Zdzisław Dzedzej

Dimension of the solution set for differential inclusions

Our paper is naturally divided into two short sections. In the first part we make some observations on results obtained by J.Saint Raymond in [7], [8] concerning topological dimension of a fixed point set of convex-valued contraction. Slightly modifying the proofs, we generalize his theorems to the case of arbitrary closed convex subsets of a Banach space. In the second section we apply the previous result to the solution set of a Cauchy problem with set-valued right-hand side. We prove that it has an infinite dimension if only values of the right-hand side function are at least onedimensional.

Equivariant degree of convex-valued maps applied to set-valued BVP

An equivariant degree is defined for equivariant completely continuous multivalued vector fields with compact convex values. Then it is applied to obtain a result on existence of solutions to a second order BVP for differential inclusions carrying some symmetries.

Borsuk-Ulam type theorems on product spaces II

A generalization of the theorem of Zhong on the product of spheres to multivalued maps is given. We prove also a stronger result of Bourgin-Yang type.

On the Nielsen fixed point theory for multivalued mappings

We present J. Jezierski’s approach to the Nielsen fixed point theory for a broad class of multivalued mappings [Je1]. We also describe some generalizations and different techniques existing in the literature. 1. Notations and definitions. Let X,Y be metric spaces. By a multivalued mapping Φ : X → Y we mean a transformation Φ : X → 2 with nonempty compact values. Many notions known for singlevalued transformations can be generalized to multivalued mappings. For A ⊂ X the image of A is the set Φ(A) = ⋃ x∈A Φ(x). The set ΓΦ = {(x, y) : y ∈ Φ(x)} is called the graph of Φ. There are several notions of continuity. Definition 1. The mapping Φ is lower semicontinuous (lsc) (respectively upper semicontinuous (usc)) if for every open subset V ⊂ Y the set Φ−1(V ) = {x ∈ X : Φ(x) ∩ V 6= ∅} (respectively Φ−1 + (V ) = {x ∈ X : Φ(x) ⊂ V }) is an open subset of X. If Φ is both lsc and usc, then we say that Φ is continuous. In the singlevalued case these three notions coincide. For basic properties and examples of usc (lsc) mappings we refer the reader to [AC] or [Gor]. In order to have a nontrivial fixed point theory we have to consider special classes of multivalued mappings. Definition 2. A subset A ⊂ X satisfies the ?-property if it is nonempty, connected and there exists an open neighbourhood U of A such that each loop in U is homotopic (with fixed ends) in X to a constant loop. 1991 Mathematics Subject Classification: 55M20. Research supported by UG grant BW 5100-5-0057-6. The paper is in final form and no version of it will be published elsewhere.