Zbigniew Gajda

Subadditive Multifunctions and Hyers-Ulam Stability

A multifunction F from an Abelian semigroup (S,+) into the family of all nonempty closed convex subsets of a Banach space (X, ║·║) is called subadditive provided that F(x+y) ⊂ F(x) + F(y) for all x, y ∈ S. We show that if all the values of a subadditive multifunction F are uniformly bounded then F admits an additive selection, i.e. a homomorphism a: S → X such that a(x) ∈ F(x) for all x ∈ S. An abstract version of this result is also presented. The subject is motivated by (and strictly related to) the Hyers-Ulam stability problem.

On stability of the Cauchy equation on semigroups

SummaryWe say that Hyerss theorem holds for the class of all complex-valued functions defined on a semigroup (S, +) (not necessarily commutative) if for anyf:S → ℂ such that the set {f(x + y) − f(x) − f(y): x, y ∈ S} is bounded, there exists an additive functiona:S → ℂ for which the functionf − a is bounded.Recently L. Székelyhidi (C. R. Math. Rep. Acad. Sci. Canada8 (1986) has proved that the validity of Hyerss theorem for the class of complex-valued functions onS implies its validity for functions mappingS into a semi-reflexive locally convex linear topological spaceX. We improve this result by assuming sequential completeness of the spaceX instead of its semi-reflexiveness. Our assumption onX is essentially weaker than that of Székelyhidi. Theorem.Suppose that Hyerss theorem holds for the class of all complex-valued functions on a semigroup (S, +) and let X be a sequentially complete locally convex linear topological (Hausdorff) space. If F: S → X is a function for which the mapping (x, y) → F(x + y) − F(x) − F(y) is bounded, then there exists an additive function A : S → X such that F — A is bounded.

On functional equations associated with characters of unitary representations of groups

SummaryThis is mainly concerned with the following functional equation:(*)

Christensen measurable solutions of generalized Cauchy functional equations

f\left( x \right)f\left( y \right) = \frac{1}{{\left| \Phi \right|}}\sum\limits_{\varphi \in \Phi } {M_t \left( {f\left( {xt\varphi \left( y \right)t^{ - 1} } \right)} \right)} , x,y \in G,

On quadratic functionals and some properties of Hamel bases

whereG is a topological group,M stands for the invariant mean on the space of almost periodic functions defined onG and Ф is a finite set of morphisms ofG. It is shown that almost periodic solutions of this equation can be expressed by means of characters of finite-dimensional irreducible unitary representations of the groupG. In the case whereG is compact, we study an integral version of Eq. (*) and we determine its solutions in the space

Local stability of the functional equation characterizing polynomial functions

\mathfrak{L}_1