Some functional equations involving means of associative functions are investigated.

In 1940 S. M. Ulam (Intersci. Publ., Inc., New York 1960) imposed at the University of Wisconsin the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist”. According to P. M. Gruber (Trans. Amer. Math. Soc. 245 (1978), 263–277) the afore-mentioned problem of S. M. Ulam belongs to the following general problem or Ulam type problem: “Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this objects...

We are investigating quasigroup functional equation classification up to parastrophic equivalence [Sokhatsky F.M.: On classification of functional equations on quasigroups, Ukrainian Math. J. 56 (2004), no. 4, 1259–1266 (in Ukrainian)]. If functional equations are parastrophically equivalent, then their functional variables can be renamed in such a way that the obtained equations are equivalent, i.e., their solution sets are equal. There exist five classes of generalized distributive-like quasigroup...

Let $G$ be a group and $H$ an abelian group. Let $J^{*}(G,H)$ be the set of solutions $f:G\to H$ of the Jensen functional equation $f\left(xy\right)+f\left(x{y}^{-1}\right)=2f\left(x\right)$ satisfying the condition $f\left(xyz\right)-f\left(xzy\right)=f\left(yz\right)-f\left(zy\right)$ for all $x,y,z\in G$. Let $Q^{*}(G,H)$ be the set of solutions $f:G\to H$ of the quadratic equation $f\left(xy\right)+f\left(x{y}^{-1}\right)=2f\left(x\right)+2f\left(y\right)$ satisfying the Kannappan condition $f\left(xyz\right)=f\left(xzy\right)$ for all $x,y,z\in G$. In this paper we determine solutions of the Whitehead equation on groups. We show that every solution $f:G\to H$ of the Whitehead equation is of the form $4f=2\varphi +2\psi$, where $2\varphi \in J^{*}(G,H)$ and $2\psi \in Q^{*}(G,H)$. Moreover, if $H$ has the additional property that $2h=0$ implies $h=0$ for all $h\in H$, then every...

We obtain a result on the existence of a solution with big graph of functional equations of the form g(x,𝜑(x),𝜑(f(x)))=0 and we show that it is applicable to some important equations, both linear and nonlinear, including those of Abel, Böttcher and Schröder. The graph of such a solution 𝜑 has some strange properties: it is dense and connected, has full outer measure and is topologically big.

Let X be a reflexive Banach space and (Ω,,μ) be a probability measure space. Let T: M(μ;X) → M(μ;X) be a linear operator, where M(μ;X) is the space of all X-valued strongly measurable functions on (Ω,,μ). We assume that T is continuous in the sense that if (fₙ) is a sequence in M(μ;X) and $li{m}_{n\to \infty }fₙ=f$ in measure for some f ∈ M(μ;X), then also $li{m}_{n\to \infty }Tfₙ=Tf$ in measure. Then we consider the functional equation f = (T-I)h, where f ∈ M(μ;X) is given. We obtain several conditions for the existence of h ∈ M(μ;X) satisfying...

In the present paper some complex vector functional equations of higher order without parameters and with complex parameters are solved.