Criteria for monotonicity properties of Musielak-Orlicz spaces equipped with the Amemiya norm

Criteria for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity, and uniform monotonicity of Musielak-Orlicz spaces over any σ-finite and complete measure space, endowed with the Amemiya norm are given. The fact that the spaces are considered over arbitrary σ-finite measure space is essential because, as it is shown in Example 3, the Musielak-Orlicz spaces need not be strictly monotone even if their restrictions to the nonatomic part and the purely atomic part are strictly monotone.     

In some symmetric spaces monotonicity properties can be reduced to the cone of rearrangements

Geometric properties being the rearrangement counterparts of strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in some symmetric spaces are considered. The relationships between strict monotonicity, upper local uniform monotonicity restricted to rearrangements and classical monotonicity properties (sometimes under some additional assumptions) are showed. It is proved that order continuity and lower uniform monotonicity properties for rearrangements of symmetric spaces together are equivalent to the classical lower local uniform monotonicity for any symmetric space over a \({\sigma}\)-finite complete and non-atomic measure space. It is also showed that in the case of order continuous symmetric spaces over a \({\sigma}\)-finite and complete measure space, upper local uniform monotonicity and its rearrangement counterpart shortly called ULUM* coincide. As an application of this result, in the case of a non-atomic complete finite measure a new proof of the theorem which is already known in the literature, giving the characterization of upper local uniform monotonicity of Orlicz–Lorentz spaces, is presented. Finally, it is proved that every rotund and reflexive space X such that both X and X* have the Kadec-Klee property is locally uniformly rotund. Some other results are also given in the first part of Sect. 2.     

MONOTONE POINTS IN ORLICZ-BOCHNER SEQUENCE SPACES

In Orlicz-Bochner sequence spaces endowed with Orlicz norm and Luxemburg norm, points of lower monotonicity, upper monotonicity, lower local uniform monotonicity and upper local uniform monotonicity are characterized.     

Points of monotonicity in Orlicz-Lorentz function spaces

In Orlicz-Lorentz function space ?_M,ω[0,γ), points of upper monotonicity, lower monotonicity, upper local uniform monotonicity (γ<∞) and lower local uniform monotonicity (γ<∞) are characterized. As the corollary, we can easily obtain the criteria for strict monotonicity, upper local uniform monotonicity (γ<∞) and lower local uniform monotonicity (γ<∞) of ?_M,ω[0,γ).     

Monotonicity and best approximation in Orlicz-Sobolev spaces with the Luxemburg norm

Criteria for strict monotonicity, upper (lower) locally uniform monotonicity and uniform monotonicity of Orlicz-Sobolev spaces with the Luxemburg norm are given. Some applications to best approximation are presented.     

Orlicz-Sobolev空间单调性及其在最佳逼近中的应用

改进了Hudzik,Kurc关于最佳逼近中的结果,给出了赋Orlicz范数的Orlicz- Sobolev空间具有一致单调性、局部一致单调性和严格单调性的充要条件、单调系数的数值,以及在最佳逼近中的应用.     

Local monotonicity structure of Calderón-Lozanovskiǐ spaces

We first prove that if x is an element on the unit sphere of arbitrary Köthe space E, x is strickly positive μ-a.e. and x is an LM-point, then x is an UM-point. Criteria for lower and upper monotone points in Calderón-Lozanovskiǐ spaces E_? are presented. Points of lower local uniform monotonicity and upper local uniform monotonicity in E_? are also considered. Some sufficient conditions and necessary conditions for these properties of a given point x in S(E_?⁺) are given.     

Basic topological and geometric properties of Cesàro-Orlicz spaces

Necessary and sufficient conditions under which the Cesàro-Orlicz sequence spaceces _ϕ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesàro-Orlicz spacesces _ϕ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements inces _ϕ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spacesces _ϕ are given.     

Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators

In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a Banach space with locally monotone operators, which is a generalization of the classical result for monotone operators. In particular, we show that local monotonicity implies pseudo-monotonicity. The main results are applied to PDE of various types such as porous medium equations, reaction–diffusion equations, the generalized Burgers equation, the Navier–Stokes equation, the 3D Leray-α model and the p-Laplace equation with non-monotone perturbations.     

Uniqueness and transport density in Monge's mass transportation problem

Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures on , find the measure preserving map s(x) between them which minimizes the average distance transported. Here distance can be induced by the Euclidean norm, or any other uniformly convex and smooth norm on . Although the solution is never unique, we give a geometrical monotonicity condition singling out a particular optimal map s(x). Furthermore, a local definition is given for the transport cost density associated to each optimal map. All optimal maps are then shown to lead to the same transport density . Received: 18 December 2000 / Accepted: 11 May 2001 / Published online: 19 October 2001     

Uniform nonsquareness and locally uniform nonsquareness in Orlicz–Bochner function spaces and applications

In this paper, criteria for uniform nonsquareness and locally uniform nonsquareness of Orlicz–Bochner function spaces equipped with the Orlicz norm are given. Although, criteria for uniform nonsquareness and locally uniform nonsquareness in Orlicz function spaces were known, we can easily deduce them from our main results. Moreover, we give a sufficient condition for an Orlicz–Bochner function space to have the fixed point property.     

BLO spaces associated with the Ornstein–Uhlenbeck operator

Let (Rn,|⋅|,dγ) be the Gauss measure metric space, where Rn denotes the n-dimensional Euclidean space, |⋅| the Euclidean norm and for all x∈Rn the Gauss measure. In this paper, for any a∈(0,∞), the authors introduce some BLOa(γ) space, namely, the space of functions with bounded lower oscillation associated with a given class of admissible balls with parameter a. Then the authors prove that the noncentered local natural Hardy–Littlewood maximal operator is bounded from BMO(γ) of Mauceri and Meda to BLOa(γ). Moreover, a characterization of the space BLOa(γ), via the local natural maximal operator and BMO(γ), is given. The authors further prove that a class of maximal singular integrals, including the corresponding maximal operators of both imaginary powers of the Ornstein–Uhlenbeck operator and Riesz transforms of any order associated with the Ornstein–Uhlenbeck operator, are bounded from L^∞(γ) to BLOa(γ).     

Points of monotonicity in F-normed Orlicz function spaces

Aequationes mathematicae - In this paper points of lower strict monotonicity, upper strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity of Orlicz function...     

On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces

Generalized Orlicz–Lorentz sequence spaces λ_φ generated by Musielak‐Orlicz functions φ satisfying some growth and regularity conditions (see [28] and [33]) are investigated. A regularity condition δ^λ ₂ for φ is defined in such a way that it guarantees many positive topological and geometric properties of λ_φ. The problems of the Fatou property, the order continuity and the Kadec–Klee property with respect to the uniform convergence of the space λ_φ are considered. Moreover, some embeddings between λ_φ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of λ_φ, their subspaces of order continuous elements and finite dimensional subspaces are presented. This paper generalizes the results from [19], [4] and [17]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

In this paper we introduce two mappings associated with the lower and upper semi-inner product (·, ·) _i and (·, ·) _S and with semi-inner products [·, ·] (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.     

Rotundity and monotonicity properties of selected Cesàro function spaces

Positivity - We study rotundity, strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in some classes of Cesàro function spaces. We present full criteria...     

Unified boundedness,periodicity, and stability in ordinary and functional differential equations

Summary We discuss a unified theory of periodicity of dissipative ordinary and functional differential equations in terms of uniform boundedness. Sufficient conditions for the uniform boundedness are given by means of Liapunov functionals having a weighted norm as an upper bound. The theory is developed for ordinary differential equations, equations with bounded delay, and equations with infinite delay.On leave from Anhui University, Hefei, Anhui, People's Republic of China     

Quadrature Rules for the Surface Integral of the Unit Sphere Based on Extremal Fundamental Systems

Quadrature rules for the surface integral of the unit Sphere S^r–1 based on an extremal fundamental system, i.e., a nodal system which provides fundamental Lagrange interpolatory polynomials with minimal uniform norm, are investigated. Such nodal systems always exist; their construction has been given in earlier work. Here the main results is that the corresponding interpolatory quadrature for the space of homogeneous polynomials of degree two is equally weighted for arbitrary r, and hence positive. For the full quadratic polynomial space we can prove positivity of the weights, only.     

Comparison principles applied to obstacle problems with penalties

Penalty methods form a well known technique to embed elliptic variational inequality problems into a family of variational equations (cf. [6], [13], [17]). Using the specific inverse monotonicity properties of these problems L _∞-bounds for the convergence can be derived by means of comparison solutions. Lagrange duality is applied to estimate parameters involved.

For piecewise linear finite elements applied on weakly acute triangulations in combination with mass lumping the inverse monotonicity of the obstacle problems can be transferred to its discretization. This forms the base of similar error estimations in the maximum norm for the penalty method applied to the discrete problem.

The technique of comparison solutions combined with the uniform boundedness of the Lagrange multipliers leads to decoupled convergence estimations with respect to the discretization and penalization parameters.