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Delta-exponential mappings in Banach algebras. (English) Zbl 0905.46030
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).
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].
##### MSC:
 46G05 Derivatives of functions in infinite-dimensional spaces 39B52 Functional equations for functions with more general domains and/or ranges
##### Keywords:
delta-convex mappings; Hyers-Ulam stability; superstability