique; ZARCO GARCIA, ANA MARIA
2013-01-01
For an arbitrary open subset U subset of R-d or U subset of C-d and a continuous function v : U ->]0,infinity[ we show that the space h(v0) (U) of weighed harmonic functions is almost isometric to a (closed) subspace of c(0), thus extending a theorem due to Bonet and Wolf for spaces of holomorphic functions H-v0 (U) on open sets U subset of C-d. Inspired by recent work of Boyd and Rueda, we characterize in terms of the extremal points of the dual of h(v0) (U) when h(v0) (U) is isometric to a ...
A convergence rates result for an iteratively regularized Gauss–Newton–Halley method in Banach space
The use of second order information on the forward operator often comes at a very moderate additional computational price in the context of parameter identification problems for differential equation models. On the other hand the use of general (non-Hilbert) Banach spaces has recently found much interest due to its usefulness in many applications. This motivates us to extend the second order method from Kaltenbacher (2014 Numer. Math. at press), (see also Hettlich and Rundell 2000 SIAM J. Numer. Anal. 37 587620) to a Banach space setting and analyze its convergence. We here show rates results for a particular source condition and different exponents in the formulation of Tikhonov regularization in each step. This includes a complementary result on the (first order) iteratively regularized Gauss–Newton method in case of a one-homogeneous data misfit term, which corresponds to exact penalization. The results clearly show the possible advantages of using second order information, which get most pronounced in this exact penalization case. Numerical simulations for an inverse source problem for a nonlinear elliptic PDE illustrate the theoretical findings. (paper)
Shi-sheng ZHANG
2009-01-01
The purpose of this paper is to study the weak convergence problems of the implicity iteration process for Lipschitzian pseudocontractive semi-groups in the general Banach spaces. The results presented in this paper extend and improve the corresponding results of some people.
无
2010-01-01
This paper investigates the existence and multiplicity of nonnegative solutions to a singular nonlinear boundary value problem of second order differential equations with integral boundary conditions in a Banach space. The arguments are based on the construction of a nonempty bounded open convex set and fixed point index theory. Our nonlinearity possesses singularity and first derivative which makes it different with that in [10].
Rongfei Lin
2013-01-01
Full Text Available Newton-Kantorovich and Smale uniform type of convergence theorem of a deformed Newton method having the third-order convergence is established in a Banach space for solving nonlinear equations. The error estimate is determined to demonstrate the efficiency of our approach. The obtained results are illustrated with three examples.
Jiancai Huang
2012-01-01
Full Text Available We introduce an implicit and explicit iterative schemes for a finite family of nonexpansive semigroups with the Meir-Keeler-type contraction in a Banach space. Then we prove the strong convergence for the implicit and explicit iterative schemes. Our results extend and improve some recent ones in literatures.
A type (4) space in (FR)-classification
Argyros, Spiros A; Pelczar-Barwacz, Anna
2012-01-01
We present a reflexive Banach space with an unconditional basis which is quasi-minimal and tight by range, i.e. of type (4) in Ferenczi-Rosendal list within the framework of Gowers' classification program of Banach spaces. The space is an unconditional variant of the Gowers Hereditarily Indecomposable space with asymptotically unconditional basis.
On Well-posed Mutually Nearest and Mutually Furthest Point Problems in Banach Spaces
Chong LI; Ren Xing NI
2004-01-01
Let G be a non-empty closed (resp. bounded closed) boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X. Let K(X) denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance. Moreover, let KG(X) denote the closure of the set {A ∈ K(X): A∩G =φ}. We prove that the set of all A ∈KG(X) (resp. A ∈ K(X)), such that the minimization (resp. maximization) problem min(A, G) (resp. max(A, G)) is well posed, contains a dense Gδ-subset of KG(X) (resp. K(X)), thus extending the recent results due to Blasi, Myjak and Papini and Li.
Models of CT dose profiles in Banach space; with applications to CT Dosimetry
Weir, Victor J
2015-01-01
This paper consists of two parts.In the first part, the scatter components of computed tomograpahy dose profiles are modeled using various functions including the solution to Riccati's differential equation. These scatter functions are combined with primary components such as a trapezoidal function and a constructed function that uses the analytic continuation of Heaviside step function. A mathematical theory is developed in Banach space. The modeled function, which is the product of the scatter and primary functions, is used to accurately fit data from the O-arm cone beam imaging system. In a second part of the paper, an approach to dosimtery is developed that shows that the results obtained from the use of a pencil shaped ion chamber is equivalent to that from a farmer chamber. This result is verified by presenting some preliminary experimental data measured in a 64 slice Siemens Sensation scanner.
A Banach space-valued ergodic theorem for amenable groups and applications
Pogorzelski, Felix
2012-01-01
In this paper we study unimodular amenable groups. The first part is devoted to results on the existence of uniform families of quasi tilings for these groups. In light of that, constructions of Ornstein and Weiss are extended by quantitative estimates for the covering properties of the corresponding decompositions. Afterwards, we apply the developed methods to obtain an abstract ergodic theorem for a class of functions mapping subsets of the group into some Banach space. Moreover, applications of this convergence result are studied: the uniform existence of the integrated density of states (IDS) for operators on amenable Cayley graphs; the uniform existence of the IDS for operators on discrete structures being quasi-isometric to some amenable group; the approximation of L2-Betti numbers on cellular CW-complexes; the existence of certain densities of clusters in a percolated Cayley graph.
Suppose E is a real uniformly smooth Banach space and K is a nonempty closed convex and bounded subset of E, T:K → K is a Lipschitz pseudo-contraction. It is proved that the Picard iterates of a suitably defined operator converges strongly to the unique fixed point of T. Furthermore, this result also holds for the slightly larger class of Lipschitz strong hemi-contractions. Related results deal with strong convergence of the Picard iterates to the unique solution of operator equations involving Lipschitz strongly accretive maps. Apart from establishing strong convergence, our theorems give existence, uniqueness and convergence-rate which is at least as fast as a geometric progression. (author). 51 refs
Hybrid methods for accretive variational inequalities involving pseudocontractions in Banach spaces
Chen Rudong
2011-01-01
Full Text Available Abstract We use strongly pseudocontractions to regularize a class of accretive variational inequalities in Banach spaces, where the accretive operators are complements of pseudocontractions and the solutions are sought in the set of fixed points of another pseudocontraction. In this paper, we consider an implicit scheme that can be used to find a solution of a class of accretive variational inequalities. Our results improve and generalize some recent results of Yao et al. (Fixed Point Theory Appl, doi:10.1155/2011/180534, 2011 and Lu et al. (Nonlinear Anal, 71(3-4, 1032-1041, 2009. 2000 Mathematics subject classification 47H05; 47H09; 65J15
Improved bounds in the scaled Enflo type inequality for Banach spaces
Giladi, Ohad
2010-01-01
It is shown that if (X,||.||_X) is a Banach space with Rademacher type p \\ge 1, then for every integer n there exists an even integer m X, \\Avg_{x,\\e}[||f(x+ m\\e/2)-f(x)}||_X^p] < C(p,X) m^p\\sum_{j=1}^n\\Avg_x[||f(x+e_j)-f(x)||_X^p], where the expectation is with respect to uniformly chosen x \\in Z_m^n and \\e \\in \\{-1,1\\}^n, and C(p,X) is a constant that depends on p and the Rademacher type constant of X. This improves a bound of m < Cn^{3-2/p} that was obtained in [Mendel, Naor 2007]. The proof is based on an augmentation of the "smoothing and approximation" scheme, which was implicit in [Mendel, Naor 2007].
Song, Guohui; Zhang, Haizhang
2011-01-01
A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically. Following this direction, we illustrate how reproducing kernel Banach spaces with the l1 norm can be applied to improve the learn...
Perturbations of Bessel Sequences of Order p in a Banach Space%Banach空间中P阶Bessel列的扰动
王秋芬; 曹怀信; 武海辉
2011-01-01
应用算子论方法研究Banach空间X中p(1＜P＜∞)阶Bessel列的扰动问题,对X中的任一p阶Bessel列f={fi}i∈I,定义了有界线性算子Tf:X*→lp,建立了从全体p阶Bessel列组成的Banach空间BpX(Ⅰ)到算子空间B(X*,lp)上的等距线性同构α:f→Tf,并给出了p阶Bessel列的扰动定理.%With the aid of operator theory, some perturbations of Bessel sequences of order p in a Banach space X were discussed. For a Bessel sequence f = {fi} i∈I of order p in X, we defined a bounded linear operator Tf from X* into lp and established a linear isometry isomorphism a from the space BpX (I) of all the Bessel sequences of order p in a Banach space X into the operator space B(X*,lp) defined by α(f) = Tr. In the light of operator theory, some results on perturbations of Bessel sequences of order p were given, which are helpful to the study of frames of order p for a Banach space.
Quasi-geostrophic equations with initial data in Banach spaces of local measures
Sadek Gala
2005-06-01
Full Text Available This paper studies the well posedness of the initial value problem for the quasi-geostrophic type equations $$displaylines{ partial_{t}heta+u ablaheta+( -Delta ^{gamma}heta =0 quad hbox{on }mathbb{R}^{d}imes] 0,+infty[cr heta( x,0 =heta_{0}(x, quad xinmathbb{R}^{d} }$$ where 0 less than $gammaleq 1$ is a fixed parameter and the velocity field $u=(u_{1},u_{2},dots,u_{d} $ is divergence free; i.e., $ abla u=0$. The initial data $heta_{0}$ is taken in Banach spaces of local measures (see text for the definition, such as Multipliers, Lorentz and Morrey-Campanato spaces. We will focus on the subcritical case 1/2 less than $gammaleq1$ and we analyse the well-posedness of the system in three basic spaces: $L^{d/r,infty}$, $dot {X}_{r}$ and $dot {M}^{p,d/r}$, when the solution is global for sufficiently small initial data. Furtheremore, we prove that the solution is actually smooth. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions.
Sokhuma, K.; A. Kaewkhao
2010-01-01
Let be a nonempty compact convex subset of a uniformly convex Banach space , and let and be a single-valued nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume in addition that and for all . We prove that the sequence of the modified Ishikawa iteration method generated from an arbitrary by , , where and , are sequences of positive numbers satisfying , , converges strongly to a common fixed point of and ; that is, there exists such tha...
M. Eshaghi Gordji
2009-01-01
Full Text Available We establish the general solution of the functional equation f(nx+y+f(nx−y=n2f(x+y+n2f(x−y+2(f(nx−n2f(x−2(n2−1f(y for fixed integers n with n≠0,±1 and investigate the generalized Hyers-Ulam stability of this equation in quasi-Banach spaces.
Vara Prasad KNVV
2006-01-01
Full Text Available Let be an arbitrary real Banach space and a nonempty, closed, convex (not necessarily bounded subset of . If is a member of the class of Lipschitz, strongly pseudocontractive maps with Lipschitz constant , then it is shown that to each Mann iteration there is a Krasnosleskij iteration which converges faster than the Mann iteration. It is also shown that the Mann iteration converges faster than the Ishikawa iteration to the fixed point of .
Let E be a real uniformly convex Banach space whose dual space E* satisfies the Kadec- Klee property, K be a closed convex nonempty subset of E . Let T1, T2, . . . , Tm : K → K be asymptotically nonexpansive mappings of K into E with sequences (respectively) {kin}n=1∞ satisfying kin → 1 as n → ∞, i = 1, 2 , ...,m and Σn=1∞(kin - 1) in}n=1∞ be a sequence in [ε, 1 - ε ], for each i element of { 1, 2 , . . . ,m} (respectively). Let {xn} be a sequence generated for m ≥ 2 by, x1 element of K, xn+1 = (1 - α1n)xn + α1nT1nyn+m-2, yn+m-2 = (1 - α2n)xn + α2nT2nyn+m-3, ..., yn = (1 - αmn)xn + αmnTmnxn , n ≥ 1. Let Intersectioni=1m F (Ti) ≠ 0 . Then, {xn} converges weakly to a common fixed point of the family {Ti}i=1m. Under some appropriate condition on the family {Ti}i=1m, a strong convergence theorem is also roved. (author)
Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces
Austin, Tim; Tessera, Romain
2010-01-01
Let $\\H$ denote the discrete Heisenberg group, equipped with a word metric $d_W$ associated to some finite symmetric generating set. We show that if $(X,\\|\\cdot\\|)$ is a $p$-convex Banach space then for any Lipschitz function $f:\\H\\to X$ there exist $x,y\\in \\H$ with $d_W(x,y)$ arbitrarily large and \\begin{equation}\\label{eq:comp abs} \\frac{\\|f(x)-f(y)\\|}{d_W(x,y)}\\lesssim \\left(\\frac{\\log\\log d_W(x,y)}{\\log d_W(x,y)}\\right)^{1/p}. \\end{equation} We also show that any embedding into $X$ of a ball of radius $R\\ge 4$ in $\\H$ incurs bi-Lipschitz distortion that grows at least as a constant multiple of \\begin{equation}\\label{eq:dist abs} \\left(\\frac{\\log R}{\\log\\log R}\\right)^{1/p}. \\end{equation} Both~\\eqref{eq:comp abs} and~\\eqref{eq:dist abs} are sharp up to the iterated logarithm terms. When $X$ is Hilbert space we obtain a representation-theoretic proof yielding bounds corresponding to~\\eqref{eq:comp abs} and~\\eqref{eq:dist abs} which are sharp up to a universal constant.
Lp-DICHOTOMY OF LINEAR DIFFERENTIAL EQUATIONS IN AN ARBITRARY BANACH SPACE
A. KOSSEVA; S. KOSTADINOV; K. SCHNEIDER
2003-01-01
The notion of Lp-dichotomy for linear differential equations with possibly unbounded oper-ator is introduced. By help of Banach fixed point theorem sufficient conditions for the existenceof bounded solutions of nonlinear differential equations with an Lp-dichotomous linear part areobtained.
Li Shan LIU
2001-01-01
In this paper, we will prove that Ky Fan's Theorem (Math. Z. 112(1969), 234-240) is true for1-set-contractive maps defined on a bounded closed convex subset K in a Banach space with intK ≠φ.This class of 1-set-contractive maps includes condensing maps, nonexpansive maps, semicontractivemaps, LANE maps and others. As applications of our theorems, some fixed point theorems of non-self-maps are proved under various well-known boundary conditions. Our results are generalizations andimprovements of the recent results obtained by many authors.
K. N. V. V. Vara Prasad
2007-01-01
Full Text Available Let E be an arbitrary real Banach space and K a nonempty, closed, convex (not necessarily bounded subset of E. If T is a member of the class of Lipschitz, strongly pseudocontractive maps with Lipschitz constant LÃ¢Â‰Â¥1, then it is shown that to each Mann iteration there is a Krasnosleskij iteration which converges faster than the Mann iteration. It is also shown that the Mann iteration converges faster than the Ishikawa iteration to the fixed point of T.
Ring homomorphisms on real Banach algebras
Sin-Ei Takahasi
2003-08-01
Full Text Available Let B be a strictly real commutative real Banach algebra with the carrier space ÃŽÂ¦B. If A is a commutative real Banach algebra, then we give a representation of a ring homomorphism ÃÂ:AÃ¢Â†Â’B, which needs not be linear nor continuous. If A is a commutative complex Banach algebra, then ÃÂ(A is contained in the radical of B.
In this paper we investigate the relation between weak convergence of a sequence {μn} of probability measures on a Polish space S converging weakly to the probability measure μ and continuous, norm-bounded functions into a Banach space X. We show that, given a norm-bounded continuous function f:S→X, it follows that limn∞ ∫Sf, dμn = ∫Sf, dμ —the limit one has for bounded and continuous real (or complex)—valued functions on S. This result is then applied to the stability theory of Feynman’s operational calculus where it is shown that the theory can be significantly improved over previous results.
D. A. Robbins
1994-12-01
Full Text Available We study bundles of Banach algebras ÃÂ€:AÃ¢Â†Â’X, where each fiber Ax=ÃÂ€Ã¢ÂˆÂ’1({x} is a Banach algebra and X is a compact Hausdorff space. In the case where all fibers are commutative, we investigate how the Gelfand representation of the section space algebra ÃŽÂ“(ÃÂ€ relates to the Gelfand representation of the fibers. In the general case, we investigate how adjoining an identity to the bundle ÃÂ€:AÃ¢Â†Â’X relates to the standard adjunction of identities to the fibers.
Boubakari Ibrahimou
2013-01-01
maximal monotone with and . Using the topological degree theory developed by Kartsatos and Quarcoo we study the eigenvalue problem where the operator is a single-valued of class . The existence of continuous branches of eigenvectors of infinite length then could be easily extended to the case where the operator is multivalued and is investigated.
Banach空间范数的 k-点态粗性和 k-粗性%k- pointwise roughness and k- roughness on Banach spaces
义德日胡; 苏雅拉图
2012-01-01
对 Banach 空间范数引入了 k-点态粗和 k-粗的概念，利用 Banach 空间理论的方法，给出了 x ∈ S(X)为范数的 k-粗糙点和 X 的范数是 k-粗的一些充分必要条件，证明了(k +1)-粗糙点是 k-粗糙点以及 k-粗糙点与 Fr´echet 可微性的一些结果。特别地，在 k =1的情形下蕴含了关于范数的粗糙点、点态粗范数和粗范数的相应结果% In this paper, the k- pointwise rough and k- rough norm on Banach space are introduced. By the method of Banach space theory, the some necessary and suﬃcient conditions of k- rough point of norm and k-rough norm of X are given respectively. We proved that (k + 1)- rough point is k- rough point and obtained some results related with k- rough point and Fr´echet differentiability. In particular, when k = 1 our results contains the results of about rough point of norm, pointwise rough norm and rough norm.
Metatext Phenomenon: Mode of Irony in Reflexive-Interpretative Space
Kuznetsova, Anna V.
2013-01-01
The article presents metatext and correlative interaction of irony in the reflexive-interpretative space of the literary text, clarifies the status of metatext space as linguo-cognitive activity of text producer. Irony serves as a system factor, forming metatext within the linguo-rhetoric script of text. Context plays special role in singling out of linguistic and discursive irony; multilevel character of irony is determined by context breadth, required for its decoding. The integrated nature...
Modification of Otolith Reflex Asymmetries Following Space Flight
Clarke, Andrew H.; Schoenfeld, Uwe; Wood, Scott J.
2011-01-01
We hypothesize that changes in otolith-mediated reflexes adapted for microgravity contribute to perceptual, gaze and postural disturbances upon return to Earth s gravity. Our goal was to determine pre- versus post-fight differences in unilateral otolith reflexes that reflect these adaptive changes. This study represents the first comprehensive examination of unilateral otolith function following space flight. Ten astronauts participated in unilateral otolith function tests three times pre-flight and up to four times after Shuttle flights from landing day through the subsequent 10 days. During unilateral centrifugation (UC, +/- 3.5cm at 400deg/s), utricular function was examined by the perceptual changes reflected by the subjective visual vertical (SVV) and by video-oculographic measurement of the otolith-mediated ocular counter-roll (OOR). Unilateral saccular reflexes were recorded by measurement of collic Vestibular Evoked Myogenic Potential (cVEMP). Although data from a few subjects were not obtained early post-flight, a general increase in asymmetry of otolith responses was observed on landing day relative to pre-flight baseline, with a subsequent reversal in asymmetry within 2-3 days. Recovery to baseline levels was achieved within 10 days. This fluctuation in the asymmetry measures appeared strongest for SVV, in a consistent direction for OOR, and in an opposite direction for cVEMP. These results are consistent with our hypothesis that space flight results in adaptive changes in central nervous system processing of otolith input. Adaptation to microgravity may reveal asymmetries in otolith function upon to return to Earth that were not detected prior to the flight due to compensatory mechanisms.
Some Extensions of Banach's Contraction Principle in Complete Cone Metric Spaces
Raja P
2008-01-01
Full Text Available Abstract In this paper we consider complete cone metric spaces. We generalize some definitions such as -nonexpansive and -uniformly locally contractive functions -closure, -isometric in cone metric spaces, and certain fixed point theorems will be proved in those spaces. Among other results, we prove some interesting applications for the fixed point theorems in cone metric spaces.
Networks for the weak topology of Banach and Fréchet spaces
Gabriyelyan, S.; Kąkol, Jerzy; Kubiś, Wieslaw; Marciszewski, W.
2015-01-01
Roč. 432, č. 2 (2015), s. 1183-1199. ISSN 0022-247X R&D Projects: GA ČR(CZ) GA14-07880S Institutional support: RVO:67985840 Keywords : Fréchet space * space of continuous functions * locally convex space Subject RIV: BA - General Mathematics Impact factor: 1.120, year: 2014 http://www.sciencedirect.com/science/article/pii/S0022247X15006836
One-sided interpolation of injective tensor products of Banach spaces
Michels, Carsten
2007-01-01
The Kouba formula for the complex interpolation of injective tensor products requires all spaces involved to be at least of cotype $2$. We show that this can be weakened when one side of the tensor products is fixed.
CONTINUITY AND LINEARITY OF ADDITIVE DERIVATIONS OF NEST ALGEBRAS ON BANACH SPACES
HANDEGUANG
1996-01-01
This paper discusses the problem concerning the continuity and linearity of additive derivation of nest algebras on normed spaces. It is proved that erevy linear derivation of a nest algebra algN is continuous provided that one of the following conditions is satisfied:(1)0+（包含）0.(2)X-（包含于）X.(3)there exists a non-trivial idempotnet p in algN such that the range of p belongs to N, It is also proved that every additive derivation of a nest algebra is automatically linear if the underlying normed space is infinite dimemnsional.
Variational Formulas of Poincaré-Type Inequalities in Banach Spaces of Functions on the Line
Mu Fa CHEN
2002-01-01
Motivated from the study on logarithmic Sobolev, Nash and other functional inequalities,spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimatesfor the optimal constants in the inequalities are presented. As a typical application, the logarithmicSobolev constant is carefully examined.
New Fixed Point Results with PPF Dependence in Banach Spaces Endowed with a Graph
Hussain, N.; S. Khaleghizadeh; P. Salimi; Akbar, F.
2013-01-01
We introduce the concept of an ${\\alpha }_{c}$ -admissible non-self-mappings with respect to ${\\eta }_{c}$ and establish the existence of PPF dependent fixed and coincidence point theorems for ${\\alpha }_{c}{\\eta }_{c}$ - $\\psi $ -contractive non-self-mappings in the Razumikhin class. As applications of our PPF dependent fixed point and coincidence point theorems, we derive some new fixed and coincidence point results for $\\psi $ -contractions whenever the range space is endowed with a graph ...
Botelho, Fabio
2014-01-01
This book introduces the basic concepts of real and functional analysis. It presents the fundamentals of the calculus of variations, convex analysis, duality, and optimization that are necessary to develop applications to physics and engineering problems. The book includes introductory and advanced concepts in measure and integration, as well as an introduction to Sobolev spaces. The problems presented are nonlinear, with non-convex variational formulation. Notably, the primal global minima may not be attained in some situations, in which cases the solution of the dual problem corresponds to an appropriate weak cluster point of minimizing sequences for the primal one. Indeed, the dual approach more readily facilitates numerical computations for some of the selected models. While intended primarily for applied mathematicians, the text will also be of interest to engineers, physicists, and other researchers in related fields.
$\\sigma $ -Approximately Contractible Banach Algebras
Momeni, M; Yazdanpanah, T.; Mardanbeigi, M. R.
2012-01-01
We investigate $\\sigma $ -approximate contractibility and $\\sigma $ -approximate amenability of Banach algebras, which are extensions of usual notions of contractibility and amenability, respectively, where $\\sigma $ is a dense range or an idempotent bounded endomorphism of the corresponding Banach algebra.
娄健; 何欣枫; 何震
2009-01-01
In this paper,we introduce and study a new system of variational inclusions involving (H,η)-monotone operators in Banach space.Using the resolvent operator associated with (H,η)-monotone operators,we prove the existence and uniqueness of solutions for this new system of variational inclusions.We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the iterative sequence generated by the algorithm.
Fréchet differential of a power series in Banach algebras
Benedetto Silvestri
2010-01-01
Full Text Available We present two new forms in which the Fréchet differential of a power series in a unitary Banach algebra can be expressed in terms of absolutely convergent series involving the commutant \\(C(T : A \\mapsto [A,T]\\. Then we apply the results to study series of vector-valued functions on domains in Banach spaces and to the analytic functional calculus in a complex Banach space.
Nonlinear potentials in function spaces
Rao, Murali; Vondraček, Zoran
2002-01-01
We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capa...
An inner product for a Banach-algebra
An inner product is defined on a commutative Banach algebra with an essential involution and the resultant inner product space is shown to be a topological algebra. Several conditions for its completeness are established and moreover, a decomposition theorem is proved. It is shown that every commutative Banach algebra with an essential involution has an auxiliary norm which turns it into an A*-algebra. (author). 6 refs
A Generation Condition of Integrated Bisemigroup on Banach Space%Banach空间积分双半群的生成条件
许跟起; 邵琛
2002-01-01
研究Banach空间中积分双半群的生成条件. 利用算子A的豫解算子,给出了积分双半群T(t)的生成定理. 结果表明:如果对任意的x∈X,f∈X*,以及()|λ|≤δ,λ∈ρ(A),有∈Lp(R),则存在算子族S(t), t∈R,S(t)强连续且满足积分双半群的定义.%The generation condition of integrated bisemigroup in Banach space is studied in this paper. Using the resolvent of operator A, a generation theorem of integrated bisemigroup T(t) is proved. The result is showed that if ()|λ|≤δ,λ∈ρ(A), and satisfy 〈∈Lp(R), for all x∈X, f∈X*, then there exists a family operators S(t), t∈R, which is strongly continuous and satisfies the definition of integrated bisemigroup.
Banach Algebras Associated to Lax Pairs
Glazebrook, James F.
2015-04-01
Lax pairs featuring in the theory of integrable systems are known to be constructed from a commutative algebra of formal pseudodifferential operators known as the Burchnall- Chaundy algebra. Such pairs induce the well known KP flows on a restricted infinite-dimensional Grassmannian. The latter can be exhibited as a Banach homogeneous space constructed from a Banach *-algebra. It is shown that this commutative algebra of operators generating Lax pairs can be associated with a commutative C*-subalgebra in the C*-norm completion of the *-algebra. In relationship to the Bose-Fermi correspondence and the theory of vertex operators, this C*-algebra has an association with the CAR algebra of operators as represented on Fermionic Fock space by the Gelfand-Naimark-Segal construction. Instrumental is the Plücker embedding of the restricted Grassmannian into the projective space of the associated Hilbert space. The related Baker and tau-functions provide a connection between these two C*-algebras, following which their respective state spaces and Jordan-Lie-Banach algebras structures can be compared.
Spectral synthesis for Banach Algebras II
Feinstein, J. F.; Somerset, D. W. B.
1999-01-01
This paper continues the study of spectral synthesis and the topologies $\\tau_{\\infty}$ and $\\tau_r$ on the ideal space of a Banach algebra, concentrating particularly on the class of Haagerup tensor products of C$^*$-algebras. For this class, it is shown that spectral synthesis is equivalent to the Hausdorffness of $\\tau_{\\infty}$. Under a weak extra condition, spectral synthesis is shown to be equivalent to the Hausdorffness of $\\tau_r$.
Fundamental aspects of vector-valued Banach limits
Garcia-Pacheco, F. J.; Perez-Fernandez, F. J.
2016-04-01
This paper is divided into four parts. In the first we study the existence of vector-valued Banach limits and show that a real Banach space with a monotone Schauder basis admits vector-valued Banach limits if and only if it is 1-complemented in its bidual. In the second we prove two vector-valued versions of Lorentz' intrinsic characterization of almost convergence. In the third we show that the unit sphere in the space of all continuous linear operators from \\ell_∞(X) to X which are invariant under the shift operator on \\ell_∞(X) cannot be obtained via compositions of surjective linear isometries with vector-valued Banach limits. In the final part we show that if X enjoys the Krein-Milman property, then the set of vector-valued Banach limits is a face of the unit ball in the space of all continuous linear operators from \\ell_∞(X) to X which are invariant under the shift operator on \\ell_∞(X).
From a special class of systems has been used the linear homogeneous differential equations with impulse effect in Minkowski space field theory with time dependent boundary conditions, i.e. those of moving mirrors. The field theoretical approach for studing the properties of the vacuum starts from an analysis of the behaviour of local field quantities in Minkowski space with uniformly moving mirrors. For the impulsive moving mirror model is the real process of interaction between the quantum field and the external mirror a subject to disturbances in its evolution acting in time very short compared with the entire duration of the process. The stability of the process in the stability of the vacuum state energy. 7 refs
Zhang Peiguo
2011-01-01
Full Text Available Abstract By obtaining intervals of the parameter λ, this article investigates the existence of a positive solution for a class of nonlinear boundary value problems of second-order differential equations with integral boundary conditions in abstract spaces. The arguments are based upon a specially constructed cone and the fixed point theory in cone for a strict set contraction operator. MSC: 34B15; 34B16.
2008-01-01
Let (Ω,A,μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S,X) a random normed space over K with base (Ω,A,μ). Denote the support of (S,X) by E, namely E is the essential supremum of the set {A ∈ A : there exists an element p in S such that Xp(ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit ball S*(1) = {f ∈ S* : Xf* 1} of the random conjugate space (S*,X*) of (S,X) is compact under the random weak star topology on (S*,X*) iff E∩A=: {E∩A | A ∈ A} is essentially purely μ-atomic (namely, there exists a disjoint family {An : n ∈ N} of at most countably many μ-atoms from E∩A such that E =∪n∞=1 An and for each element F in E∩A, there is an H in the σ-algebra generated by {An : n ∈ N} satisfying μ(F △H) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding classical case. Further, Banach-Bourbaki-Kakutani-mulian (briefly, BBKS) theorem in a complete random normed module is established as follows: If (S,X) is a complete random normed module, then the random closed unit ball S(1) = {p ∈ S : Xp 1} of (S,X) is compact under the random weak topology on (S,X) iff both (S,X) is random reflexive and E∩A is essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James theorem and
GUO TieXin
2008-01-01
Let(Ω,A,μ)be a probability space,K the scalar field R of real numbers or C of complex numbers,and(S,X)a random normed space over K with base(Ω,A,μ).Denote the support of(S,X)by E,namely E is the essentiaI supremum of the ser {A∈A:there exists an element p in S such that Xp(ω)＞0 for almost all ω in A}. In this paper,Banach-Alaoglu theorem in a random normed space is first established as follows:The random closed unit ball S*(1)={f∈S*:X*f≤1}of the random conjugate space(S*,x*)of(S,X)is compact under the random weak star topology on (S*,X*)iff E∩A=:{E∩A|A∈A}is essentially purely μ-atomic(namely,there exists a disjoint family{An:n∈N}of at most countably many μ-atoms from E∩A such that E=∪∞n=1 An and for each element F in E∩A,there is an H in the σ-algebra generated by{An:n∈N)satisfying μ(F△H)=0),whose proof forces us to provide a key topological skill,and thus is much more involved than the corresponding classical case.Further,Banach-Bourbaki-Kakutani-Smulian(briefly,BBKS)theorem in a complete random normed module is established as follows:If(S,X)is a complete random normed module,then the random closed unit ball S(1)={p∈S:Xp≤1}of(S,X)is compact under the random weak topology on(S,X)iff both(S,X)is random reflexive and E∩A is essentially purely μ-atomic.Our recent work shows that the famous classical James theorem still holds for an arbitrary complete random normed module,namely a complete random normed module is random reflexive iff the random norm of an arbitrary almost surely bounded random linear functional on it is attainable on its random closed unit ball,but this paper shows that the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they possess extremely simple stratification structure,namely their supports are essentially purely μ-atomic.Combining the James theorem and BBKS theorem in complete random normed modules leads directly to an interesting
Moduli and Characteristics of Monotonicity in Some Banach Lattices
Foralewski, P.; Hudzik, H.; Kaczmarek, R.; Krbec, Miroslav
-, - (2010), s. 852346. ISSN 1687-1812 R&D Projects: GA AV ČR IAA100190804; GA MŠk LC06052 Institutional research plan: CEZ:AV0Z10190503 Keywords : Banach lattice * characteristics of monotonicity * Orlicz function space * Orlicz sequence space Subject RIV: BA - General Mathematics http://www. fixed pointtheoryandapplications.com/content/2010/1/852346
Derivations into Duals of Ideals of Banach Algebras
M E Gorgi; T Yazdanpanah
2004-11-01
We introduce two notions of amenability for a Banach algebra $\\mathcal{A}$. Let be a closed two-sided ideal in $\\mathcal{A}$, we say $\\mathcal{A}$ is -weakly amenable if the first cohomology group of $\\mathcal{A}$ with coefficients in the dual space * is zero; i.e., $H^1(\\mathcal{A},I^*) =\\{0\\}$, and, $\\mathcal{A}$ is ideally amenable if $\\mathcal{A}$ is -weakly amenable for every closed two-sided ideal in $\\mathcal{A}$. We relate these concepts to weak amenability of Banach algebras. We also show that ideal amenability is different from amenability and weak amenability. We study the -weak amenability of a Banach algebra $\\mathcal{A}$ for some special closed two-sided ideal .
张育梅; 程毅; 王靖华
2012-01-01
利用同伦方法证明了一类发展方程反周期解在Banach空间中的存在性和唯一性.先构造同伦方程,再对方程做先验估计.最后通过定义解算子,运用拓扑度方法,给出了发展方程反周期解存在的充分条件.%We proved the existence and uniqueness of a class of evolution equation in Banach space using homotopy methods. We constructed a homotopy equation and made α priori, estimate to this equation. Under the definition solution operator conditions, a sufficient condition of existence of solutions was obtained by means of topological degree method.
赵吕慧子; 孙经先
2011-01-01
The definition of a class of new operators, convex-power 1-set-contraction operators in Banach spaces is giv en , and the existence of fixed points of this class of operators is studied. By using methods of approximation by opera tors, the fixed point theorems of Rothe and Altman type convex-power 1-set-contraction operators is obtained, which generalize fixed point theorems of 1-set-contraction operators.%在Banach空间中给出了一类新算子——凸幂1集压缩算子的定义,研究了这类新算子不动点的存在性问题,利用算子逼近的方法,获得了Rothe及Altman型凸幂1集压缩算子的不动点定理,推广了1集压缩算子的不动点定理.
Stability of functional equations in Banach algebras
Cho, Yeol Je; Rassias, Themistocles M; Saadati, Reza
2015-01-01
Some of the most recent and significant results on homomorphisms and derivations in Banach algebras, quasi-Banach algebras, C*-algebras, C*-ternary algebras, non-Archimedean Banach algebras and multi-normed algebras are presented in this book. A brief introduction for functional equations and their stability is provided with historical remarks. Since the homomorphisms and derivations in Banach algebras are additive and R-linear or C-linear, the stability problems for additive functional equations and additive mappings are studied in detail. The latest results are discussed and examined in stability theory for new functional equations and functional inequalities in Banach algebras and C*-algebras, non-Archimedean Banach algebras, non-Archimedean C*-algebras, multi-Banach algebras and multi-C*-algebras. Graduate students with an understanding of operator theory, functional analysis, functional equations and analytic inequalities will find this book useful for furthering their understanding and discovering the l...
Abbas NAJATI
2009-01-01
In this paper, we prove the generalized Hyers-Ulam stability of homomorphisms in quasi-Banach algebras associated with the following Pexiderized Jensen functional equation f(x+y/2+z)-g(x-y/2+z)=h(y).This is applied to investigating homomorphisms between quasi-Banach algebras. The concept of the generalized Hyers-Ulam stability originated from Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soe., 72, 297-300 (1978).
Several complex variables and Banach algebras
This paper aims to present certain applications of the theory of holomorphic functions of several complex variables to the study of commutative Banach algebras. The material falls into the following sections: (A) Introcution to Banach algebras (this will not presuppose any knowledge of the subject); (B) Groups of differential forms (mainly concerned with setting up a useful language); (C) Polynomially convex domains. (D) Holomorphic functional calculus for Banach algebras; (E) Some applications of the functional calculus. (author)
Approximately -Jordan Homomorphisms on Banach Algebras
Karimi T
2009-01-01
Full Text Available Let , and let be two rings. An additive map is called -Jordan homomorphism if for all . In this paper, we establish the Hyers-Ulam-Rassias stability of -Jordan homomorphisms on Banach algebras. Also we show that (a to each approximate 3-Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique 3-ring homomorphism near to , (b to each approximate -Jordan homomorphism between two commutative Banach algebras there corresponds a unique -ring homomorphism near to for all .
Erratum to: “Polynomial algebras on classical Banach spaces”
D'Alessandro, Stefania; Hájek, Petr Pavel; Johanis, M.
2015-01-01
Roč. 207, č. 2 (2015), s. 1003-1012. ISSN 0021-2172 R&D Projects: GA ČR(CZ) GAP201/11/0345; GA MŠk(CZ) 7AMB12FR003 Institutional support: RVO:67985840 Keywords : Banach space * polynomial algebra Subject RIV: BA - General Mathematics Impact factor: 0.787, year: 2014 http://link.springer.com/article/10.1007%2Fs11856-015-1155-y
Segal algebras in commutative Banach algebras
INOUE, Jyunji; TAKAHASI, Sin-Ei
2014-01-01
The notion of Reiter's Segal algebra in commutative group algebras is generalized to a notion of Segal algebra in more general classes of commutative Banach algebras. Then we introduce a family of Segal algebras in commutative Banach algebras under considerations and study some properties of them.
Reflexive Aero Structures for Enhanced Survivability Project
National Aeronautics and Space Administration — Cornerstone Research Group Inc. (CRG) proposes to develop an advanced reflexive structure system to increase the survivability of aerostructures. This reflexive...
刘玉波; 张凤敏; 苟长义
2009-01-01
首先证明了取值于Banach空间上强连续的向量值函数是可积分的;然后用初等方法证明了向量值函数的柯西积分公式和高阶导数公式;最后讨论了取值于lp(p≥1)空间上的向量值函数解析、可积、柯西积分公式和高阶导数公式与其各分量函数的关系和表示形式,并且给出了有限维赋范线性空间上的向量值函数连续、解析、可积、柯西定理、柯西积分公式和高阶导数公式与其各分量函数的关系和表示形式.%The strongly continuous vector valued functions in Banach space are proved integrable. The Cauchy integral for-mula and higher.derivative formula of vector-valued functions are proved using the elementary method. The relationship be-tween vector valued functions in the space of lp (p≥1) and their componential functions about analyticity, integrability, the Cauchy integral formula and higher derivative formula are discussed. And the relationship between vector valued functions in finite dimensional normed linear space and their componential functions about continuity, analyticity, integrability, Cauchys theorem, the Cauchy integral formula and higher derivative formula are also given.
The Banach-Mazur-Schmidt game and the Banach-Mazur-McMullen game
Fishman, Lior; Reams, Vanessa; Simmons, David
2015-01-01
We introduce two new mathematical games, the Banach-Mazur-Schmidt game and the Banach-Mazur-McMullen game, merging well-known games. We investigate the properties of the games, as well as providing an application to Diophantine approximation theory, analyzing the geometric structure of certain Diophantine sets.
Aileen Collier
2016-05-01
Full Text Available Background: There is international consensus of the need for improved palliative and end-of-life care in hospital settings. What is less clear is how such improvements might be realised in practice. Research and practice improvement methodologies need to acknowledge the relational, spiritual, moral and ethical as well as physical dimensions of death and dying if improvements in care are to be achieved. Aims and objectives: The aim of this article is to explore the potential of video-reflexive ethnography as a practice development methodology to improve care of people with a life-limiting illness in the hospital setting. Methods: The study used video-reflexive ethnography and was underpinned by an indigenous research ethical framework. Findings: Study findings highlight the potential of video-reflexive ethnography as a practice development methodology. The reach of video extended internally and externally beyond immediate practice research sites to make hospital dying tangible. The research acted as a disruptive innovation, foregrounding peoples’ (patients and families expertise as well as that of healthcare workers. For some patient and family participants, the research offered a visual legacy. Conclusions: The theories underpinning video-reflexive ethnography and practice development are closely aligned; the former has potential as a practice development methodology to promote person-centred palliative and end-of-life care. The underpinning philosophical, ethical and values framework through which it is applied, along with the skills and aptitude of facilitation, are critical if its potential is to be realised. Implications for practice development: The delivery of person-centred end-of-life care may be facilitated by: Healthcare workers seeing themselves and those they care for differently Healthcare organisations seeing their employees as well as patients and families differently Researchers also being prepared to see themselves differently
An elementary approach to a lattice-valued Banach-Stone Theorem
Jinxi Chen
2007-01-01
Let $ X $ and $ Y $ be compact Hausdorff spaces, and $ E $ be a nonzero real Banach lattice. In this note, we give an elementary proof of a lattice-valued Banach-Stone theorem by Cao, Reilly and Xiong [3] which asserts that if there exists a Riesz isomorphism $ \\Phi: C(X,E)\\rightarrow C(Y,\\mathbb{R}) $ such that $ \\Phi(f) $ has no zeros if $ f $ has none, then $ X $ is homeomorphic to $ Y $ and $ E $ is Riesz isomorphic to $ \\mathbb{R} $.
Peterka, Robert J.
1993-01-01
Recent studies have identified significant correlations between space motion sickness susceptibility and measures of disconjugate torsional eye movements recorded during parabolic flights. These results support an earlier proposal which hypothesized that an asymmetry of otolith function between the two ears is the cause of space motion sickness. It may be possible to devise experiments that can be performed in the 1 g environment on earth that could identify and quantify the presence of asymmetric otolith function. This paper summarizes the known physiological and anatomical properties of the otolith organs and the properties of the torsional vestibulo-ocular reflex which are relevant to the design of a stimulus to identify otolith asymmetries. A specific stimulus which takes advantage of these properties is proposed.
Wood, Scott J.; Clarke, A. H.; Rupert, A. H.; Harm, D. L.; Clement, G. R.
2009-01-01
Two joint ESA-NASA studies are examining changes in otolith-ocular reflexes and motion perception following short duration space flights, and the operational implications of post-flight tilt-translation ambiguity for manual control performance. Vibrotactile feedback of tilt orientation is also being evaluated as a countermeasure to improve performance during a closed-loop nulling task. Data is currently being collected on astronaut subjects during 3 preflight sessions and during the first 8 days after Shuttle landings. Variable radius centrifugation is utilized to elicit otolith reflexes in the lateral plane without concordant roll canal cues. Unilateral centrifugation (400 deg/s, 3.5 cm radius) stimulates one otolith positioned off-axis while the opposite side is centered over the axis of rotation. During this paradigm, roll-tilt perception is measured using a subjective visual vertical task and ocular counter-rolling is obtained using binocular video-oculography. During a second paradigm (216 deg/s, less than 20 cm radius), the effects of stimulus frequency (0.15 - 0.6 Hz) are examined on eye movements and motion perception. A closed-loop nulling task is also performed with and without vibrotactile display feedback of chair radial position. Data collection is currently ongoing. Results to date suggest there is a trend for perceived tilt and translation amplitudes to be increased at the low and medium frequencies on landing day compared to pre-flight. Manual control performance is improved with vibrotactile feedback. One result of this study will be to characterize the variability (gain, asymmetry) in both otolith-ocular responses and motion perception during variable radius centrifugation, and measure the time course of post-flight recovery. This study will also address how adaptive changes in otolith-mediated reflexes correspond to one's ability to perform closed-loop nulling tasks following G-transitions, and whether manual control performance can be improved
Banach function algebras and certain polynomially norm-preserving maps
Hosseini, Maliheh; Sady, Fereshteh
2012-01-01
Let $A$ and $B$ be Banach function algebras on compact Hausdorff spaces $X$ and $Y$, respectively. Given a non-zero scalar $\\alpha$and $s,t\\in \\Bbb N$ we characterize the general form of suitable powers of surjective maps $T, T': A \\longrightarrow B$ satisfying $\\|(Tf)^s (T'g)^t-\\alpha\\|_Y=\\|f^s g^t-\\alpha \\|_X$, for all $f,g \\in A$, where $\\|\\cdot \\|_X$ and $\\|\\cdot \\|_Y$ denote the supremum norms on $X$ and $Y$, respectively. A similar result is given for the case where $T...
Eventually and asymptotically positive semigroups on Banach lattices
Daners, Daniel; Glück, Jochen; Kennedy, James B.
2016-09-01
We develop a theory of eventually positive C0-semigroups on Banach lattices, that is, of semigroups for which, for every positive initial value, the solution of the corresponding Cauchy problem becomes positive for large times. We give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators, complementing existing results on spaces of continuous functions. This enables us to treat a range of new examples including the square of the Laplacian with Dirichlet boundary conditions, the bi-Laplacian on Lp-spaces, the Dirichlet-to-Neumann operator on L2 and the Laplacian with non-local boundary conditions on L2 within the one unified theory. We also introduce and analyse a weaker notion of eventual positivity which we call "asymptotic positivity", where trajectories associated with positive initial data converge to the positive cone in the Banach lattice as t → ∞. This allows us to discuss further examples which do not fall within the above-mentioned framework, among them a network flow with non-positive mass transition and a certain delay differential equation.
Spectral theory of linear operators and spectral systems in Banach algebras
Müller, Vladimir
2003-01-01
This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach algebras. It presents a survey of results concerning various types of spectra, both of single and n-tuples of elements. Typical examples are the one-sided spectra, the approximate point, essential, local and Taylor spectrum, and their variants. The theory is presented in a unified, axiomatic and elementary way. Many results appear here for the first time in a monograph. The material is self-contained. Only a basic knowledge of functional analysis, topology, and complex analysis is assumed. The monograph should appeal both to students who would like to learn about spectral theory and to experts in the field. It can also serve as a reference book. The present second edition contains a number of new results, in particular, concerning orbits and their relations to the invariant subspace problem. This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach alg...
Tensor products of commutative Banach algebras
Shobha Madan
1982-09-01
Full Text Available Let A1, A2 be commutative semisimple Banach algebras and A1Ã¢ÂŠÂ—Ã¢ÂˆÂ‚A2 be their projective tensor product. We prove that, if A1Ã¢ÂŠÂ—Ã¢ÂˆÂ‚A2 is a group algebra (measure algebra of a locally compact abelian group, then so are A1 and A2. As a consequence, we prove that, if G is a locally compact abelian group and A is a comutative semi-simple Banach algebra, then the Banach algebra L1(G,A of A-valued Bochner integrable functions on G is a group algebra if and only if A is a group algebra. Furthermore, if A has the Radon-Nikodym property, then the Banach algebra M(G,A of A-valued regular Borel measures of bounded variation on G is a measure algebra only if A is a measure algebra.
Admissibility of Linear Systems in Banach Spaces
GUO Fa-ming
2005-01-01
In this paper, infinite-time p-admissibility of unbounded operators is introduced and the Co-semigroup characterization of the infinite-time p-admissibility of unbounded observation operators is given. Moreover, the analogous result for the infinite-time p-admissibility of unbounded control operators is presented.
... area into the arm may be present (these nerves are called brachial plexus). A Moro reflex in an older infant, child, or adult is ... be done to examine the child's muscles and nerves. Diagnostic ... absent reflex, may include: Shoulder x-ray Tests for disorders ...
(2-1)-Ideal amenability of triangular banach algebras
S Etemad; M Ettefagh
2015-05-01
Let $\\mathcal{A}$ and $\\mathcal{B}$ be two unital Banach algebras and let $\\mathcal{M}$ be an unital Banach $\\mathcal{A}$, $\\mathcal{B}$-module. Also, let $\\mathcal{T}=\\left[\\begin{smallmatrix} \\mathcal{A} & \\mathcal{M}\\\\ & \\mathcal{B}\\end{smallmatrix}\\right]$ be the corresponding triangular Banach algebra. Forrest and Marcoux (Trans. Amer. Math. Soc. 354 (2002) 1435–1452) have studied the -weak amenability of triangular Banach algebras. In this paper, we investigate (2-1)-ideal amenability of $\\mathcal{T}$ for all ≥ 1. We introduce the structure of ideals of these Banach algebras and then, we show that (2-1)-ideal amenability of $\\mathcal{T}$ depends on (2-1)-ideal amenability of Banach algebras $\\mathcal{A}$ and $\\mathcal{B}$.