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1

Modelling a solution of a homogeneous parabolic equation with random initial condition from Lp(Ω)

Kuchinka Katalin, Slyvka-Tylyshchak Ganna

Przegląd Elektrotechniczny

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2023

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R. 99, nr 5

77--82

EN

In this paper, a model of solutions to the heat equation with initial conditions from the Orlicz space Lp(Ω) of random variables is built. The constructed model approximates the solution of a homogeneous parabolic equation with given reliability and accuracy in some Orlicz space.

PL

W pracy zbudowano model rozwiązan równania ciepła z warunkami początkowymi z przestrzeni Orlicza Lp(Ω) zmiennych losowych. Skonstruowany model przybliza rozwiązanie jednorodnego równania parabolicznego z zadaną niezawodnoscią i dokładnoscią w pewnej przestrzeni Orlicza.

2

Summing multi-norms defined by Orlicz spaces and symmetric sequence space

Blasco O.

Commentationes Mathematicae

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2016

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Vol 56, No. 1

145--168

EN

We develop the notion of the (X1,X2)-summing power-norm based on a~Banach space E, where X1 and X2 are symmetric sequence spaces. We study the particular case when X1 and X2 are Orlicz spaces ℓΦ and ℓΨ respectively and analyze under which conditions the (Φ,Ψ)-summing power-norm becomes a~multinorm. In the case when E is also a~symmetric sequence space L, we compute the precise value of ∥(δ1,⋯,δn)∥n(X1,X2), where (δk) stands for the canonical basis of L, extending known results for the (p,q)-summing power-norm based on the space ℓr which corresponds to X1=ℓp, X2=ℓq, and E=ℓr.

3

Effective energy integral functionals for thin films in the Orlicz-Sobolev space setting

Laskowski W., Nguyen H. T.

Demonstratio Mathematica

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2013

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Vol. 46, nr 3

585--604

EN

We consider an elastic thin film as a bounded open subset ω of R2. First, the effective energy functional for the thin film ω is obtained, by Γ-convergence and 3D-2D dimension reduction techniques applied to the sequence of re-scaled total energy integral functionals of the elastic cylinders (…) as the thickness ε goes to 0. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function for the elastic cylinders has the growth prescribed by an Orlicz convex function M. Here M is assumed to be non-power-growth-type and to satisfy the conditions (…) and (…) (that is equivalent to the reflexivity of Orlicz and Orlicz–Sobolev spaces generated by M). These results extend results of H. Le Dret and A. Raoult for the case M(t) = (…) for some (…).

4

On nonlinear differential equations in generalized Musielak-Orlicz spaces

Kozlowski W. M.

Commentationes Mathematicae

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2013

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Vol. 53, No. 2

13--33

EN

We consider ordinary differential equations u′(t)+(I−T)u(t)=0, where an unknown function takes its values in a given modular function space being a generalization of Musielak-Orlicz spaces, and T is nonlinear mapping which is nonexpansive in the modular sense. We demonstrate that under certain natural assumptions the Cauchy problem related to this equation can be solved. We also show a process for the construction of such a solution. This result is then linked to the recent results of the fixed point theory in modular function spaces.

5

On (∆2) condition in density-type topologies

Filipczak M., Terepeta M.

Demonstratio Mathematica

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2011

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Vol. 44, nr 2

423--432

EN

We discuss properties of density-type topologies Tψ connected with condition (∆2) similar to the condition considered in the theory of Orlicz spaces. Density-type topologies Tψ introduced in [5] may not be invariant under multiplication by a number. This property is strictly connected with the condition, which we call (∆2), by analogy with well known condition introduced in Orlicz spaces. Like in the theory of Orlicz spaces, (∆2) condition causes that the considered topologies are more convenient for examination and have simpler properties. Moreover, the power functions are also of great importance as a handy instrument. Recall some basic facts. Let (Ω, Σ, μ) be a measure space and A be a family of all functions φ: [0, ∞) → [0, ∞) which are continuous, nondecreasing, such that φ(0) = 0, φ(x) > 0 for x > 0 and limx→∞ φ(x) = ∞.

6

On the Existence of Common Fixed Points for Semigroups of Nonlinear Mappings in Modular Function Spaces

Kozłowski W. M.

Commentationes Mathematicae

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2011

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Vol. 51, [Z] 1

81-98

EN

Let C be a ρ-bounded, ρ -closed, convex subset of a modular function space Lρ. We investigate the existence of common fixed points for semigroups of nonlinear mappings Tt : C → C, i.e. a family such that T0(x) = x, Ts+t = Ts(Tt(x)), where each Tt is either ρ -contraction or ρ -nonexpansive. We also briefly discuss existence of such semigroups and touch upon applications to differential equations.

7

Topological properties of spaces of linear operators on non-locally convex Orlicz spaces

Oelke A.

Commentationes Mathematicae

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2010

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Vol. 50, [Z] 2

103-108

EN

We study the topological properties of the space L(L^Φ,X) of all continuous linear operators from an Orlicz space L^Φ (an Orlicz function Φ is not necessarily convex) to a Banach space X.We provide the space L(L^Φ,X) with the Banach space structure. Moreover, we examine the space Ls(L^Φ,X) of all singular operators from L^Φ to X.

8

Uniformly ž-Continuous Topologies on Orlicz-Bochner Spaces

Feledziak K.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2006

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Vol. 46, [Z] 1

93-111

EN

We examine the topological properties of Orlicz-Bochner spaces L^[fi](X) (over a -finite measure space [...], where ' is an Orlicz function (not necessarily convex) and X is a real Banach space. We continue the study of some class of locally convex topologies on L^[fi](X), called uniformly ž-continuous topologies. In particular, the generalized mixed topology [...] (in the sense of Turpin) is considered.

9

The exact values of nonsquare constants for a class of Orlicz spaces

Wang J.

Opuscula Mathematica

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2005

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Vol. 25, no. 2

325-332

EN

We extend the MDelta-condition from [10] and introduce the PhiDelta-condition at zero. Next we discuss nonsquare constant in Orlicz spaces generated by an N-function Phi(u) which satisfy PhiDelta-condition. We obtain exact value of nonsquare constant in this class of Orlicz spaces equipped with the Luxemburg norm.

10

Weak sequential completeness and compactness in topological function spaces

Nowak M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2004

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Vol. 44, nr spec.

155-165

EN

Let (E, r) be a Hausdorff locally convex-solid function space (over a cr-finite measure space) and let E* stand for its topological dual. It is proved that the space (E, r) is weakly sequentially complete if and only if r is a c-Lebesgue and cr-Levy topology. In particular, a characterization of weak sequential completeness of Or-licz spaces L* in terms of Orlicz functions is given. Moreover, it is proved that the Eberlein-Smulian type theorem remains valid for a locally convex space (E, o~(E, E*)). A characterization of conditional and relative weak compactness in (E, r) is obtained.

11

Orthogonally complemented hyperplanes in Orlicz function spaces

Chen S., Hudzik H., Wang Y.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2004

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Vol. 44, nr spec.

71-82

EN

In this paper, necessary and sufficient conditions for a closed hyperplane in Orlicz space to be generalized orthogonally complemented are given for both the Orlicz and the Luxemburg norm. The concept of strongly generalized orthogonally complemented subspace in Banach space is defined and criteria for such subspaces in Orlicz space for both norms are given.

12

Contractive projections onto isometric copies of L1(ν) in strictly monotone Banach lattices

Wójtowicz M.

Bulletin of the Polish Academy of Sciences. Mathematics

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2003

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Vol. 51, no 1

1-12

EN

Let Y be a Banach lattice and X a strictly monotone sublattice of Y. Every isometric copy of an L1-space in X is 1-complemented in Y (Theorem l). This is an extension of the classical result of Pełczyński for Y = X = L1(my), and of Dor for Y = X being q-concave. We also study the consequences of the existence of isometric copies of l1 in strictly monotone E[phi](my)-spaces, where E[phi](my) denotes the ideal of an Orlicz space L[phi](my) of the elements with absolutely continuous norm.

13

On nonlinear integral equations in some functions spaces

Bardaro C., Musielak J., Vinti G.

Demonstratio Mathematica

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2002

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Vol. 35, nr 3

583-592

EN

There are established some conditions for existence of solutions of a nonlinear integral equation Tf =f+g, where T is a convolution-type integral operator.

14

Integral of function with values in complete modular space

Liskowski M.

Demonstratio Mathematica

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2002

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Vol. 35, nr 2

347-356

EN

The theorem on existence of an integral of a function with values in a modular space and some fundamental properties of this integral are given.

15

Orthogonal uniform convexity in Köthe spaces and Orlicz spaces

Kolwicz P.

Bulletin of the Polish Academy of Sciences. Mathematics

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2002

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Vol. 50, no 4

395-411

EN

We study a geometric property in Köthe spaces which is called orthogonal uniform convexity (UC┴). It was introduced in [19]. We prove that the class of Köthe spaces with property (UC┴) is a proper subset of the class of uniformly monotone and P-convex Köthe spaces. Next we consider connections between (UC┴) and property (β) of Rolewicz. We shown that the implication (UC┴) → (β) is true in any Köthe sequence space. Moreover, we find criteria for Orlicz function (sequence) spaces to be orthogonally uniformly convex. As a corollary we get that there holds (UC) → (UC┴) → (β) in any Köthe sequence space and the converse of any of these implications is not true. Furthermore, the implications (UC) → (β) → (UC┴) hold in any Köthe function space and the second one cannot be reversed.

16

Compact midpoint local uniform convexity in Orlicz spaces

Li X., Cui Y.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2002

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[Z] 42(1)

53-62

EN

In this paper, we introduce a new geometric property in Banach spaces, namely compact midpoint local uniform convexity. Criteria for this property in Orlicz spaces are given for both norms in the function case as well as in the sequence case.

17

On theorem of Grothendieck type

Narloch A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2001

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[Z] 41

143-146

18

On certain nonlinear integral operators

Misiak A., Ryż A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2001

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[Z] 41

135-142

19

On a pair of geometric parameters of Orlicz norm

Yan Y.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2001

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[Z] 41

257-263

20

Modular function spaces and control functions of almost everywhere convergence

Kita H., Miyamoto T., Yoneda K.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2001

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[Z] 41

99-133

EN

The modular spaces have been studied by many authors. In this paper we consider the modular function spaces which provide a generalization of Banach function spaces, and discuss some relations between properties of these spaces and control functions of almost everywhrere convergence of functions in modular function spaces. We give a necessary and sufficient condition for a control function to be in the same modular function space as functions appearing in almost everywhere convergence.