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Characterization of inclusion among Riesz−Medvedev variation spaces

100%

Aziz W., Ereú T.

Commentationes Mathematicae

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2016

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tom 56

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nr 2

EN

We present a characterization of inclusion among Riesz−Medvedev bounded variation spaces, i.e., we shall present necessary and sufficient conditions for the Young functions $\varphi_1$ and $\varphi_2$ so that $RV_{\varphi_{1}}[a,b]\subset RV_{\varphi{_{2}}}[a,b]$ or $RV^*_{\varphi_{1}}[a,b]\subset RV^*_{\varphi_{2}}[a,b]$.

2

Existence theorem for the Hammerstein integral equation

80%

Cichoń M., Kubiaczyk I.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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1996

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tom 16

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nr 2

171-177

EN

In this paper we prove an existence theorem for the Hammerstein integral equation $x(t) = p(t) + λ ∫_I K(t,s)f(s,x(s))ds$, where the integral is taken in the sense of Pettis. In this theorem continuity assumptions for f are replaced by weak sequential continuity and the compactness condition is expressed in terms of the measures of weak noncompactness. Our equation is considered in general Banach spaces.

3

Measure valued solutions for systems governed by neutral differential equations on Banach spaces and their optimal control

80%

Ahmed N.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2013

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tom 33

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nr 1

89-109

EN

In this paper we consider the question of existence of measure valued solutions for neutral differential equations on Banach spaces when there is no mild solutions. We prove the existence of measure solutions and their regularity properties. We consider also control problems of such systems and prove existence of optimal feedback controls for some interesting a-typical control problems.

4

On the mutually non isomorphic $l_{p}(l_{q})$

80%

Cembranos P., Mendoza J.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2016

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tom 36

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nr 1

117-127

EN

In this note we survey the partial results needed to show the following general theorem: ${l_{p}(l_{q}) : 1 ≤ p,q ≤ +∞}$ is a family of mutually non isomorphic Banach spaces. We also comment some related facts and open problems.

5

Solutions of the Hammerstein equations in $BV_\varphi(I_{a}^{b},\, \mathbb{R})

71%

Aziz W., Guerrero J., Azócar L., Merentes N.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2016

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tom 36

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nr 2

207-229

EN

In this paper we study existence and uniqueness of solutions for the Hammerstein equation \[ u(x)= v(x) + \lambda \int_{I_{a}^{b}}K(x,y)f(y,u(y))dy \] \noi in the space of function of bounded total $\varphi$-variation in the sense of Hardy-Vitali-Tonelli, where $\lambda\in \mathbb{R}$, $K:I_a^b\times I_a^b \longrightarrow \mathbb{R}$ and $f:I_a^b\times \mathbb{R} \longrightarrow \mathbb{R}$ are suitable functions. The existence and uniqueness of solutions are proved by means of the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle.

6

Asymptotically hilbertian modular Banach spaces: examples of uncountable categoricity

71%

Henson C. W., Raynaud Y.

Commentationes Mathematicae

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2016

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tom 56

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nr 1

EN

We give a criterion ensuring that the elementary class of a modular Banach space $E$ (that is, the class of Banach spaces, some ultrapower of which is linearly isometric to an ultrapower of $E$) consists of all direct sums $E\oplus_m H$, where $H$ is an arbitrary Hilbert space and $\oplus_m$ denotes the modular direct sum. Also, we give several families of examples in the class of Nakano direct sums of finite dimensional normed spaces that satisfy this criterion. This yields many new examples of uncountably categorical Banach spaces, in the model theory of Banach space structures.

7

Series representation of compact linear operators in Banach spaces

71%

Edmunds D. E., Lang J.

Commentationes Mathematicae

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2016

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tom 56

|

nr 1

EN

Let $p\in(1,\infty)$ and $I=(0,1)$; suppose that $T\colon L_{p}(I)\rightarrow L_{p}(I)$ is a~compact linear map with trivial kernel and range dense in $L_{p}(I)$. It is shown that if the Gelfand numbers of $T$ decay sufficiently quickly, then the action of $T$ is given by a series with calculable coefficients. The special properties of $L_{p}(I)$ enable this to be established under weaker conditions on the Gelfand numbers than in earlier work set in the context of more general spaces.

8

Approximate weak invariance for semilinear differential inclusions in Banach spaces

71%

Lazu A., Postolache V.

Open Mathematics

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2011

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tom 9

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nr 5

1143-1155

EN

In this paper we give a criterion for a given set K in Banach space to be approximately weakly invariant with respect to the differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A generates a C 0-semigroup and F is a given multi-function, using the concept of a tangent set to another set. As an application, we establish the relation between approximate solutions to the considered differential inclusion and solutions to the relaxed one, i.e., x′(t) ∈ Ax(t) + $\overline {co} $ F(x(t)), without any Lipschitz conditions on the multi-function F.

9

The Orthogonal Projection and the Riesz Representation Theorem

71%

Narita K., Endou N., Shidama Y.

Formalized Mathematics

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2015

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tom 23

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nr 3

243-252

EN

In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor RUSp2RNSp, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real Hilbert spaces. Furthermore, we defined the continuity and the Lipschitz the continuity of functionals on real Hilbert spaces. Referring to the article [15], we also defined some definitions on real Hilbert spaces and proved some theorems for defining dual spaces of real Hilbert spaces. As to the properties of all definitions, we proved that they are equivalent properties of functionals on real normed spaces. In Sec. 2, by the definitions [11], we showed properties of the orthogonal complement. Then we proved theorems on the orthogonal decomposition of elements of real Hilbert spaces. They are the last two theorems of existence and uniqueness. In the third and final section, we defined the kernel of linear functionals on real Hilbert spaces. By the last three theorems, we showed the Riesz representation theorem, existence, uniqueness, and the property of the norm of bounded linear functionals on real Hilbert spaces. We referred to [36], [9], [24] and [3] in the formalization.

10

Weak Convergence and Weak Convergence

71%

Narita K., Shidama Y., Endou N.

Formalized Mathematics

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2015

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tom 23

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nr 3

231-241

EN

In this article, we deal with weak convergence on sequences in real normed spaces, and weak* convergence on sequences in dual spaces of real normed spaces. In the first section, we proved some topological properties of dual spaces of real normed spaces. We used these theorems for proofs of Section 3. In Section 2, we defined weak convergence and weak* convergence, and proved some properties. By RNS_Real Mizar functor, real normed spaces as real number spaces already defined in the article [18], we regarded sequences of real numbers as sequences of RNS_Real. So we proved the last theorem in this section using the theorem (8) from [25]. In Section 3, we defined weak sequential compactness of real normed spaces. We showed some lemmas for the proof and proved the theorem of weak sequential compactness of reflexive real Banach spaces. We referred to [36], [23], [24] and [3] in the formalization.

11

On Banach spaces of regulated functions

61%

Drewnowski L.

Commentationes Mathematicae

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2017

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tom 57

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nr 2

EN

For a relatively compact subset $S$ of the real line $\BR$, let $R(S)$ denote the Banach space (under the sup norm) of all regulated scalar functions defined on $S$. The purpose of this paper is to study those closed subspaces of $R(S)$ that consist of functions that are left-continuous, right-continuous, continuous, and have a (two-sided) limit at each point of some specified disjoint subsets of $S$. In particular, some of these spaces are represented as $C(K)$ spaces for suitable, explicitly constructed, compact spaces $K$.

12

Vector and operator valued measures as controls for infinite dimensional systems: optimal control

61%

Ahmed N.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2008

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tom 28

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nr 1

95-131

EN

In this paper we consider a general class of systems determined by operator valued measures which are assumed to be countably additive in the strong operator topology. This replaces our previous assumption of countable additivity in the uniform operator topology by the weaker assumption. Under the relaxed assumption plus an additional assumption requiring the existence of a dominating measure, we prove some results on existence of solutions and their regularity properties both for linear and semilinear systems. Also presented are results on continuous dependence of solutions on operator and vector valued measures, and other parameters determining the system which are then used to prove some results on control theory including existence and necessary conditions of optimality. Here the operator valued measures are treated as structural controls. The paper is concluded with some examples from classical and quantum mechanics and a remark on future direction.

13

Stochastic diffrential equations on Banach spaces and their optimal feedback control

61%

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2012

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tom 32

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nr 1

87-109

EN

In this paper we consider stochastic differential equations on Banach spaces (not Hilbert). The system is semilinear and the principal operator generating a C₀-semigroup is perturbed by a class of bounded linear operators considered as feedback operators from an admissible set. We consider the corresponding family of measure valued functions and present sufficient conditions for weak compactness. Then we consider applications of this result to several interesting optimal feedback control problems. We present results on existence of optimal feedback operators.

Ograniczanie wyników

Characterization of inclusion among Riesz−Medvedev variation spaces

Existence theorem for the Hammerstein integral equation

Measure valued solutions for systems governed by neutral differential equations on Banach spaces and their optimal control

On the mutually non isomorphic $l_{p}(l_{q})$

Solutions of the Hammerstein equations in $BV_\varphi(I_{a}^{b},\, \mathbb{R})

Asymptotically hilbertian modular Banach spaces: examples of uncountable categoricity

Series representation of compact linear operators in Banach spaces

Approximate weak invariance for semilinear differential inclusions in Banach spaces

The Orthogonal Projection and the Riesz Representation Theorem

Weak Convergence and Weak Convergence

On Banach spaces of regulated functions

Vector and operator valued measures as controls for infinite dimensional systems: optimal control

Stochastic diffrential equations on Banach spaces and their optimal feedback control