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Copies of the sequence space ω in F-lattices with applications to Musielak−Orlicz spaces

Wójtowicz M., Wiśniewska H.

Commentationes Mathematicae

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2016

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Vol 56, No. 1

103--117

EN

Let E be a fixed real function F-space, i.e., E is an order ideal in L0(S,Σ,μ) endowed with a monotone F-norm ∥∥ under which E is topologically complete. We prove that E contains an isomorphic (topological) copy of ω, the space of all sequences, if and only if E contains a lattice-topological copy W of ω. If E is additionally discrete, we obtain a much stronger result: W can be a projection band; in particular, E contains a~complemented copy of ω. This solves partially the open problem set recently by W. Wnuk. The property of containing a copy of ω by a Musielak−Orlicz space is characterized as follows. (1) A sequence space ℓΦ, where Φ=(φn), contains a copy of ω iff infn∈Nφn(∞)=0, where φn(∞)=limt→∞φn(t). (2) If the measure μ is atomless, then ω embeds isomorphically into LM(μ) iff the function M∞ is positive and bounded on some set A∈Σ of positive and finite measure, where M∞ (s)=limn→∞M(n,s), s∈S. In particular, (1)' ℓφ does not contain any copy of ω, and (2)' Lφ(μ), with μ atomless, contains a~copy W of ω iff φ is bounded, and every such copy W is uncomplemented in Lφ(μ).

2

Local structure of generalized Orlicz−Lorentz function spaces

Kolwicz P.

Commentationes Mathematicae

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2015

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Vol 55, No. 2

211--227

EN

We study the local structure of a separated point x in the generalized Orlicz-Lorentz space Λφ which is a symmetrization of the respective Musielak-Orlicz space Lφ. We present criteria for an LM point and a UM point, and sufficient conditions for a point of order continuity and an LLUM point, in the space Λφ. We prove also a characterization of strict monotonicity of the space Λφ .

3

Asymptotically isometric copies of c0 in Musielak-Orlicz spaces

Narloch A., Szymaszkiewicz L.

Opuscula Mathematica

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2014

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Vol. 34, no. 1

161--169

EN

Criteria in order that a Musielak-Orlicz function space Lφ as well as Musielak-Orlicz sequence space lφ contains an asymptotically isometric copy of c0 are given. These results extend some results of [Y.A. Cui, H. Hudzik, G. Lewicki, Order asymptotically isometric copies of c0 in the subspaces of order continuous elements in Orlicz spaces, Journal of Convex Analysis 21 (2014)] to Musielak-Orlicz spaces.

4

On the boundedness of sets in Musielak-Orlicz spaces

Shragin I. V.

Commentationes Mathematicae

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2013

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Vol. 53, No. 2

305--312

EN

The notions of metrical and topological vector boundedness of sets are considered in Musielak-Orlicz spaces. The space is called right if both these notions are equivalent. Necessary conditions of the rightness are established, and certain sufficient conditions are found.

5

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.

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

Characterization of the uniform convexity of the Besicovitch-Musielak-Orlicz spaces of almost periodic functions

Morsli M., Smaali M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2006

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Vol. 46, [Z] 2

215-231

EN

We introduce the new class of Besicovitch-Musielak-Orlicz spaces of almost periodic functions B'a.p..The uniform convexity of this space is characterized in terms of it’s generating functional [fi]

8

On some approximation problems in Musielak-Orlicz spaces of multifunctions

Kasperski A.

Demonstratio Mathematica

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2004

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Vol. 37, nr 2

393--406

EN

We introduce the Musielak-Orlicz spaces of multifunctions Xmphi and Xc,m,phi. We prove that these spaces are complete. Also, we get some convergence and approximation theorems in these spaces.

9

On property (B) in Musielak-Orlicz sequence spaces of Bochner type

Kolwicz P.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2000

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[Z] 40

103-126

EN

It is proved that the Musielak-Orlicz sequence space of Bochner type l_f(X) has property (B) if and only if both spaces lip and X have it also. It is considered the case of Luxemburg and Orlicz norm as well. In particular, the Lebesgue-Bochner sequence space l_p(X) has property (B) iff X has property (B)). As a corollary we also conclude that in Musielak-Orlicz sequence spaces equipped with the Luxemburg (Orlicz) norm property (B), nearly uniform convexity, drop property and reflexivity are in pairs equivalent. Such a theorem was proved directly in [4] in the case of Orlicz sequence space.

10

The necessary and sufficient conditions for triviality of a Musielak-Orlicz space and its subspace of finite elements

Shragin I.V.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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1999

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[Z] 39

167-172

EN

It is considered a Musielak-Orlicz space L generated by a function M:T x X -[0, ] (T is a measure space, X is a topological vector space) and its subspace E of finite elements. The necessary and sufficient conditions are obtained under which L or E contains only the zero function.

11

On uniform convexity of some spaces of multifunctions

Kasperski A.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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1999

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[Z] 39

87-95

EN

We introduce the appropriate space of multifunctions and the notions of uniform convexity of some subsets of this space. We get some remarks and theorems on uniform convexity of some subsets of this space. We prove that Musielak-Orlicz space of multifunctions is uniformly convex if the as-sumptions of Theorem 11.6 from [6] hold.