General Banach Spaces Research Articles

Ball Comparison between Three Sixth Order Methods for Banach Space Valued Operators

Three methods of sixth order convergence are tackled for approximating the solution of an equation defined on the finitely dimensional Euclidean space. This convergence requires the existence of derivatives of, at least, order seven. However, only derivatives of order one are involved in such methods. Moreover, we have no estimates on the error distances, conclusions about the uniqueness of the solution in any domain, and the convergence domain is not sufficiently large. Hence, these methods have limited usage. This paper introduces a new technique on a general Banach space setting based only the first derivative and Lipschitz type conditions that allow the study of the convergence. In addition, we find usable error distances as well as uniqueness of the solution. A comparison between the convergence balls of three methods, not possible to drive with the previous approaches, is also given. The technique is possible to use with methods available in literature improving, consequently, their applicability. Several numerical examples compare these methods and illustrate the convergence criteria.

Positive Semigroups Using Resolvent Estimate Bounds on Sharp Growth Rates

In this paper, we study growth rates for strongly continuous semigroups. We fixate that a growth rate for the resolvent estimate on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the semigroup is asymptotically analytic, or if the semigroupis positive and the underlying space is an -space or a space of continuous functions. Also proved variations of the main results on fractional domains; these are valid on more general Banach spaces by Jan Rozendaal and Mark Veraar. In the second part apply the main theorem to prove optimality in a classical example of a perturbed wave equation which shows unusual sequence of spectral behavior.

The split common null point problem for Bregman generalized resolvents in two Banach spaces

In this paper, we first consider the split common null point problem in two Banach spaces. Then, using the Bregman generalized resolvents of maximal monotone operators, we prove strong convergence theorems of Halpern type iteration for finding a solution of the split common null point problem in two Banach spaces. As an application of our result, we study the split equilibrium problem in general Banach spaces and approximate a solution of the problem for the first time. Our new technique is based on basic properties of a Bregman distance induced by a Bregman function without using Bregman projection or the requirement of Mosco convergence of the sequences produced by the method. It is well known that the Bregman distance does not satisfy either the symmetry property or the triangle inequality which are required for standard distances. So, the results of the paper improve and extend many recent results in the literature.

Bounds for uniform resolvent conditions

For a bounded linear operator on a Banach space, the uniform resolvent condition implies the absolute summability of the powers of the operator. In this paper, we study the bounds for the absolute sum of the powers of an operator that satisfies the uniform resolvent condition. Some known bounds on general Banach spaces as well as on finite-dimensional Banach spaces are improved.

WEAK AND STRONG CONVERGENCE THEOREMS FOR NONEXPANSIVE SEMIGROUPS IN A BANACH SPACE SATISFYING OPIAL'S CONDITION

�In this paper, we study the weak convergence of Mann’s type iteration procedure and the existence of nonexpansive retractions for commutative semigroups in Banach spaces which satisfy Opial’s condition. Further, we introduce an implicit iteration procedure for nonexpansive semigroups and then prove a strong convergence theorem for the nonexpansive semigroups in general Banach spaces.

A study of orthogonality of bounded linear operators

We study Birkhoff-James orthogonality and isosceles orthogonality of bounded linear operators between Hilbert spaces and Banach spaces. We explore Birkhoff-James orthogonality of bounded linear operators in light of a new notion introduced by us and also discuss some of the possible applications in this regard. We also study isosceles orthogonality of bounded (positive) linear operators on a Hilbert space and some of the related properties, including that of operators having disjoint support. We further explore the relations between Birkhoff-James orthogonality and isosceles orthogonality in a general Banach space.

Coupled Hilfer and Hadamard random fractional differential systems with finite delay in generalized Banach spaces

Coupled Hilfer and Hadamard random fractional differential systems with finite delay in generalized Banach spaces

The existence and compactness of the set of solutions for a nonlinear integrodifferential equation in N variables in a Banach space

The paper is devoted to the study of a nonlinear integrodifferential equation in N variables with values in a general Banach space. By applying fixed point theorems in a suitable Banach space under appropriate conditions for subsets to be relatively compact, we prove the existence and the compactness of the set of solutions. In order to illustrate the results, we give two examples

Feedback Systems on a Reflexive Banach Space—Linearization

The aim of our work is to formulate and demonstrate the results of the normality, the Lipschitz continuity, of a nonlinear feedback system described by the monotone maximal operators and hemicontinuous, defined on real reflexive Banach spaces, as well as the approximation in a neighborhood of zero, of solutions of a feedback system [A,B] assumed to be non-linear, by solutions of another linear, This approximation allows us to obtain appropriate estimates of the solutions. These estimates have a significant effect on the study of the robust stability and sensitivity of such a system see [1] [2] [3]. We then consider a linear FS , and prove that, if ; , with the respective solutions of FS’s [A,B] and corresponding to the given (u,v) in . There exists,, positive real constants such that, . These results are the subject of theorems 3.1, ... , 3.3. The proofs of these theorems are based on our lemmas 3.2, ... , 3.5, devoted according to the hypotheses on A and B, to the existence of the inverse of the operator I+BA and . The results obtained and demonstrated along this document, present an extension in general Banach space of those in [4] on a Hilbert space H and those in [5] on a extended Hilbert space .

Functional Calculus for the Series of Semigroup Generators Via Transference

In this paper, apply an established transference principle to obtain the boundedness of certain functional calculi for the sequence of semigroup generators. It is proved that if be the sequence generates 0- semigroups on a Hilbert space, then for each the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on right half–plane. The bounded of this calculus grows at most logarithmically as. As a consequence decay at ∞. Then showed that each sequence of semigroup generator has a so-called (strong) m-bounded calculus for all m∈ℕ, and that this property characterizes the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called semigroups, the Hilbert space results actually hold in general Banach spaces.

Generalized representer theorems in Banach spaces

In this paper, we generalize the representer theorems in Banach spaces by the theory of nonsmooth analysis. The generalized representer theorems assure that the regularized learning models can be constructed by the nonconvex loss functions, the generalized training data, and the general Banach spaces which are nonreflexive, nonstrictly convex, and nonsmooth. Specially, the sparse representations of the regularized learning in 1-norm reproducing kernel Banach spaces are shown by the generalized representer theorems.

Existence results for systems of coupled impulsive neutral functional differential equations driven by a fractional Brownian motion and a Wiener process

Abstract In this paper, we prove some results on the existence and uniqueness of mild solutions for a system of semilinear impulsive differentials with infinite fractional Brownian motions and a Wiener process. Our approach is based on a new version of fixed point theorem, due to Krasnoselskii, in generalized Banach spaces.

Coupled implicit Caputo fractional q-difference systems

This paper deals with some existence, uniqueness, and Ulam stability results for a coupled implicit Caputo fractional q-difference system in Banach and generalized Banach spaces. Some applications are made of some fixed point theorems for the existence and uniqueness of solutions. Next we prove that our problem is generalized Ulam–Hyers–Rassias stable. Some illustrative examples are given in the last section.

Selfadjoint operators on real or complex Banach spaces

Selfadjoint operators on Hilbert spaces have been extended to more general scopes such as certain real Banach spaces and complex Banach spaces endowed with a continuous Hermitian bilinear form. Here we propose a definition of selfadjoint operator that works for both real and complex Banach spaces and that naturally extends the classical concept of selfadjointness for operators on Hilbert spaces. We strongly rely on selectors of the duality mapping, obtaining a very natural extension in the case of smooth Banach spaces. We provide nontrivial examples of selfadjoint operators on nonHilbert smooth Banach spaces and on nonHilbert general Banach spaces. Finally, we extend the classical theory on eigenvalues and supporting vectors of selfadjoint positive operators on Hilbert spaces to the scope of reflexive, strictly convex and smooth Banach spaces.

ON THE GENERALIZED BANACH SPACES

For any non-negative real number $\epsilon_{0}$, we shall introduce a concept of the $\epsilon_{0}$-Cauchy sequence in a normed linear space $V$ and also introduce a concept of the $\epsilon_{0}$-completeness in those spaces. Finally we introduce a concept of the generalized Banach spaces with these concepts.

New Fixed Point Results in Generalized Banach Space

In this paper, we prove some fixed point results dealing with rational type contractions on generalized Banach Space. Our results in the new cotext extend and completement fixed point theorems of contractive mappings. Also we present certain common point result for such contraction.

Hilbert generated Banach spaces need not have a norming Markushevich basis

We construct a zero dimensional uniform Eberlein compact space K such that C(K) is a Hilbert generated Banach space which does not admit any norming Markushevich basis. This solves a classical problem from the seventies.

Bounded reflexivity of kernels of elementary operators

Kernels of elementary operators of length two give rise to many interesting classes of operators. In this paper we study bounded reflexivity of these kernels. For general Banach spaces, a sufficient condition is given for the kernels to be boundedly reflexive, in particular this assures that the commutant of any Banach space operator is boundedly reflexive. We also show that the bounded reflexive cover of the set of all Toeplitz matrices in Mm×n is the entire Mm×n. For finite-dimensional Banach spaces, using Kronecker canonical form for matrix pencils, we obtain some necessary and sufficient conditions for the kernels to be boundedly reflexive. These conditions enable us to easily determine bounded reflexivity by simply computing rank or nullity of matrices.

Random Coupled Hilfer and Hadamard Fractional Differential Systems in Generalized Banach Spaces

This article deals with some existence and uniqueness result of random solutions for some coupled systems of Hilfer and Hilfer–Hadamard fractional differential equations with random effects. Some applications are made of generalizations of classical random fixed point theorems on generalized Banach spaces.

Statistical margin error bounds for L1-norm support vector machines

Comparing with Lp-norm (1<p<+∞) Support Vector Machines (SVMs), the L1-norm SVM enjoys the nice property of simultaneously performing classification and feature selection. Margin error bounds for SVM on Hilbert spaces (or on more general q-uniformly smooth Banach spaces) have been obtained in the literature to justify the strategy of maximizing the margin in SVM. In this paper, we devote to estimating the margin error bound for L1-norm SVM methods and giving a geometrical interpretation for the result. We show that the fat-shattering dimension of the Banach spaces ℓ1 and ℓ∞ are both infinite. Therefore, we establish margin error bounds for the SVM on finite dimensional spaces with L1-norm, thus supplying statistical justification for the large margin classification of L1-norm SVM on finite dimensional spaces. To complete the theory, corresponding results for the L∞-norm SVM are also presented.