On Functions with the Cauchy Difference Bounded by a Functional

Włodzimierz Fechner. "On Functions with the Cauchy Difference Bounded by a Functional." Bulletin of the Polish Academy of Sciences. Mathematics 52.3 (2004): 265-271. <http://eudml.org/doc/280650>.

@article{WłodzimierzFechner2004,
abstract = { K. Baron and Z. Kominek [2] have studied the functional inequality f(x+y) - f(x) - f(y) ≥ ϕ (x,y), x, y ∈ X, under the assumptions that X is a real linear space, ϕ is homogeneous with respect to the second variable and f satisfies certain regularity conditions. In particular, they have shown that ϕ is bilinear and symmetric and f has a representation of the form f(x) = ½ ϕ(x,x) + L(x) for x ∈ X, where L is a linear function. The purpose of the present paper is to consider this functional inequality under different assumptions upon X, f and ϕ. In particular we will give conditions which force biadditivity and symmetry of ϕ and the representation f(x) = ½ ϕ(x,x) - A(x) for x ∈ X, where A is a subadditive function. },
author = {Włodzimierz Fechner},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {functional inequality; subadditive; quadratic and biadditive functionals; linear space},
language = {eng},
number = {3},
pages = {265-271},
title = {On Functions with the Cauchy Difference Bounded by a Functional},
url = {http://eudml.org/doc/280650},
volume = {52},
year = {2004},
}

TY - JOUR
AU - Włodzimierz Fechner
TI - On Functions with the Cauchy Difference Bounded by a Functional
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2004
VL - 52
IS - 3
SP - 265
EP - 271
AB - K. Baron and Z. Kominek [2] have studied the functional inequality f(x+y) - f(x) - f(y) ≥ ϕ (x,y), x, y ∈ X, under the assumptions that X is a real linear space, ϕ is homogeneous with respect to the second variable and f satisfies certain regularity conditions. In particular, they have shown that ϕ is bilinear and symmetric and f has a representation of the form f(x) = ½ ϕ(x,x) + L(x) for x ∈ X, where L is a linear function. The purpose of the present paper is to consider this functional inequality under different assumptions upon X, f and ϕ. In particular we will give conditions which force biadditivity and symmetry of ϕ and the representation f(x) = ½ ϕ(x,x) - A(x) for x ∈ X, where A is a subadditive function.
LA - eng
KW - functional inequality; subadditive; quadratic and biadditive functionals; linear space
UR - http://eudml.org/doc/280650
ER -