Remarks on Lipschitzian mappings and some fixed point theorems.
A separation theorem for -convex functions.
A simultaneous system of functional inequalities and mappings which are weakly of a constant sign.
Geometrical convexity and generalizations of the Bohr-Mollerup theorem on the gamma function.
A sandwich with convexity.
Integrable solutions of functional equations
ContentsIntroduction............................................................................................................................................................................... 50. Explanatory notes, definitions and a lemma................................................................................................................. 51. Some fixed point theorems..................................................................................................................................................
Functional inequality characterizing nonnegative concave functions in (0, ?)k.
Remark on BV-solutions of a functional equation connected with invariant measures.
Functional inequality characterizing nonnegative concave functions in (0, ?)k. (Summary).
On a ?-Wright convexity and the converse of Minkowsi's inequality.
On a characterization of -norm
On subaddidive and Ψ-additive mappings
On subadditive functions and ψ-additive mappings
In [4], assuming among others subadditivity and submultiplicavity of a function ψ: [0, ∞)→[0, ∞), the authors proved a Hyers-Ulam type stability theorem for “ψ-additive” mappings of a normed space into a normed space. In this note we show that the assumed conditions of the function ψ imply that ψ=0 and, consequently, every “ψ-additive” mapping must be additive
The converse of the Hölder inequality and its generalizations
Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then there exist...
Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem
A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions , k ≥ 2, denoted by , is considered. Some properties of , including “associativity” assumed in the Kolmogorov-Nagumo theorem, are shown. Convex and affine functions involving this type of means are considered. Invariance of a quasi-arithmetic mean with respect to a special mean-type mapping built of generalized means is applied in solving a functional equation. For a sequence of...
A functional equation related to an equality of means problem
The functional equation (F(x)-F(y))/(x-y) = (G(x)+G(y))(H(x)+H(y)) where F,G,H are unknown functions is considered. Some motivations, coming from the equality problem for means, are presented.
Invariance identity in the class of generalized quasiarithmetic means
An invariance formula in the class of generalized p-variable quasiarithmetic means is provided. An effective form of the limit of the sequence of iterates of mean-type mappings of this type is given. An application to determining functions which are invariant with respect to generalized quasiarithmetic mean-type mappings is presented.
Affine and convex functions with respect to the logarithmic mean
The class of all functions f:(0,∞) → (0,∞) which are continuous at least at one point and affine with respect to the logarithmic mean is determined. Some related results concerning the functions convex with respect to the logarithmic mean are presented.
Fixed points and iterations of mean-type mappings
Let (X, d) be a metric space and T: X → X a continuous map. If the sequence (T n)n∈ℕ of iterates of T is pointwise convergent in X, then for any x ∈ X, the limit is a fixed point of T. The problem of determining the form of µT leads to the invariance equation µT ○ T = µT, which is difficult to solve in general if the set of fixed points of T is not a singleton. We consider this problem assuming that X = I p, where I is a real interval, p ≥ 2 a fixed positive integer and T is the mean-type mapping...
Fixed point theorems for contractive mappings in metric spaces
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