| Title:
|
Solution of Whitehead equation on groups (English) |
| Author:
|
Faĭziev, Valeriĭ A. |
| Author:
|
Sahoo, Prasanna K. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
138 |
| Issue:
|
2 |
| Year:
|
2013 |
| Pages:
|
171-180 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $G$ be a group and $H$ an abelian group. Let $J^{*} (G, H)$ be the set of solutions $f \colon G \to H$ of the Jensen functional equation $f(xy)+f(xy^{-1}) = 2f(x)$ satisfying the condition $f(xyz) - f(xzy) = f(yz)-f(zy)$ for all $x, y , z \in G$. Let $Q^{*} (G, H)$ be the set of solutions $f \colon G \to H$ of the quadratic equation $f(xy)+f(xy^{-1}) = 2f(x) + 2f(y)$ satisfying the Kannappan condition $f(xyz) = f(xzy)$ for all $x, y, z \in G$. In this paper we determine solutions of the Whitehead equation on groups. We show that every solution $f \colon G \to H$ 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 solution $f \colon G \to H$ of the Whitehead equation is of the form $2f = \varphi + \psi $, where $\varphi \in J^*(G,H)$ and $2\psi (x) = B(x, x)$ for some symmetric bihomomorphism $B \colon G \times G \to H$. (English) |
| Keyword:
|
homomorphism |
| Keyword:
|
Fréchet functional equation |
| Keyword:
|
Jensen functional equation |
| Keyword:
|
symmetric bihomomorphism |
| Keyword:
|
Whitehead functional equation |
| MSC:
|
39B52 |
| idZBL:
|
Zbl 06221247 |
| idMR:
|
MR3112363 |
| DOI:
|
10.21136/MB.2013.143289 |
| . |
| Date available:
|
2013-05-27T14:25:33Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143289 |
| . |
| Reference:
|
[1] Kannappan, Pl.: Quadratic functional equation and inner product spaces.Results Math. 27 (1995), 368-372. Zbl 0836.39006, MR 1331110, 10.1007/BF03322841 |
| Reference:
|
[2] Kannappan, Pl.: Functional Equations and Inequalities with Applications.Springer Monographs in Mathematics, Springer, New York (2009). Zbl 1178.39032, MR 2524097 |
| Reference:
|
[3] Ng, C. T.: Jensen's functional equation on groups.Aequationes Math. 39 (1990), 85-99. Zbl 0688.39007, MR 1044167, 10.1007/BF01833945 |
| Reference:
|
[4] Friis, P. de Place, Stetkær, H.: On the quadratic functional equation on groups.Publ. Math. Debrecen 69 (2006), 65-93. MR 2228477 |
| Reference:
|
[5] Whitehead, J. H. C.: A certain exact sequence.Ann. Math. (2) 52 (1950), 51-110. Zbl 0037.26101, MR 0035997, 10.2307/1969511 |
| Reference:
|
[6] Yang, D.: The quadratic functional equation on groups.Publ. Math. Debrecen 66 (2005), 327-348. Zbl 1100.39028, MR 2137773 |
| . |
