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In the paper, we will present some important results from the theory of vector-valued invariant means. Discussing the results contained in the author's paper Badora (Ann Pol Math 58:147–159, 1993), we will add new facts that were previously not published. [ABSTRACT FROM AUTHOR]

We investigate the functional equation f (p (x)) = q (f (x)) where p and q are given real functions. In the paper "On solvability of f (p (x)) = q (f (x)) for given real functionsp, q, Aequat. Math. 90 (2016), 471 - 494", we solved the problem of the solvability of f (p (x)) = q (f (x)) under the assumption that p, q are strictly increasing continuous real functions. Now, we extend the solutions of this problem for any strictly monotonous continuous real functions p, q. Thereby, we use the methods of the just mentioned paper. Further, we present computations of the so called characteristics of the given functions p, q using the results of this paper and, finally, present a quite short algorithm with input p, q and output 'solvable/not solvable'. [ABSTRACT FROM AUTHOR]

Let X be an abelian metric group with an invariant metric, Y be a real normed space and K be a convex cone in Y. We prove that a K-subadditive (K-superadditive) compact- and convex-valued map F : X → C C (Y) , for which the functionals f y ∗ (x) = inf y ∗ (F (x)) are lower (upper, resp.) semicontinuous for any real continuous and non-negative on K functional y ∗ , has to be locally K-bounded on X. Our results refer to the papers Banakh and Jabłońska (Israel J Math 230:361–386, 2019), Jabłońska and Nikodem (Math Inequal Appl 22:1081–1089, 2019) and Nikodem (Aequationes Math 62:175–183, 2001). [ABSTRACT FROM AUTHOR]

The aim of this paper is to fill in the gaps in the formulation and the proof of a theorem contained in the paper by K.Ozeki (Aequ Math 25:247–252, 1982) published in this journal. We also give a short proof of this theorem and use it to obtain certain information about the factorization of polynomials of the form f (x) - f (y) . [ABSTRACT FROM AUTHOR]

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F (x + y) - F (x) - F (y) = y f (x) + x f (y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation. [ABSTRACT FROM AUTHOR]

The paper is a continuation of Korolev and Laurinčikas (Aequ Math 93:859–873, 2019), where theorems on the approximation of analytic functions by shifts ζ (s + i h t k) , h > 0 , k ∈ N , and t k are the Gram points, were obtained. In this paper, it is proved, that the set of shifts ζ (s + i h t k) has a positive density in short intervals [ N , N + M ] with M = o (N) . [ABSTRACT FROM AUTHOR]

For a given p-variable mean M:Ip→I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M :I^p \rightarrow I$$\end{document} (I is a subinterval of R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}$$\end{document}), following (Horwitz in J Math Anal Appl 270(2):499–518, 2002) and (Lawson and Lim in Colloq Math 113(2):191–221, 2008), we can define (under certain assumptions) its (p+1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(p+1)$$\end{document}-variable β\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta $$\end{document}-invariant extension as the unique solution K:Ip+1→I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K :I^{p+1} \rightarrow I$$\end{document} of the functional equation K(M(x2,⋯,xp+1),M(x1,x3,⋯,xp+1),⋯,M(x1,⋯,xp))=K(x1,⋯,xp+1),for allx1,⋯,xp+1∈I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned}&K\big (M(x_2,\dots ,x_{p+1}),M(x_1,x_3,\dots ,x_{p+1}),\dots ,M(x_1,\dots ,x_p)\big )\\&\quad =K(x_1,\dots ,x_{p+1}), \text { for all }x_1,\dots ,x_{p+1} \in I \end{aligned}$$\end{document}in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions. [ABSTRACT FROM AUTHOR]

We examine the family of all (at most) three-point symmetric quadratures on [ - 1 , 1 ] which are exact on polynomials of order 3 to find all possible inequalities between them in the class of 3-convex functions. Next we optimise them by using convex combinations of the quadratures considered. We find the optimal quadrature and use it to construct the adaptive method of approximate integration. An effective method to estimate the error of this method is also given. It needs a considerably fewer number of subdivisions of the interval of integration than the classical adaptive methods as well as the method developed by the second-named author in his recent paper (Wa̧sowicz in Aequ Math 94(5):887–898, 2020). [ABSTRACT FROM AUTHOR]

It is known that if a bivariate mean K is symmetric, continuous and strictly increasing in each variable, then for every mean M there is a unique mean N such that K is invariant with respect to the mean-type mapping M , N , which means that K ∘ M , N = K and N is called a K-complementary mean for M (Matkowski in Aequ Math 57(1):87–107, 1999). This paper extends this result for a large class of nonsymmetric means. As an application, the limits of the sequences of iterates of the related mean-type mappings are determined, which allows us to find all continuous solutions of some functional equations. [ABSTRACT FROM AUTHOR]

It was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function. [ABSTRACT FROM AUTHOR]

According to the famous and pioneering result of Laurent Schwartz, any closed translation invariant linear space of continuous functions on the reals is synthesizable from its exponential monomials. Due to a result of D. I. Gurevič there is no straightforward extension of this result to higher dimensions. Following Székelyhidi (Acta Math Hungar 153(1):120–142, 2017), with the aid of Gelfand pairs and K-spherical functions, K-synthesizability of K-varieties can be described. In this paper we contribute to this direction in the special case when K is the symmetric group of order d. [ABSTRACT FROM AUTHOR]

A scientific law can be a faithful representation of the physical world only if its form is invariant with respect to changes in the unit or units. This is referred to as the 'meaningfulness condition.' This condition is powerful. If we require it, the mathematical form of a scientific or geometric law may be derivable from some abstract constraint, possibly verifiable by a thought experiment or a trivial argument. We discuss five examples of such abstract constraints in this paper: In each case, just one or a couple of meaningful mathematical representations are possible. In this paper, we derive the possible meaningful representations in five examples. These results are obtained under some general conditions, in addition to those listed in 1-5. Other meaningful representations may be possible under different additional conditions. [ABSTRACT FROM AUTHOR]

In (Brzdęk and Schwaiger in Aeq Math 92: 975–991, 2018) solutions of far reaching generalizations of the so-called radical functional equation f (p (π (x) + π (y))) = f (x) + f (y) have been investigated. These investigations are continued here by analysing the corresponding stability results, which have been the main subject of several recent papers. We propose a very general and uniform approach. [ABSTRACT FROM AUTHOR]

In this paper, we show that the hyperstability of the general linear equation recently proved by Piszczek (Aequationes Math 88:163–168, 2014) is a direct consequence of that of the Cauchy equation proved earlier by Brzdȩk (Acta Math Hung 141:58–67, 2013). [ABSTRACT FROM AUTHOR]

In this paper, we prove a fixed point theorem for a system of maps on the finite product of metric spaces. Our result generalizes the result of Matkowski (Bull Acad Pol Sci Sér Sci Math Astron Phys 21:323-324, 1973), Cosentino and Vetro (Filomat 28(4):715-722, 2014) and Hardy and Rogers (Can Math Bull 16(2):201-206, 1973) and other results in the literature. Moreover, we have an application for a system of functional equations and an example to illustrate our result. [ABSTRACT FROM AUTHOR]

The n-dimensional integer lattice, denoted by Zn, is the subset of Rn consisting of those points whose coordinates are all integers. In this expository paper, many concrete, intuitive, and geometric results concerning the integer lattice Zn are presented, most of them together with new elementary or streamlined proofs. Some of the presented results are new, and others are improved versions of old results. [ABSTRACT FROM AUTHOR]

This paper gives a purely set-theoretical extension theorem on iterative roots, which generalizes the results about PM functions by Zhang and his collaborators. Applying this extension theorem, we construct all iterative roots with infinitely many discontinuities for a class of Markov maps. [ABSTRACT FROM AUTHOR]

Recently, Dong (Aequ Math 86:269-277, ) has proved the generalized stability of the functional equation $${\|f(x + y)\| = \|f(x) + f(y)\|}$$ under the assumption that X, the domain of f, is an Abelian group. In this paper, we prove a generalization of this result by removing the commutativity assumption of X. [ABSTRACT FROM AUTHOR]

One famous open problem in graph theory is the Graceful Tree Conjecture, which states that every finite tree has a graceful labeling. In 1973, Kotzig (Util Math 4:261-290, ) proved that if a leaf of a long enough path is identified with any vertex of an arbitrary tree, the resulting tree is graceful. In this paper, we prove that if the center of a large enough star is identified with any vertex of an arbitrary tree, the resulting tree is graceful, and we also provide an upper bound for the size of the star. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we study the connections between generalized Aczél-Jabotinsky differential equations and Aczél-Jabotinsky differential equations. [ABSTRACT FROM AUTHOR]

In this paper we investigate continuity properties of functions $${f : \mathbb {R}_+ \to \mathbb {R}_+}$$ that satisfy the ( p, q)-Jensen convexity inequality where H stands for the pth power (or Hölder) mean. One of the main results shows that there exist discontinuous multiplicative functions that are ( p, p)-Jensen convex for all positive rational numbers p. A counterpart of this result states that if f is ( p, p)-Jensen convex for all $${p \in P \subseteq \mathbb {R}_+}$$ , where P is a set of positive Lebesgue measure, then f must be continuous. [ABSTRACT FROM AUTHOR]

In this paper we determine the general solution of some special cases of the generalized Hosszú functional equation on intervals [0,1] and (0,1). [ABSTRACT FROM AUTHOR]

Given rational vector spaces V, W a mapping $${f \colon V \to W}$$ is called a generalized polynomial of degree at most n, if there are homogeneous generalized polynomials f of degree i such that $${f = \sum_{i = 0}^n f_i}$$ . Homogeneous generalized polynomials f of degree i are mappings of the form $${f_i (x) = f_i^*(x, x, \ldots , x)}$$ with $${f_i^* \colon V^i \to W i}$$ -linear. In the literature one may find quite a lot of functional equations such that their general solution is of the form f or $${f_n + f_{n - 1}}$$ where n is a small positive integer ( ≤ 6 or ≤ 4 respectively). In this paper, given an arbitrary positive integer n and an arbitrary subset $${L \subseteq \{0, 1, \ldots, n\}}$$ such that $${n \in L}$$ , a method is described to find (many) functional equations, such that their general solution is given by $${\sum_{i \in L} f_i}$$ . For the cases $${L = \{n\}}$$ and $${L = \{n - 1, n\}}$$ additional equations are given. [ABSTRACT FROM AUTHOR]