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Content available remote Summing multi-norms defined by Orlicz spaces and symmetric sequence space
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.
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 (…).
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.
Content available remote On (∆2) condition in density-type topologies
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) = ∞.
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 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.
Content available remote Uniformly ž-Continuous Topologies on Orlicz-Bochner Spaces
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.
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.
Content available remote Weak sequential completeness and compactness in topological function spaces
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.
Content available remote Orthogonally complemented hyperplanes in Orlicz function spaces
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.
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.
Content available remote On nonlinear integral equations in some functions spaces
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.
Content available remote Integral of function with values in complete modular space
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.
Content available remote Orthogonal uniform convexity in Koethe spaces and Orlicz spaces
We study a geometric property in Koethe spaces which is called orthogonal uniform convexity (UC1). It was introduced in [19]. We prove that the class of Koethe spaces with property (UC1) is a proper subset of the class of uniformly monotone and P-convex Koethe spaces. Next we consider connections between (UC1) and property (Beta) of Rolewicz. We shown that the implication (UC1) -> (Beta) is true in any Koethe 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) -> (UC1)-> (Beta) in any Koethe sequence space and the converse of any of these implications is not true. Furthermore, the implications (UC) -> (Beta) -> (UC1) hold in any Koethe function space and the second one cannot be reversed.
Content available remote Compact midpoint local uniform convexity in Orlicz spaces
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.
Content available remote Modular function spaces and control functions of almost everywhere convergence
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.
Content available remote Closedness of the set of extreme points in Orlicz spaces with Orlicz norm
In this paper the problem of closedness of the set of exreme points of the unit ball of Orlicz spaces equipped with Orlicz norm over a non-negative, atomless, [ro]-finite and complete measure space in discussed.
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