A stability result for the Pexider equation will be derived from a stability theorem published in [9] for the Cauchy functional equation. Then we discuss the quality of some constants occuring in this context; as a model case we consider functions defined on the multiplicative semigroup {1, 0}.

In this note we generalize the concept of orderings of higher level to multifields and multirings. We show how part of the standard Artin-Schreier theory for orderings of higher level of fields and rings extends to multifields and multirings.

Invertible operators preserving singularity of matrices were studied in [3] and [4] under assumption that operators are linear. In the present paper the linearity of operators is not assumed: we assume only that operators are of the form F = (f_i,j), where f_i,j : ????→???? and ???? is a field, i,j∈{1, 2, . . . , n}. If n≥3, then in the matrix space M_n(????) operators preserving singularity of matrices must be as in [1]. If n≤2, then operators may be nonlinear. In this case the forms of the operators are presented.

In this note we prove that every affine transformation in ℝⁿ of the form f(x) = Ax+a without fixed points has an invariant straight line if and only if when g(a)∈Im(g◦g), where g(x) = (A−I)x.

In this paper, the stochastic modification of the bicriterial minimum cost flow problem is presented. After the problem’s formulation two approximate algorithms based on sandwich method for a convex curve approximation are presented. The obtained results are illustrated by examples.

Some functional equations related to homographic functions and their characterization are presented.

We show that the set of solutions of the initial-value problem
u(τ)=a,   u'(t) = g(t,u(t)) + k(t,u(t)),   τ≤t≤T,
in a Banach space is compact and connected, whenever g and k are bounded and continuous functions such that g is one-sided Lipschitz and k is Lipschitz with respect to the Kuratowski measure of noncompactness. The existence of solutions is already known from Sabina Schmidt [10].

Report of Meeting: The Tenth Debrecen–Katowice Winter Seminar on Functional Equations and Inequalities, Zamárdi (Hungary), February 3–6, 2010.