Krasnoselski-Mann Iterations for Hierarchical Fixed Point Problems for a Finite Family of Nonself Mappings in Banach Spaces

This paper deals with a method for approximating a solution of the following problem: find hierarchically a common fixed point of a finite family of nonself nonexpansive mappings with respect to a nonexpansive self mapping on a closed convex subset of a smooth and reflexive Banach space X, which admits a weakly sequentially continuous duality mapping. First, we prove a weak convergence theorem which extends and improves one recent result proved by Yao and Liou (see Inverse Problems 24 (2008), doi:). Secondly, when the self mapping is a contraction, we prove, under different restrictions on parameters, a strong convergence result which generalize some recent results in the literature.     

Quantification of Pełczyński's property (V)

A Banach space X has Pełczyński's property (V) if for every Banach space Y every unconditionally converging operator is weakly compact. In 1962, Aleksander Pełczyński showed that spaces for a compact Hausdorff space K enjoy the property (V), and some generalizations of this theorem have been proved since then. We introduce several possibilities of quantifying the property (V). We prove some characterizations of the introduced quantitative versions of this property, which allow us to prove a quantitative version of Pelczynski's result about spaces and generalize it. Finally, we study the relationship of several properties of operators including weak compactness and unconditional convergence, and using the results obtained we establish a relation between quantitative versions of the property (V) and quantitative versions of other well known properties of Banach spaces.     

Convergence and Data Dependency of Normal−S Iterative Method for Discontinuous Operators on Banach Space

We study convergence, rate of convergence and data dependency of normal?S iterative method for a fixed point of a discontinuous operator T on a Banach space. We also prove some Collage type theorems for T. The main aim here is to show that there is a close relationship between the concepts of data dependency of fixed points and the collage theorems and show that the latter provides better estimate. Numerical examples in support of the results obtained are also given.     

Untypical methods of convergence acceleration

Iterated Aitken’s method is one of classical procedures which permit to accelerate series or sequences convergence. It may be a starting point of constructing better methods in some classes of series whose important parameters are known. Such untypical modifications are here proposed and investigated. They based on a common idea and refer to two kinds of series; cf. Section 2 (series with rational coefficients, hypergeometric series and many others) and Section 3 (so-called quasi-geometric series). The second kind of series is associated with a class of infinite products whose convergence may be also accelerated. Behaviour of Levin’s and Weniger’s methods depends on a parameter β. In Section 4 its role is investigated and possibility of an improvement of their initial steps is showed.     

Coefficient Condition forL1-Convergence of Walsh–Fourier Series

In this paper we give a condition with respect to Walsh–Fourier coefficients that implies theL₁-convergence of the corresponding Walsh–Fourier series. We show that theL₁-convergence class induced by this condition contains each one of the previously known convergence classes as a proper subset. We also show that our condition implies not only theL₁-convergence but also the convergence in the dyadic Hardy norm if the function represented by the series belongs to the dyadic Hardy space.     

Power bounded operators and supercyclic vectors II

We show that each power bounded operator with spectral radius equal to one on a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant cone if belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators.

For non-reflexive Banach spaces these results remain true; however, the non-supercyclic vector (invariant cone, respectively) relates to the adjoint of the operator.

     

On mackey convergence in locally convex spaces

After some introductory propositions, we give a dual characterization of those locally convex spaces which satisfy the Mackey convergence condition or the fast convergence condition by means of Schwartz topologies. Making use of the universal Schwartz space (l _∞ ,τ(l _∞ ,l ₁)) we prove some representation theorems for bornological and ultrabornological spaces, that is, every bornological spaceE is a dense subspace of an inductive limit lim indE _a, a∈A, ofseparable Banach spacesE _a, and every Mackey null sequence inE is a null sequence in someE _a. IfE is ultrabornological, thenE can be represented as lim indE _a,a∈A, allE _a separable Banach spaces, such that every fast null sequence inE is a null sequence in someE _a.     

Convergence of Some Greedy Algorithms in Banach Spaces

We consider some theoretical greedy algorithms for approximation in Banach spaces with respect to a general dictionary. We prove convergence of the algorithms for Banach spaces which satisfy certain smoothness assumptions. We compare the algorithms and their rates of convergence when the Banach space is L_p(\mathbbT^d)L_p(\mathbb{T}^d) ($1相似文献   

On the convergence of Fourier series of computable Lebesgue integrable functions

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of L^p‐computable functions (computable Lebesgue integrable functions) with a size notion, by introducing L^p‐computable Baire categories. We show that L^p‐computable Baire categories satisfy the following three basic properties. Singleton sets {f } (where f is L^p‐computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of L^p‐computable functions is not meager. We give an alternative characterization of meager sets via Banach‐Mazur games. We study the convergence of Fourier series for L^p‐computable functions and show that whereas for every p > 1, the Fourier series of every L^p‐computable function f converges to f in the L^p norm, the set of L¹‐computable functions whose Fourier series does not diverge almost everywhere is meager (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coefficient Quantization in Banach Spaces

Let (e _i ) be a dictionary for a separable infinite-dimensional Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a ‘finite alphabet’. We investigate several approximation properties of this type and connect them to the Banach space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c ₀. We also show that, for every ε>0, the unit ball of every separable infinite-dimensional Banach space X contains a dictionary (x _i ) such that the additive group generated by (x _i ) is (3+ε)⁻¹-separated and 1/3-dense in X.      

A Multivalued History-Dependent Operator and Implicit Evolution Inclusions. I

Siberian Mathematical Journal - On the space of continuous functions from a line segment to a reflexive Banach space, we consider some operator whose values are closed convex...     

On completion of fuzzy metric spaces

Completions of fuzzy metric spaces (in the sense of George and Veeramani) are discussed. A complete fuzzy metric space Y is said to be a˜fuzzy metric completion of a˜given fuzzy metric space X if X is isometric to a˜dense subspace of Y. We present an example of a˜fuzzy metric space that does not admit any fuzzy metric completion. However, we prove that every standard fuzzy metric space has an (up to isometry) unique fuzzy metric completion. We also show that for each fuzzy metric space there is an (up to uniform isomorphism) unique complete fuzzy metric space that contains a˜dense subspace uniformly isomorphic to it.     

The banach algebra A and its properties

Beurling’s algebra is considered. A* arises quite naturally in problems of summability of the Fourier series at Lebesgue points, whereas Wiener’s algebra A of functions with absolutely convergent Fourier series arises when studying the norm convergence of linear means. Certainly, both algebras are used in some other areas. A* has many properties similar to those of A, but there are certain essential distinctions. A* is a regular Banach algebra, its space of maximal ideals coincides with[−π, π], and its dual space is indicated. Analogs of Herz’s and Wiener-Ditkin’s theorems hold. Quantitative parameters in an analog of the Beurling-Pollard theorem differ from those for A. Several inclusion results comparing the algebra A* with certain Banach spaces of smooth functions are given. Some special properties of the analogous space for Fourier transforms on the real axis are presented. The paper ends with a summary of some open problems.     

On the convergence of double Fourier series of functions in L P[0, 1]2, where p=(p 1, p 2)

Pointwise convergence of double trigonometric Fourier series of functions in the Lebesgue space L _p[0,2]²was studied by M. I. Dyachenko. In this paper, we also consider the problems of the convergence of double Fourier series in Pringsheim"s sense with respect to the trigonometric as well as the Walsh systems of functions in the Lebesgue space L _p[0,1]², p=(p₁,p₂), endowed with a mixed norm, in the particular case when the coefficients of the series in question are monotone with respect to each of the indices. We shall obtain theorems which generalize those of M. I. Dyachenko to the case when p is a vector. We shall also show that our theorems in the case of trigonometric Fourier series are best possible.     

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.     

Stampacchia’s Property,Self-duality and Orthogonality Relations

We show that if the conclusion of the well known Stampacchia Theorem on variational inequalities holds on a real Banach space X, then X is isomorphic to a Hilbert space. Motivated by this, we obtain a relevant result concerning self-dual Banach spaces and investigate some connections between properties of orthogonality relations, self-duality and Hilbert space structure. Moreover, we revisit the notion of the cosine of a linear operator and show that it can be used to characterize real Banach spaces that are isomorphic to a Hilbert space. Finally, we present some consequences of our results to quadratic forms and to evolution triples.     

Infinite dimensional Banach spaces of functions with nonlinear properties

The aim of this paper is to show that there exist infinite dimensional Banach spaces of functions that, except for 0, satisfy properties that apparently should be destroyed by the linear combination of two of them. Three of these spaces are: a Banach space of differentiable functions on ?ⁿ failing the Denjoy‐Clarkson property; a Banach space of non Riemann integrable bounded functions, but with antiderivative at each point of an interval; a Banach space of infinitely differentiable functions that vanish at infinity and are not the Fourier transform of any Lebesgue integrable function (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights: Average Case Setting

We study multivariate linear problems in the average case setting with respect to a zero-mean Gaussian measure whose covariance kernel has a finite-order weights structure. This means that the measure is concentrated on a Banach space of d-variate functions that are sums of functions of at most q ^* variables and the influence of each such term depends on a given weight. Here q ^* is fixed whereas d varies and can be arbitrarily large. For arbitrary finite-order weights, based on Smolyak’s algorithm, we construct polynomial-time algorithms that use standard information. That is, algorithms that solve the d-variate problem to within ε using of order function values modulo a power of ln ε ⁻¹. Here p is the exponent which measures the difficulty of the univariate (d=1) problem, and the power of ln ε ⁻¹ is independent of d. We also present a necessary and sufficient condition on finite-order weights for which we obtain strongly polynomial-time algorithms, i.e., when the number of function values is independent of d and polynomial in ε ⁻¹. The exponent of ε ⁻¹ may be, however, larger than p. We illustrate the results by two multivariate problems: integration and function approximation. For the univariate case we assume the r-folded Wiener measure. Then p=1/(r+1) for integration and for approximation.      

A universal reflexive space for the class of uniformly convex Banach spaces

We show that there exists a separable reflexive Banach space into which every separable uniformly convex Banach space isomorphically embeds. This solves problems raised by J. Bourgain [B] in 1980 and by W. B. Johnson in 1977 [Jo]. We also give intrinsic characterizations of separable reflexive Banach spaces which embed into a reflexive space with a block q-Hilbertian and/or a block p-Besselian finite dimensional decomposition. Dedicated to the memory of V. I. Gurarii Research supported by the National Science Foundation.     

Approximating curves for non-expansive mappings and accretive operators

In this article, under the hypotheses that E is a reflexive Banach space which has a weakly continuous duality mapping J _? with gauge function ?, we investigate several alternative viscosity approximation methods for finding some common fixed points of infinite non-expansive mappings, and prove two strong convergence theorems.