Let ℙ := (P_t)_t>o be a strongly continuous contraction semigroup of symmetric operators on L²(m). Let β be a Bochner subordinator and let ℙ^β be the subordinated semigroup of ℙ by means of β, i.e. P_t^β := ∫₀^∞ P_sβ_t(ds). We give in this paper an energy formula for the ℙ^β-potentials with finite energy in terms of the ℙ-exit laws and of β. We deduce an explicit energy formula for the α-potentials.

In this note we show that every affine transformation in the Euclidean space ℝⁿ, which has no fixed points and fulfils the inequality |f(x)f(y)| ≤ |xy| for any x and y has invariant straight line.

Two dual sequence functions describing some kind of local convexity and dimension of subspaces of linear metric spaces are introduced. It is shown that the functions give a useful tool in the investigations of fixed point properties of the Schauder type.

We study quasi-continuous functions on the product of two spaces provided they are separately continuous. We apply our results to actions of (semi-) groups on topological spaces and to the problem of the uniqueness of extensions of separately continuous functions.

The purpose of this paper is to establish the boundedness for some multilinear operators generated by Marcinkiewicz integral operators and Lipschitz functions on Hardy and Herz-Hardy spaces.

In the present paper we will prove the theorem concerning the mixed stability of the d'Alembert functional equation, i.e. we will show that if ɛ > 0,
s ≥ 1, δ = [2^s + \sqrt{2^2s+16ɛ+8}]/4, X is a real normed space and f: X→ℂ satisfies the inequality
|f(x+y) + f(x-y) - 2f(x)f(y)| ≤ ɛ(∥x∥^s + ∥y∥^s)
for all x,y∈X, then |f(x)| ≤ δ∥x∥^s for all x∈X such that ∥x∥ ≥ 1, or f(x+y) + f{x-y) = 2f(x)f(y) for all x,y∈X.

We prove that for given sets A₀,...,A_n ⊂ Rⁿ such as D_i ⊂ A_i for each i = 0,...,n exists a point x∈D such as d(x,A₀) = ... = d(x,A_n)- This proof gives an algorithm of finding the point x.

Report of Meeting. The Fifth Katowice-Debrecen Winter Seminar On Functional Equations and Inequalities February 2-5, 2005, Będlewo, Poland.