An Introduction to the Theory of Functional Equations and Inequalities

The properties of functional inclusions and Hyers–Ulam stability

“…Ulam in a talk at a conference at the Wisconsin University in 1940 and it represents the starting point of the Hyers-Ulam stability theory of functional equations (see [8,9]). The subject was later strongly developed by many authors, see for example: [1,4,11,12,16].…”

Convexity with respect to families of means

“…The basic properties and of these classes of functions are described in details in the monographs [2] and [9]. A functions a : R → R is said to be a derivation if it is additive and satisfies the Leibniz Rule, that is, if d fulfills the following functional equation…”

“…Moreover, by taking an algebraic base of R, any real-valued function can uniquely be extended to a derivation (cf. [9]). The next theorem is one of the main results of this paper.…”

“…Jensen convex functions, in general, are not necessarily continuous. In view of the existence of a Hamel base of R considered as a vector space over Q, one can construct a discontinuous additive function ( [4], [9]), such a function is automatically a discontinuous Jensen convex function. On the other hand, certain weak regularity properties of Jensen convex functions ensure that they are also convex and henceforth continuous.…”

An application of the Choquet theorem to the study of randomly-superinvariant measures

“…Taking into account that F (x) is a non-decreasing function, we have that (1.7) holds if and only if F (x) is convex (see [5]). Moreover, it is not difficult to check that ∆ h µ(B) ≥ 0 holds for all B ∈ B(R) if and only if this condition is satisfied for all B of the form (1.6).…”