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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...
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.
We show that any quasi-arithmetic mean and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations
and
are the constant ones.
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.
The paper discusses the asymptotic properties of solutions of the scalar functional differential equation
of the advanced type. We show that, given a specific asymptotic behaviour, there is a (unique) solution which behaves in this way.
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.
We give a characterization of the globally Lipschitzian composition operators acting in the space
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