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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).
Content available remote Local structure of generalized Orlicz−Lorentz function spaces
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 .
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.
Content available remote On the boundedness of sets in Musielak-Orlicz spaces
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.
Content available remote On nonlinear differential equations in generalized Musielak-Orlicz spaces
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.
Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space L_ρ. We investigate the existence of common fixed points for semigroups of nonlinear mappings Tt : C→ C, i.e. a family such that To(x) = x, Ts+t = Ts(Tt(x)), where each T_t is either ρ-contraction or -nonexpansive. We also briefly discuss existence of such semigroups and touch upon applications to differential equations.
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]
Content available remote On some approximation problems in Musielak-Orlicz spaces of multifunctions
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.
Content available remote On property (B) in Musielak-Orlicz sequence spaces of Bochner type
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.
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.
Content available remote On uniform convexity of some spaces of multifunctions
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.
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