The aim of this work is to provide a complete answer to the question of whether two functions can be separated by an additive mapping, knowing that these functions satisfy a system of two inequalities describing the relationships between the value on the sum of two arguments and the sum of the values on these arguments.

Let $$I\subset \mathbb {R}$$ be an interval. A function $$M:I^{2}\rightarrow I$$ is said to be weakly associative, if $$\begin{aligned} M\left( M\left( x,y\right) ,x\right) =M\left( x,M\left( y,x\right) \right) , \qquad x,y\in I. \end{aligned}$$ One can easily check that every weighted quasi-arithmetic mean, i.e. a function $$M:I^{2}\rightarrow I$$ given by $$ M\left( x,y\right) =f^{-1}\left( pf\left( x\right) +\left( 1-p\right) f\left( y\right) \right) , $$ where $$f:I\rightarrow \mathbb {R}$$ is a continuous and strictly monotonic function and $$p\in \left[ 0,1\right] $$ , satisfies the above condition, so it is weakly associative. We give the characterization of weakly associative functions in the class of some generalized weighted quasi-arithmetic means. Moreover, we characterize premeans which are rational functions of degree at most 2 and weakly associative.

Abstract Let $$k\ge 1$$ k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V(G). A signed double Roman k-dominating function (SDRkDF) on a graph G is defined in [Signed double Roman k-domination in graphs, Australas. J. Combin. 72 (2018), 82–105] as a function $$f :V(G) \rightarrow \{-1,1,2,3\}$$ f : V ( G ) → { - 1 , 1 , 2 , 3 } satisfying the conditions that $$\sum _{x\in N[v]}f(x)\ge k$$ ∑ x ∈ N [ v ] f ( x ) ≥ k for each vertex $$v\in V(G)$$ v ∈ V ( G ) , where N[v] is the closed neighborhood of v, every vertex u for which $$f(u)=-1$$ f ( u ) = - 1 is adjacent to at least one vertex v for which $$f(v)=3$$ f ( v ) = 3 or adjacent to two vertices x and y with $$f(x)=f(y)=2$$ f ( x ) = f ( y ) = 2 , and every vertex u with $$f(u)=1$$ f ( u ) = 1 is adjacent to vertex v with $$f(v)\ge 2$$ f ( v ) ≥ 2 . The weight of an SDRkDF f is $$\textrm{w}(f) = \sum _{v\in V(G)}f(v)$$ w ( f ) = ∑ v ∈ V ( G ) f ( v ) . The signed double Roman k-domination number $$\gamma _{\textrm{sdR}}^k(G)$$ γ sdR k ( G ) of G is the minimum weight among all SDRkDF on G. In this paper we continue the study of the signed double Roman k-domination number of graphs, and we present new bounds on $$\gamma _{\textrm{sdR}}^k(G)$$ γ sdR k ( G ) . In addition, we determine the signed double Roman k-domination number of some classes of graphs. Some of our results are extensions of well-known properties of the signed double Roman domination number, $$\gamma _{\textrm{sdR}}(G)=\gamma _{\textrm{sdR}}^1(G)$$ γ sdR ( G ) = γ sdR 1 ( G ) , introduced and investigated in [1, 2].

This is the second of three sequels to (Ostaszewski in Aequat Math 90:427–448, 2016)—the third of the resulting quartet—concerning the real-valued continuous solutions of the multivariate Goldie functional equation (GFE) below of Levi–Civita type. Following on from the preceding paper (Bingham and Ostaszewski in Homomorphisms from Functional Equations: II. The Goldie Equation, arXiv:1910.05816), in which these solutions are described explicitly, here we characterize (GFE) as expressing homomorphy (in all but some exceptional “improper” cases) between multivariate Popa groups, defined and characterized earlier in the sequence. The group operation involves a form of affine addition (with local scalar acceleration) similar to the circle operation of ring theory. We show the affine action in (GFE)may be replaced by a general continuous acceleration yielding a functional equation (GGE) which it emerges has the same solution structure as (GFE). The final member of the sequence (Bingham and Ostaszewski, The Gołąb–Schinzel and Goldie functional equations in Banach algebras, arXiv:2105.07794) considers the richer framework of a Banach algebra which allows vectorial acceleration, giving the closest possible similarity to the circle operation.