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The issue of distributivity of aggregation operations is crucial for many different areas such as utility theory and integration theory. Of special interest are aggregation operations with annihilator. This paper is focused on the problem of distributivity for some so called associative, commutative aggregation operations with annihilator a, known as associative a-CAOA, and uninorms. The full characterization of distributive pairs for T-uninorms, S-uninorms and bi-uninorms is given.

This paper deals with functional equations in the form of $f(x) + g(y) = h(x,y)$ where $h$ is given and $f$ and $g$ are unknown. We will show that if $h$ is a Borel measurable function associated with characterizations of the uniform or Cauchy distributions, then there is no measurable solutions of the equation. Our proof uses a characterization of the Dirac measure and it is also applicable to the arctan equation., Comment: 7 pages, to appear in Aequationes mathematicae

In this survey paper we present some recent results in the iteration theory. Mainly, we focus on the problems concerning real iteration groups (flows) and semigroups (semiflows) such as existence, regularity and embeddability. We also discuss some issues associated to the problem of embedddability, i.e. iterative roots and approximate iterability. The topics of this paper are: (1) measurable iteration semigroups; (2) embedding of diffeomorphisms in regular iteration semigroups in $${{\mathbb{R}}^n}$$ space; (3) iteration groups of fixed point free homeomorphisms on the plane; (4) embedding of interval homeomorphisms with two fixed points in a regular iteration group; (5) commuting functions and embeddability; (6) iterative roots; (7) the structure of iteration groups on an interval; (8) iteration groups of homeomorphisms of the circle; (9) approximately iterable functions; (10) set-valued iteration semigroups; (11) iterations of mean-type mappings; (12) Hayers–Ulam stability of the translation equation. Most of the results presented here was obtained by the means of functional equations. We indicate the relations between the iteration theory and functional equations.

Multi-edge trees as introduced in a recent paper of Dziemia\'nczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. The distribution of the number of vertices as well as the height are analysed asymptotically.

Since the monograph by Amir that appeared in 1986, a lot of attention has been given to the problem of characterizing, by means of properties of the norms, when a Banach space is indeed a Hilbert space, i.e., when the norm derives from an inner product. In this paper, similar investigations will be carried out for smooth spaces instead of inner product spaces. We consider the approximate orthogonalities in real normed spaces. We show that the relations approximate semi-orthogonality and approximate ρ+-orthogonality are generally incomparable (unless the normed space is smooth). As a result, we give a characterization of smooth spaces in terms of those approximate orthogonalities.

In this paper we prove that if a generalized polynomial function f satisfies the condition f(x) f(y) = 0 for all solutions of the equation x 2 + y 2 = 1, then f is identically equal to 0.

Let X be a real linear space. We characterize solutions \({f, g : X \rightarrow \mathbb{R}}\) of the equation f(x + g(x)y) = f(x)f(y), where f is continuous on rays. Our result refers to papers by Brzdȩk (Acta Math Hungar 101:281–291, 2003), Chudziak (Aequat Math, doi:10.1007/s00010-013-0228-4, 2013) and Jablonska (J Math Anal Appl 375:223–229, 2011).

The aim of this paper is to solve the orthogonality equation with two unknown functions. This problem was posed by J. Chmieli´nski during two international conferences.

In this paper we consider stability in the Ulam–Hyers sense, and in other similar senses, for the five equivalent definitions of one-dimensional dynamical systems.

A graceful labeling of a graph G with q edges is an injective assignment of labels from {0, 1, . . . , q} to the vertices of G so that when each edge is assigned the absolute value of the difference of the vertex labels it connects, the resulting edge labels are distinct. A labeling of the first kind for coronas \({C_n \odot K_1}\) occurs when vertex labels 0 and q = 2n are assigned to adjacent vertices of the n-gon. A labeling of the second kind occurs when q = 2n is assigned to a pendant vertex. Previous research has shown that all coronas \({C_n \odot K_1}\) have a graceful labeling of the second kind. In this paper we show that all coronas \({C_n \odot K_1}\) with \({n \equiv 3, 4\, {\rm (mod\, 8)}}\) also have a graceful labeling of the first kind.

This is an expository paper in which we present some simple observations on the stability of some inhomogeneous functional equations. In particular, we state several stability results for the inhomogeneous Cauchy equation $$f(x+y)=f(x)+f(y)+d(x,y)$$ and for the inhomogeneous forms of the Jensen and linear functional equations.