<Emphasis Type="Italic">ϕ</Emphasis>-pairs and common fixed points in cone metric spaces

In this paper we introduce a contractive condition, called ϕ-pair, for two mappings in the framework of cone metric spaces and we prove a theorem which assures existence and uniqueness of common fixed points for ϕ-pairs. Also we obtain a result on points of coincidence. These results extend and generalize well-known comparable results in the literature. The authors are supported by Università degli Studi di Palermo, R. S. ex 60%.     

A simple characterization of weighted Sobolev spaces with bounded multiplication operator

In this paper we give a simple characterization of weighted Sobolev spaces (with piecewise monotone weights) such that the multiplication operator is bounded: it is bounded if and only if the support of μ₀ is large enough. We also prove some basic properties of the appropriate weighted Sobolev spaces. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials.     

The Dichotomy Between Traces on d-sets F in R^n and the Density of D（R^n／Г） in Function Spaces

A space Apq^s （R^n） with A ： B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp（Г）, where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D（R^n／Г） is dense in it. Related dichotomy numbers are introduced and calculated.     

Smoothness of solutions of almost hypoelliptic equations

A two-dimensional linear differential operator P(D) = P(D ₁, D ₂) is called almost hypoelliptic if all derivatives D ^α P of the characteristic polynomial P(ζ) = P(ζ ₁, ζ ₂) are estimated by P(ζ). Assuming that {Ω _κ = (x ₁, x ₂) ∈ E ² : |x ₁| < κ, x ₂ ∈ R ¹}, the paper proves that if the width κ of the strip Ω _κ exceeds some C = C(P) > 0, then all solutions {u} of the almost hypoelliptic equation P(D)u = 0 in a Sobolev space are infinitely smooth functions with respect to x ₁.     

On the Space bvp（F） of Sequences of p-bounded Variation of Fuzzy Numbers

Recently, the space bvp of real or complex numbers consisting of all sequences whose differences are in the space lp has been studied by Basar, Altay [Ukrainian Math. J. 55（1）（2003）, 136-147], where 1 ≤ p ≤ ∞. The main purpose of the present paper is to introduce the space bvp（F） of sequences of p-bounded variation of fuzzy numbers. Moreover, it is proved that the space bvp（F） includes the space lp（F） and also shown that the spaces bvp（F） and lp（F） axe isomorphic for 1 ≤ p ≤∞. Furthermore, some inclusion relations have been given.     

Totally skew embeddings of manifolds

We study a version of Whitney’s embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an n-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points? This problem is related to the generalized vector field problem, existence of non-singular bilinear maps, and the immersion problem for real projective spaces. We use these connections and other methods to obtain several specific and general bounds for the desired dimension.     

Non-linear ground state representations and sharp Hardy inequalities

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.     

Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws

The stability of traveling wave solutions of scalar viscous conservation laws is investigated by decomposing perturbations into three components: two far-field components and one near-field component. The linear operators associated to the far-field components are the constant coefficient operators determined by the asymptotic spatial limits of the original operator. Scaling variables can be applied to study the evolution of these components, allowing for the construction of invariant manifolds and the determination of their temporal decay rate. The large time evolution of the near-field component is shown to be governed by that of the far-field components, thus giving it the same temporal decay rate. We also give a discussion of the relationship between this geometric approach and previous results, which demonstrate that the decay rate of perturbations can be increased by requiring that initial data lie in appropriate algebraically weighted spaces.     

Inversion of the Bessel potential operator in weighted variable Lebesgue spaces

We study the inversion problem of the Bessel potential operator within the frameworks of the weighted Lebesgue spaces with variable exponent. The inverse operator is constructed by using approximative inverse operators. This generalizes some classical results to the variable exponent setting.