A variant of d'Alembert's functional equation
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| 024 | 7 | 0 | |a 10.1007/s00010-014-0253-y |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s00010-014-0253-y | ||
| 100 | 1 | |a Stetkær |D Henrik |u Department of Mathematical Sciences, University of Aarhus, Ny Munkegade 118, Building 1530, 8000, Aarhus C, Denmark |4 aut | |
| 245 | 1 | 2 | |a A variant of d'Alembert's functional equation |h [Elektronische Daten] |c [Henrik Stetkær] |
| 520 | 3 | |a 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[. | |
| 540 | |a Springer Basel, 2014 | ||
| 690 | 7 | |a d'Alembert's equation |2 nationallicence | |
| 690 | 7 | |a semigroup |2 nationallicence | |
| 690 | 7 | |a multiplicative function |2 nationallicence | |
| 773 | 0 | |t Aequationes mathematicae |d Springer Basel |g 89/3(2015-06-01), 657-662 |x 0001-9054 |q 89:3<657 |1 2015 |2 89 |o 10 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s00010-014-0253-y |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s00010-014-0253-y |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 100 |E 1- |a Stetkær |D Henrik |u Department of Mathematical Sciences, University of Aarhus, Ny Munkegade 118, Building 1530, 8000, Aarhus C, Denmark |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Aequationes mathematicae |d Springer Basel |g 89/3(2015-06-01), 657-662 |x 0001-9054 |q 89:3<657 |1 2015 |2 89 |o 10 | ||