Volume 5, Issue 2

Let ( S n : n ≥ 0) be a random walk on a hypergroup (ℝ + , ∗) of polynomial growth. We show that the possible limit laws of the form c n · S n → μ (weakly), c n > 0, are the stable laws of the Bessel–Kingman hypergroup (ℝ + , # α ) for a specific α ≥ –1/2 depending on the growth of the hypergroup. Furthermore we describe the domain of attraction (with respect to the convolution ∗) of these stable laws in terms of the regular variation of the Fourier transform.

It is shown that for each k > 1, if ƒ is a Baire one function and ƒ is the product of k bounded Darboux (quasi–continuous) functions, then ƒ is the product of k bounded Darboux (quasi–continuous) Baire one functions as well.

Two equivalent metrics can be compared, with respect to their uniform properties, in several different ways. We present some of them, and then use one of these conditions to characterize which metrics on a space induce the same lower Hausdorff topology on the hyperspace. Finally, we focus our attention to complete metrics.

Existence of weak solutions for systems of quasilinear degenerate parabolic equations with non-diagonal main part and nonlinear boundary conditions is proved. Under some restrictions we find also L ∞ - bounds for the solutions.

A σ -finite Borel measure in a topological space is called residual if each nowhere dense set has measure zero. We show that in various types of spaces without isolated points there are no residual measures. Among these spaces are e.g. σ -spaces, locally metrizable spaces, locally separable spaces, spaces that have a σ -point–finite π -base, submanifolds.

The aim of this work is to present a new result about estimation of the hypertangent normal cone of an intersection without using directionally Lipschitz assumptions.

The differential ℂ-algebra of generalized functions of J.- F. Colombeau contains the space of Schwartz distributions as a ℂ-vector subspace and the notion of ‘association’ in that algebra generalizes the equality in . This is particularly useful for evaluation of distribution products, as they are embedded in , in terms of distributions again. The paper is devoted to results on particular products of distributions with coinciding singularities, as well as to a general property of Colombeau product of distributions together with some applications. All formulas obtained, when restricted to dimension one, are easily transformed into the setting of regularized model products in classical distribution theory.

Sufficient conditions for oscillation of all solutions of a class of nonlinear impulsive differential equations of the first order with advanced argument and fixed moments of impulse effect are found.

We consider a cancer started with a single cancerous cell which spreads through an epithelial basal layer according to the Williams–Bjerknes tumour model on the lattice . We prove that the expected number μ ( t ) of cancerous cells at time t satisfies for all ρ > 2.

In this paper, the concept of metrically weakly inward set–valued mappings is introduced. The relationship between the metrically weakly inward mappings and other inward type mappings is considered in normed spaces. A positive answer of the Caristi's open problem is given. These results generalize the corresponding results in recent literature.