caa a22 4500 605508399 CHVBK 20210128100636.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00010-014-0253-y doi (NATIONALLICENCE)springer-10.1007/s00010-014-0253-y Stetkær Henrik Department of Mathematical Sciences, University of Aarhus, Ny Munkegade 118, Building 1530, 8000, Aarhus C, Denmark aut A variant of d'Alembert's functional equation [Elektronische Daten] [Henrik Stetkær] Let S be a semigroup, and let $${\sigma \in {\rm Hom}(S,S)}$$ σ ∈ Hom ( S , S ) satisfy $${\sigma \circ \sigma = {\rm id}}$$ σ ∘ σ = id . We show that any solution $${g: S \to \mathbb{C}}$$ g : S → C of the functional equation $$ g(xy) + g(\sigma(y)x) = 2g(x)g(y), \quad x, y \in S, $$ g ( x y ) + g ( σ ( y ) x ) = 2 g ( x ) g ( y ) , x , y ∈ S , has the form $${g = (\mu + \mu \circ \sigma) /2}$$ g = ( μ + μ ∘ σ ) / 2 , whereμ is a multiplicative function on S. From this we find the solutions $${f: I \times I \to \mathbb{C}}$$ f : I × I → C , where I is a semigroup, of $$ f(pr, qs) + f(sp, rq) = f(p, q)f(r, s), \quad p, q, r, s \in I, $$ f ( p r , q s ) + f ( s p , r q ) = f ( p , q ) f ( r , s ) , p , q , r , s ∈ I , thereby generalizing a result by Chung, Kannappan, Ng and Sahoo for the multiplicative semigroup I =]0, 1[. Springer Basel, 2014 d'Alembert's equation nationallicence semigroup nationallicence multiplicative function nationallicence Aequationes mathematicae Springer Basel 89/3(2015-06-01), 657-662 0001-9054 89:3<657 2015 89 10 https://doi.org/10.1007/s00010-014-0253-y text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-014-0253-y text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Stetkær Henrik Department of Mathematical Sciences, University of Aarhus, Ny Munkegade 118, Building 1530, 8000, Aarhus C, Denmark aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/3(2015-06-01), 657-662 0001-9054 89:3<657 2015 89 10