Manfred Krüppel
On an Inequality for Nonexpansive Mappings in Uniformly Convex Banach Spaces
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 51, 25-32(1997)

MSC:

47H09 Nonexpansive mappings, and their generalizations

47H19 Inequalities involving nonlinear operators, See Also {49J27, 49J40, 49K27}

Abstract: Let $C$ be a bounded convex subset of an uniformly convex Banach space $X$.
Then it is shown that there exists an increasing continuous function $h :
\Bbb R^+ \rightarrow \Bbb R^+$ depending on the diameter of $C$ so that for
any nonexpansive mapping $T:C \rightarrow X$ and any convex combination of
arbitrarily many elements $x_i$ in $C$ the inequality $h (\|\sum \lambda_i
x_i - T (\sum \lambda_i x_i)\|) \le \sum \lambda_i \| x_i - Tx_i \|$ holds.
This
inequality has several consequences in the theory of nonexpansive
mappings.
Keywords: Inequality, uniformly convex Banach space, nonexpansive mapping