Roman Ger

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 Some Trigonometric Functional Inequalities

We deal with d’Alembert’s and Wilson’s differences

Note on almost additive functions

f\left( {x + y} \right) + f\left( {x - y} \right) - 2f\left( x \right)f\left( y \right)

Additive selections and the stability of the Cauchy functional equation

and

Boundedness and continuity of additive and convex functionals

f\left( x \right)f\left( y \right) - f{\left( {\frac{{x + y}}{2}} \right)^2} + f{\left( {\frac{{x - y}}{2}} \right)^2}

On sums of linear and bounded mappings

respectively, assuming that their absolute values (or norms) are majorized by some function in a single variable. The superstability type results obtained are then used to characterize the functions