Alice Simon

On functions having Cauchy differences in some prescribed sets

SummaryAssumeE is a real topological vector space,F is a real Banach space,K is a discrete subgroup ofF andC is a symmetric, convex and compact subset ofF such thatK ∩ (6C) = {0}. If a functionh:E → F is continuous at at least one point andh(x + y) − h(x) − h(y) ∈ K + C for allx, y ∈ E, then there exists a continuous linear functiona:E → F such thath(x) − a(x) ∈ K + C for everyx ∈ E.

Eine Charakterisierung Von Polynomialen Funktionen Mittels Der Dinghasschen Intervall-Derivierten

ZusammenfassungEs sei J ein nicht-ausgeartetes Intervall von ℝ. Für eine Funktion ƒ: J → ℝ gilt Δnhƒ(x) = 0 (x, x + nh ∈ J) genau dann, wenn Dnƒ(x) = 0 (x ∈ J) ist. Dabei bezeichnet Dnƒ(x) die n-te Dinghassche Intervall-Derivierte.

Parabolic Differential Equations in Ordered Banach Spaces of Bounded Functions

Let A be a set, and let E be the Banach space of bounded functions ξ : A → R, equipped with its natural order. With a rectangle R = (a,b) × (0,T] let F(x,t,ξ) : R × E → E be a bounded, continuous function satisfying a local Hölder condition and being quasimonotone increasing with respect to ξ. Then there exists a solution u: [a,b] × [0,T] → E of the problem ut(x,t) − uxx(x,t) = F(x,t,u(x,t)) ((x,t) ∈ R), u(x,t) = 0 ((x,t) ∈ R ∖ R).