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Algebraic and topological structures on the set of mean functions and generalization of the AGM mean

Bakir Farhi (2013)

Colloquium Mathematicae

We present new structures and results on the set of mean functions on a given symmetric domain in ℝ². First, we construct on a structure of abelian group in which the neutral element is the arithmetic mean; then we study some symmetries in that group. Next, we construct on a structure of metric space under which is the closed ball with center the arithmetic mean and radius 1/2. We show in particular that the geometric and harmonic means lie on the boundary of . Finally, we give two theorems...

An extension theorem for a Matkowski-Sutô problem

Zoltán Daróczy, Gabriella Hajdu, Che Tat Ng (2003)

Colloquium Mathematicae

Let I be an interval, 0 < λ < 1 be a fixed constant and A(x,y) = λx + (1-λ)y, x,y ∈ I, be the weighted arithmetic mean on I. A pair of strict means M and N is complementary with respect to A if A(M(x,y),N(x,y)) = A(x,y) for all x, y ∈ I. For such a pair we give results on the functional equation f(M(x,y)) = f(N(x,y)). The equation is motivated by and applied to the Matkowski-Sutô problem on complementary weighted quasi-arithmetic means M and N.

An inconsistency equation involving means

Roman Ger, Tomasz Kochanek (2009)

Colloquium Mathematicae

We show that any quasi-arithmetic mean A φ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations f ( M ( x , y ) ) = A φ ( f ( x ) , f ( y ) ) and f ( A φ ( x , y ) ) = M ( f ( x ) , f ( y ) ) are the constant ones.

Asymptotic behaviour of solutions of some linear delay differential equations

Jan Čermák (2000)

Mathematica Bohemica

In this paper we investigate the asymptotic properties of all solutions of the delay differential equation y’(x)=a(x)y((x))+b(x)y(x),      xI=[x0,). We set up conditions under which every solution of this equation can be represented in terms of a solution of the differential equation z’(x)=b(x)z(x),      xI and a solution of the functional equation |a(x)|((x))=|b(x)|(x),      xI.

Asymptotic properties of differential equations with advanced argument

Jan Čermák (2000)

Czechoslovak Mathematical Journal

The paper discusses the asymptotic properties of solutions of the scalar functional differential equation y ' ( x ) = a y ( τ ( x ) ) + b y ( x ) , x ∈ [ x 0 , ∞ ) of the advanced type. We show that, given a specific asymptotic behaviour, there is a (unique) solution y ( x ) which behaves in this way.

Caratterizzazione di classi di funzioni della forma F(x,y) = φ(h(x)+k(y)).

Daniela Rusconi (1992)

Stochastica

A characterization of some classes of functions F which have a representation of the formF(x,y) = φ(h(x)+k(y))is given, when F is monotonic in each variable but not strictly monotonic. Some particular results concern classes of solutions of the bisymmetry or associativity equations.

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