Real Banach Space Research Articles

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 .

Bishop-Phelps Theorem for Normed Cones

Introduction In the last few years there is a growing interest in the theory of quasi-metric spaces and other related structures such as quasi-normed cones and asymmetric normed linear spaces, because such a theory provides an important tool in the study of several problems in theoretical computer science, approximation theory, applied physics, convex analysis and optimization. Many works on general topology and functional analysis have recently been obtained in order to extend the well-known results of the classical theory of normed linear spaces to the framework of asymmetric normed linear spaces and quasi-normed cones. ‎An abstract cone is analogous to a real vector space‎, ‎except that we take as the set‎ ‎of scalars‎. ‎ In 2004, O‎. ‎Valero introduced the normed cones and proved some closed graph and open mapping results for normed cones. Also Valero defined and studied some properties of quotient normed cones. P. Selinger studied the norm properties of a cone with its order properties and proved Hahn-Banach theorems in these cones under the appropriate conditions. Valero and his colleagues discussed the metrizability of the unit ball of the dual of a normed cone and the isometries of normed cones. Other properties are investigated in a series of papers by Romaguera, Sanchez Perez and Valero. The Bishop-Phelps theorem is a fundamental theorem in functional analysis which has many applications in the geometry of Banach spaces and optimization theory. The classical Bishop-Phelps theorem states that “the set of support functionals for a closed bounded convex subset of a real Banach space X, is norm dense in and the set of support points of is dense in the boundary of ". Indeed, E. Bishop and R. R. Phelps answer a question posed by ‎Victor Klee in 1958. We give an analogue to the normed cones‎, in fact we show that in a continuous normed cone the set of support points of a closed convex set is a dense subset of the boundary under the appropriate hypothesis. Conclusion In this paper the notion of support points of convex sets in normed cones is introduced and it is shown that in a continuous normed cone, under the appropriate conditions, the set of support points of a bounded Scott-closed convex set is nonempty. We also present a Bishop-Phelps type Theorem for normed cones../files/site1/files/52/5.pdf

Banach spaces where convex combinations of relatively weakly open subsets of the unit ball are relatively weakly open

We introduce and study Banach spaces which have property CWO, i.e., every finite convex combination of relatively weakly open subsets of their unit ball is open in the relative weak topology of the unit ball. Stability results of such spaces are established, and we introduce and discuss a geometric condition---property (co)---on a Banach space. Property (co) essentially says that the operation of taking convex combinations of elements of the unit ball is, in a sense, an open map. We show that if a finite dimensional Banach space $X$ has property (co), then for any scattered locally compact Hausdorff space $K$, the space $C_0(K,X)$ of continuous $X$-valued functions vanishing at infinity has property CWO. Several Banach spaces are proved to possess this geometric property; among others: 2-dimensional real spaces, finite dimensional strictly convex spaces, finite dimensional polyhedral spaces, and the complex space $\ell_1^n$. In contrast to this, we provide an example of a $3$-dimensional real Banach space $X$ for which $C_0(K,X)$ fails to have property CWO. We also show that $c_0$-sums of finite dimensional Banach spaces with property (co) have property CWO. In particular, this provides examples of such spaces outside the class of $C_0(K,X)$-spaces.

A strong convergence theorem for maximal monotone operators in Banach spaces with applications

An algorithm is constructed to approximate a zero of a maximal monotone operator in a uniformly convex anduniformly smooth real Banach space. The sequence of the algorithm is proved to converge strongly to a zeroof the maximal monotone map. In the case where the Banach space is a real Hilbert space, our theorem com-plements the celebrated proximal point algorithm of Martinet and Rockafellar. Furthermore, our convergencetheorem is applied to approximate a solution of a Hammerstein integral equation in our general setting. Finally,numerical experiments are presented to illustrate the convergence of our algorithm.

On split equality for finite family of generalized mixed equilibrum problem and fixed point problem in real Banach spaces

On split equality for finite family of generalized mixed equilibrum problem and fixed point problem in real Banach spaces

Complexifications of real Banach spaces and their isometries

Every norm ‖⋅‖ on a real Banach space X induces a minimal norm on the complex linear space CX=X+iX={x+iy:x,y∈X} by‖x+iy‖C=sup⁡{‖xcos⁡θ+ysin⁡θ‖:θ∈[0,2π]}. In this note we show that if X is finite-dimensional there is a decomposition X=X1⊕⋯⊕Xk into subspaces such that the isometry group of ‖⋅‖C is generated by that of ‖⋅‖ and operators of the form eiθ1In1⊕⋯⊕eiθkInk acting on CX=CX1⊕⋯⊕CXk. Various applications are given, in particular to isometries of numerical radius.

Existence results for evolution equations with superlinear growth

By combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results for semilinear differential equations involving a dissipative linear operator, generating an extendable compact $C_0$-semigroup of contractions, and a Caratheodory nonlinearity $f\colon [0,T] \times E \to F$, with $E$ and $F$ two real Banach spaces such that $E \subseteq F$, besides imposing other conditions. The case $E\neq F$ allows to treat, as an application, parabolic equations with continuous superlinear nonlinearities which satisfy a sign condition.

An existence principle for variational inequalities in Banach spaces

The present article deals with existence and uniqueness results for a variational inequality posed on a real smooth Banach space. This allows us to generalize the known results of this type given in Hilbert spaces.

Convex Stone-Weierstrass theorems and invariant convex sets

A \textit {convex polynomial} is a convex combination of the monomials $\{1, x, x^2, \ldots \}$. This paper establishes that the convex polynomials on $\mathbb R$ are dense in $L^p(\mu )$ and weak$^*$ dense in $L^\infty (\mu )$ whenever $\mu $ is a compactly supported regular Borel measure on $\mathbb {R}$ and $\mu ([-1,\infty )) = 0$. It is also shown that the convex polynomials are norm dense in $C(K)$ precisely when $K \cap [-1, \infty ) = \emptyset $, where $K$ is a compact subset of the real line. Moreover, the closure of the convex polynomials on $[-1,b]$ is shown to be the functions that have a convex power series representation. A continuous linear operator $T$ on a locally convex space $X$ is \textit {convex-cyclic} if there is a vector $x \in X$ such that the convex hull of the orbit of $x$ is dense in $X$. The previous results are used to characterize which multiplication operators on various real Banach spaces are convex-cyclic. Also, it is shown for certain multiplication operators that every nonempty closed invariant convex set is a closed invariant subspace.

Strong convergence of viscosity iterations with error terms for cosine families in Banach spaces

In this paper viscosity iterations with error terms for approximating common fixed points of a nonexpansive cosine family and a continuous pseudocontractions are firstly considered. By using the theory of cosine families and constructing some control conditions, several new strong convergence results for this iterative process are presented in the bounded subsets of real uniformly convex and uniformly Gâteaux differentiable Banach spaces, and in the unnecessarily bounded subsets of real reflexive Banach spaces which admit a weakly sequentially continuous duality mapping, respectively. The results contain the corresponding ones without errors as their special cases.

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.

Solution by iteration of split equality problem involving some families of mappings in Banach spaces

An iterative scheme is proposed for approximation of common element of set of solution of split equality fixed point problems for $$\eta $$-demimetric mappings, set of common fixed points of finite families of relatively quasi-nonexpansive mappings and set of common solutions of systems of equilibrium problems in specified real Banach spaces. Our theorems extend and compliment several existing results.

Second Existence Theorem for a Class of Variational Equations with Non-Coercive Main Part

By using “the fibering idea”, we have established a second existence theorem for the equation ∂A(u) = ∑ j=1 m [∂D j (u)] where A (main part) and D j are non-negative functionals defined in a real and reflexive Banach space and they fulfill certain conditions with A non-coercive and ∂ is the Frechet differential operator. Moreover, we analyze this equation in some concrete cases.

Complex interpolation of $\mathbb R$-norms, duality and foliations

The complex method of interpolation, going back to Calderón and Coifman et al., on the one hand, and the Alexander–Wermer–Słodkowski theorem on polynomial hulls with convex fibers, on the other hand, are generalized to a method of interpolation of real (finite-dimensional) Banach spaces and of convex functions. The underlying duality in this method is given by the Legendre transform. Our results can also be interpreted as new properties of solutions of the homogeneous complex Monge–Ampère equation.

Solvability and optimization for a class of mixed variational problems

ABSTRACTWe consider an abstract mixed variational problem governed by a nonlinear operator A and a bifunctional J, in a real reflexive Banach space X. The operator A is assumed to be continuous, Lipschitz continuous on each bounded subset of X, and generalized monotone. First, we pay attention to the unique solvability of the problem. Next, we prove a continuous dependence result of the solution with respect to the data. Based on this result, we prove the existence of at least one solution for an associated optimization problem. Finally, we apply our abstract results to the well-posedness and the optimization of an antiplane frictional contact model for nonlinearly elastic materials of Hencky type.

Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds

The aim of this paper is to show the existence and attainability of Karush–Kuhn–Tucker optimality conditions for weakly efficient Pareto points for vector equilibrium problems with the addition of constraints in the novel context of Hadamard manifolds, as opposed to the classical examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient conditions for weakly efficient Pareto points to the constrained vector optimization problem are presented. The results described in this article generalize results obtained by Gong (2008) and Wei and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed spaces, and real Banach spaces to Hadamard manifolds, respectively. This is done using a notion of Riemannian symmetric spaces of a noncompact type as special Hadarmard manifolds.

Pre-Schauder Bases in Topological Vector Spaces

A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n ∈ N in X such that for every x ∈ X there exists a unique sequence of scalars ( λ n ) n ∈ N satisfying that x = ∑ n = 1 ∞ λ n e n . Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but not in topological vector spaces). We also prove that, if a Hausdorff topological vector space admits a pre-Schauder basis and is w * -strongly torsionless, then the biorthogonal system formed by the basis and its coefficient functionals is total. Finally, we focus on Schauder bases on Banach spaces proving that every Banach space with a normalized Schauder basis admits an equivalent norm closer to the original norm than the typical bimonotone renorming and that still makes the basis binormalized and monotone. We also construct an increasing family of left-comparable norms making the normalized Schauder basis binormalized and show that the limit of this family is a right-comparable norm that also makes the normalized Schauder basis binormalized.

Variational Inclusions Involving Generalized Set-Valued Quasi-Contractive Mappings in Banach Spaces

In this paper, we give a characterization of strong convergence theorem of the Ishikawa iterative sequence with errors for the generalized set-valued variational inclusion problem involving a generalized set-valued quasi-contractive mapping in a real Banach space. First, the existence of the solution of the generalized set-valued variational inclusion problem is proved. Then, using the existence result, the Ishikawa iterative algorithm with errors is constructed and the convergence of the iterative sequences is proved. Finally, numerical experiments are proposed to support the main result. The results improve and extend some known results.

Complete convergence theorems for weighted row sums from arrays of random elements in Rademacher type p and martingale type p Banach spaces

In this correspondence, complete convergence theorems are established for weighted row sums from arrays of random elements taking values in real separable Rademacher type Banach spaces as well as real separable martingale type Banach spaces. It is assumed that A version of the Rademacher type p complete convergence theorem is also established with random variable weights. Illustrative examples are included.

Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

Abstract Let Alg 𝒩 be a nest algebra associated with the nest 𝒩 on a (real or complex) Banach space 𝕏. Suppose that there exists a non-trivial idempotent P ∈ Alg 𝒩 with range P (𝕏) ∈ 𝒩, and δ : Alg 𝒩 → Alg 𝒩 is a continuous linear mapping (generalized) left derivable at P, i.e. δ (ab) = aδ (b) + bδ (a) (δ (ab) = aδ(b) + bδ(a) − baδ(I)) for any a, b ∈ Alg 𝒩 with ab = P, where I is the identity element of Alg 𝒩. We show that is a (generalized) Jordan left derivation. Moreover, in a strongly operator topology we characterize continuous linear maps on some nest algebras Alg 𝒩 with the property that δ (P ) = 2Pδ (P ) or δ (P ) = 2P δ (P ) − Pδ (I) for every idempotent P in Alg 𝒩.