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1

The algebras of bounded and essentially bounded Lebesgue measurable functions

Mortini R., Rupp R.

Demonstratio Mathematica

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2017

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Vol. 50, nr 1

94--99

EN

Let X be a set in Rn with positive Lebesgue measure. It is well known that the spectrum of the algebra L∞(X) of (equivalence classes) of essentially bounded, complex-valued, measurable functions on X is an extremely disconnected compact Hausdorspace. We show, by elementary methods, that the spectrum M of the algebra Lb(X, C) of all bounded measurable functions on X is not extremely disconnected, though totally disconnected. Let ∆ = {δx : x ∈ X} be the set of point evaluations and let g be the Gelfand topology on M. Then (∆, g) is homeomorphic to (X, Tdis), where Tdis is the discrete topology. Moreover, ∆ is a dense subset of the spectrum M of Lb(X, C). Finally, the hull h(I), (which is homeomorphic to M(L∞(X))), of the ideal of all functions in Lb(X, C) vanishing almost everywhere on X is a nowhere dense and extremely disconnected subset of the Corona M \ ∆ of Lb(X, C).

2

BVP for Caratheodory inclusions in Hilbert spaces: sharp existence conditions and applications

Benedetti I., Panasenko E., Taddei V.

Journal of Applied Analysis

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2010

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Vol. 16, nr 2

237-258

EN

This article concerns an existence result for Floquet boundary value problems associated to semilinear differential inclusions with Carathéodory right hand side in a Hilbert space. We apply a continuation principle and we require a sharp (i.e., localized on the boundary) transversality condition. We give an application to a nonlinear partial differential inclusion with periodic conditions.

3

A note on the quasicontinuity

Grande Z.

Demonstratio Mathematica

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2006

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Vol. 39, nr 3

515-518

EN

Let (X,px) and (V, pv) be metric spaces. A function f: X - Y is said graph quasicontinuous if there is a quasicontinuous function g : X -> Y with the graph Gr(g) contained in the closure cl(Gr(f)) of Gr(f). If the space (V, pv) is compact and if there is a dense subset A C X such that the restricted function f/A is continuous then f is graph quasicontinuous. Moreover each locally bounded function f: R -> R is graph quasicontinuous.

4

Strong variational principles and generic well-posedness

Turinici M.

Demonstratio Mathematica

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2005

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Vol. 38, nr 4

935--941

EN

The strong version of Ekeland's variational principle [JMAA, 47(1974), 324-353] due to Georgiev [JMAA, 131(1988), 1-21] is deductible in a direct way from the standard one. In addition, a substitution between them is possible for many generic well-posed optimization problems.

5

Hadamard product of functions subordinated to homography

Piejko Krzysztof, Sokół Janusz, Stankiewicz Jan

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

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2004

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z. 27 [212]

49-59

EN

We consider convolution properties of regular functions using the concept of subordination. Let P(X, Y) denote the class of regular functions subordinated to the homography 1+Xz/1-Yz. It is known [10] that for some complex numbers A,B,C,D if is an element of P(A,B) and g is an element of P(C,D), then there exist X and Y such that f *p is an element of P(X,Y). In this paper we verify the reverse question: if for each h is an element of P(X, Y) it is possible to find suitable f is an element of P(A, B) and g is an element of P(C7, D) such that h=f*g.