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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...

Solution of Cauchy's functional equation on a restricted domain

Józef Tabor (1975)

Colloquium Mathematicae

Solution of distributive-like quasigroup functional equations

Fedir M. Sokhatsky, Halyna V. Krainichuk (2012)

Commentationes Mathematicae Universitatis Carolinae

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...

Solution of Whitehead equation on groups

Valeriĭ A. Faĭziev, Prasanna K. Sahoo (2013)

Mathematica Bohemica

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 $f (x y) + f (x y^{- 1}) = 2 f (x)$ satisfying the condition $f (x y z) - f (x z y) = f (y z) - f (z y)$ for all $x, y, z \in G$ . Let $Q^{*} (G, H)$ be the set of solutions $f : G \to H$ of the quadratic equation $f (x y) + f (x y^{- 1}) = 2 f (x) + 2 f (y)$ satisfying the Kannappan condition $f (x y z) = f (x z y)$ 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 $4 f = 2 ϕ + 2 ψ$ , where $2 ϕ \in J^{*} (G, H)$ and $2 ψ \in Q^{*} (G, H)$ . Moreover, if $H$ has the additional property that $2 h = 0$ implies $h = 0$ for all $h \in H$ , then every...

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