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On the functional equation x + f(y + f(x))=y + f(x + f(y))

Rätz, Jürg

In: Aequationes mathematicae, 2013, vol. 86, no. 1-2, p. 187-200

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    Title
    • On the functional equation x + f(y + f(x))=y + f(x + f(y))
    Author
    • Rätz, Jürg. Mathematisches Institut der Universität Bern, Sidlerstrasse 5, 3012, Bern, Switzerland
    Document Type
    • Postprint
    OAI-PMH Identifier
    • oai:doc.rero.ch:316747
    Summary
    For an abelian group (G,+,0) we consider the functional equation $$f : G \to G, x + f(y + f(x)) = y + f(x + f(y)) \quad (\forall x, y \in G), \quad\quad\qquad (1)$$ most times together with the condition $$f(0) = 0.\qquad\qquad\qquad\qquad\qquad (0)$$ Our main question is whether a solution of $${(1) \wedge (0)}$$ must be additive, i.e., an endomorphism of G. We shall answer this question in the negative (Example 3.14) Rätz (Aequationes Math 81:300, 2011)