Manfred Krüppel
On the nearest point projection in Hilbert spaces with application to Nonlinear Ergodic Theory
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 50, 89-94 (1997)

MSC:

47H09 Nonexpansive mappings, and their generalizations

Abstract: Let $P$ be the nearest point projection of a Hilbert space $H$ onto a closed and convex subset $K$ and $\Sigma \lambda_i x_i$ any convex combination of the points $x_i$ in $H$. We give an estimate for $||P (\Sigma \lambda_i x_i)- \Sigma \lambda_i P x_i||$. A consequence of this estimate is the following: Let $T$ be a nonexpansive selfmapping of a bounded, closed and convex subset $C$ of $H$, let $F(T)$ denote the fixed point set of $T$ which is nonempty, closed and convex, and for $x \in C$ let be $S_n x = (x+Tx+ \ldots +T^{n-1}x)/n$. Then the sequence $\{\text{Proj}_{F(T)} S_n x\}$ converges to a fixed point $p$, which is the weak limit of the sequence $\{S_nx\}$ too.
Keywords: Nearest point projection, nonexpansive mapping, inequalities, Nonlinear Ergodic Theorem, Hilbert space