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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]

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]

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]

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]

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]

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]