Marek Kuczma

An Introduction to the Theory of Functional Equations and Inequalities

Preliminaries.- Set Theory.- Topology.- Measure Theory.- Algebra.- Cauchys Functional Equation and Jensens Inequality.- Additive Functions and Convex Functions.- Elementary Properties of Convex Functions.- Continuous Convex Functions.- Inequalities.- Boundedness and Continuity of Convex Functions and Additive Functions.- The Classes A, B, ?.- Properties of Hamel Bases.- Further Properties of Additive Functions and Convex Functions.- Related Topics.- Related Equations.- Derivations and Automorphisms.- Convex Functions of Higher Orders.- Subadditive Functions.- Nearly Additive Functions and Nearly Convex Functions.- Extensions of Homomorphisms.

On some alternative functional equations

(cf. [1], [2], [4}-[8]). The most general results in this direction were claimed by P. Fischer [1]. They read as follows ( yX denotes here the class of all functions [ : X ~ Y): (A) If (X, +) is a semigroup and (Y, +, .) is an integral domain, then in the class ,,ix the functional equations (1) and (2) are equivalent. (B) If (X, +) is a semigroup and (Y, +, .) is a commutative ring, then in the class yX the functional equations (1) and (2) are equivalent if and only if Y does not contain genuine nilpotent elements. However, Fischers proof of (A) is erroneous and the results are false. In this situation, the most general correct results concerning the equivalence of (1) and (2) are due to E. Vincze [7] and to H. Swiatak and M. Hossztl [5]. The first author proved the equivalence of (1) and (2) in the class yx , where (X, +) is an arbitrary semigroup, and (Y, +, .) is a suitable number field (and hence, in particular, is of characteristic zero). H. Swiatak and M. Hosszfi have dealt with a more general equation. Their result applied to equation (1) yields the equivalence of (1) and (2) in the class yX, where (X, +) is an arbitrary group and (Y, +, .) is an integral domain of characteristic zero; however, the existence of the multiplicative unit in Y and the commutativity of the + operation in Y were not needed.

On measures connected with the Cauchy equation

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Generalizations Of A “folk-Theorem” On Simple Functional Equations In A Single Variable

It is known (“mathematical folklore”) that, to every function defined on [1,2], there exists a solution of f(2x) = 2f(x) on ]0,∞[ of which the given function is a restriction to [1,2]. With a little care in the definition on [1,2], with still a lot of arbitrariness left, the resulting solution will be continuous, even C∞ on ]0,∞[ (a behaviour markedly different from that of the Cauchy equation f(x + y) = f(x) + f(y), which has f(x) = cx as only continuous solution on ]0,∞[, even though, with y = x, it degenerates into the above equation). If 0 is added to the domain and we choose the “arbitrary function” bounded on [1,2[, then the solution will even be continuous (from the right) at 0. However, if f is supposed to be differentiable at 0 (from the right), then f(x) = cx is the only solution on [0,∞[. p In this paper we present similar and further results concerning general, Cn (n ≤ ∞), analytic, locally monotonie or γ-th order convex solutions of the somewhat more general equation f(kx) = kγf(x) (k ≠ 1 a positive, γ a real constant), which seems to be of importance in meterology. Some of the results are not quite what one expects.

Solutions of a functional equation, convex of higher order

We determine the general solutions of f(kx) = k γ f (x) (k ∈]0,∞[\{1}, γ ∈ ℝ are constants, x ∈]0,∞[variable), in particular those which are convex of order γ.

Regularly varying solutions of a linear functional equation

We are concerned with the problem of the existence and uniqueness of regularly varying (in Karamatas sense) solutions ϕ of the linear functional equation in a right neighbourhood of x = 0. Under suitable conditions on the given functions f and h , the uniqueness of solutions depends essentially on whether the series Σh ∘ f 1 converges or diverges; here f i denotes the i -th functional iterate of f . The existence of solutions may be proved under further assumptions. The case of the more general linear functional equation may be reduced to that of equation (*).

On the lower hull of convex functions

SummaryLet (X, ℱ) be a topological space. For any functionf: D→[− ∞, ∞) (whereD ⊂ X), thelower hull mf:D →[− ∞, ∞) off is defined by

Nonnegative Continuous Solutions of a Functional Inequality in a Single Variable

m_f (x) = m_{f\left| T \right.} (x) = \mathop {\sup \inf }\limits_{U \in T_x \in U \cap D} f(t),x \in D,