Results Math 72 (2017), 385–399
2016 The Author(s). This article is published
with open access at Springerlink.com
published online December 5, 2016
Results in Mathematics
On Sandwich Theorem for Delta-Subadditive
and Delta-Superadditive Mappings
Abstract. In the present paper, inspired by methods contained in Gajda
and Kominek (Stud Math 100:25–38, 1991) we generalize the well known
sandwich theorem for subadditive and superadditive functionals to the
case of delta-subadditive and delta-superadditive mappings. As a conse-
quence we obtain the classical Hyers–Ulam stability result for the Cauchy
functional equation. We also consider the problem of supporting delta-
subadditive maps by additive ones.
Mathematics Subject Classiﬁcation. 39B62, 39B72.
Keywords. Functional inequality, additive mapping, delta-subadditive
mapping, delta-superadditive mapping.
We denote by R, N the sets of all reals and positive integers, respectively,
moreover, unless explicitly stated otherwise, (Y,·) denotes a real normed
space and (S, ·) stands for not necessary commutative semigroup.
We recall that a functional f : S → R is said to be subadditive if
f(x · y) ≤ f(x)+f(y),x,y∈ S.
A functional g : S → R is called superadditive if f := −g is subadditive or,
equivalently, if g satisﬁes
g(x)+g(y) ≤ g(x · y),x,y∈ S.
If a : S → R is at the same time subadditive and superadditive then we say
that it is additive, in this case a satisﬁes the Cauchy functional equation
a(x · y)=a(x)+a(y)
The generalizations of the celebrated separation theorem of Rod´e (cf.
also K¨oning ) which represents a far-reaching generalization of the classical