On the stability of the quadratic equation on groups. (English) Zbl 1183.39018

Faĭziev, Valeriĭ A.; Sahoo, Prasanna K.

Bull. Belg. Math. Soc. - Simon Stevin 15, No. 1, 135-151 (2008).

Summary: The stability of the quadratic equation is considered on arbitrary groups. Since the quadratic equation is stable on Abelian groups, this paper examines the stability of the quadratic equation on noncommutative groups. It is shown that the quadratic equation is stable on \(n\)-Abelian groups when \(n\) is a positive integer. The stability of the quadratic equation is also established on the noncommutative group \(T(2, K)\), where \(K\) is an arbitrary commutative field. It is proved that every group can be embedded into a group in which the quadratic equation is stable.

MSC:

39B82	Stability, separation, extension, and related topics for functional equations
20F99	Special aspects of infinite or finite groups
39B52	Functional equations for functions with more general domains and/or ranges

Keywords:

Banach spaces; \(n\)-Abelian group; pseudoquadratic map; quadratic map; quasiquadratic map; quadratic functional equation; semidirect product of groups; stability of quadratic functional equation; wreath product of groups