Here are the titles and abstracts of the seminars which were held in 2006:

2006-12-19 Seminar

Ryszard Urbański
Minimal representations in Minkowski-Radström-Hörmander classes generated by pairs of closed convex and compact sets

Defining suitable topology in the collection of all closed convex subsets of a topological vector space and using ordinal cancellation law and some properties of the cone of compact sets, we obtain representation theorem.

2006-12-12 Seminar

Ryszard Urbański, Jerzy Grzybowski
On some properties of the Minkowski-Hörmander-Radström lattice

During the lecture, some properties of the Minkowski-Hörmander-Radström lattice will be shown, including Archimedeanness, existence of axial elements, orthogonality.

2006-12-05 Seminar

mgr Danuta Borowska, dr Jerzy Grzybowski
Subtraction and Sallee's elements in semigroups

We can define the Minkowski sum and the Minkowski subtraction in the cone of convex and compact sets. Properties of the the Minkowski subtraction were studied by G.T. Sallee, who formulated the definition of a family of sets, such that the intersection of any collection of translations of the given set is the summand of this set. The Minkowski subtraction was generalized to semigroups by H. Przybycień. During the lecture we will present some properties of the subtraction and we will give the generalization of the notion of a Sallee's set for the semigroups' elements. Moreover, interesting examples of semigroups will be presented.

2006-11-28 Seminar

Marcin Borkowski
On some Krasnoselskii-type theorem

We present a fixed point theorem for a sum of two multifunctions acting on a weakly compact, hyperconvex subset of a normed space. The theorem is a multivalued version of a result of D. Bugajewski.

2006-11-21 Seminar

Piotr Kasprzak
Flow invariance over closed sets under general dissipativity conditions
(Continuation of the 2006-11-14 Seminar)

2006-11-14 Seminar

Piotr Kasprzak
Flow invariance over closed sets under general dissipativity conditions
(Continuation of the 2006-11-07 Seminar)

2006-11-07 Seminar

Piotr Kasprzak
Flow invariance over closed sets under general dissipativity conditions

During the lecture some flow invariance results will be given for differential equations over Banach spaces under generalized dissipativity conditions. The technical aspect of this approach consist of constructing the approximate solutions and the parwise refinements of these via maximality procedures. The lecture is based on the paper of M. Turinici "Flow invariance over closed sets under general dissipativity conditions", Sci. Ann. Univ. Agric. Sci. Vet. Med. 47 (2004), no. 2, 35--67.

2006-10-27 Seminar

This lecture will be held on Friday on 15.00 in room B3-38

prof. dr hab. Grzegorz Lewicki (Institute of Mathematics of Jagiellonian University)
Spaces with maximal projection constant

Let $X$ be a real Banach space and $Y\subset X$ its closed subspace. Denote by $\mathcal{P}(X,Y)$ the set of all linear and continuous projections from $X$ on $Y$ . Let $\lambda(Y,X)=\inf\{\|P\|:P\in\mathcal{P}(X,Y)\}$ . For $n$ -dimensional Banach space $Y$ let $\lambda(Y) = \sup \{ \lambda(Y,X): Y \subset X \}$ . The constant $\lambda(Y)$ is called the absolute projection constant. It is a well-known fact that for $n$ -dimensional Banach space $Y$ , $\lambda(Y) = \lambda(Y, l_{\infty})$ . For fixed $n, N \in \mathbb{N}$ , $n < N$ let $\lambda^N_n(Y) = \sup\{ \lambda(Y) : Y \subset l_{\infty}^{(N)},\, \dim(Y) = n\}$ . During the lecture I will present some results and open problems concerning effective determining $n$ -dimensional spaces $Y\subset l_{\infty}^{(N)}$ such that $\lambda(Y, l_{\infty}^{(N)}) = \lambda^N_n$ .

2006-10-24 Seminar

Daria Bugajewska
$BV_{\phi}$ -solutions of nonlinear integral equations
(Continuation of the 2006-10-10 Seminar)

2006-10-17 Seminar

Hubert Przybycień
Some properties of the Minkowski sum of measurable sets

In this talk we will deal with some properties of the Minkowski sum of measurable sets. We will present the proof of Steinhaus theorem, which states that the Minkowski sum of positive measure sets contains a segment, as well as its application. Moreover we will give an example of a measurable set A, such that the set A+A is non-measurable.

2006-10-10 Seminar

Daria Bugajewska
$BV_{\phi}$ -solutions of nonlinear integral equations

In this talk we will deal with solutions of nonlinear Hammerstein and Volterra-Hammerstein integral equations in the space of functions of bounded $\phi$ -variation in the sense of Young. We will discuss the existence and in some cases the existence and uniqueness of local and global solutions in this class. Real-valued as well as vector-valued functions will be under our consideration. The method of our proofs is based on an application of the Banach contraction principle as well as the Leray-Schauder alternative for contractions.