an:01066347
Zbl 0905.46030
Ger, Roman
Delta-exponential mappings in Banach algebras
EN
Bandle, C. (ed.) et al., General inequalities 7. 7th international conference, Oberwolfach, Germany, November 13--18, 1995. Proceedings. Basel: Birkhäuser. ISNM, Int. Ser. Numer. Math. 123, 285-296 (1997).
1997
a
46G05 39B52
delta-convex mappings; Hyers-Ulam stability; superstability
Summary: An intriguing interplay between the theory of delta-convex mappings (in the sense of Veselý and Zajiček) and the Hyers-Ulam stability problems is developed by studying a functional inequality
\[
\| F(x+ y)- F(x)F(y)\|\leq f(x)f(y)- f(x+ y).\tag{\(*\)}
\]
This is an ``exponential version'' of the inequality
\[
\| F(x+ y)- F(x)- F(y)\|\leq\| x\|+\| y\|- \| x+y\|,
\]
proposed first by D. Yost and then generalized to
\[
\| F(x+ y)- F(x)- F(y)\|\leq f(x)+ f(y)- f(x+y).
\]
A superstability phenomenon in connection with \((*)\) is examined.
For the entire collection see [Zbl 0864.00057].