Proof Mining - Mathematics, Algorithms and Proofs
Proof Mining - Mathematics, Algorithms and Proofs
Proof Mining - Mathematics, Algorithms and Proofs
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<strong>Proof</strong> <strong>Mining</strong><br />
THEOREM 8 (Borwein-Reich-Shafrir,1992)<br />
For the Krasnoselski-Mann iteration (xn) starting from x ∈ C<br />
one has<br />
where rC(f) := inf<br />
x∈C<br />
xn − f(xn) n→∞<br />
→ rC(f),<br />
x − f(x).<br />
COROLLARY 9 (Ishikawa,1976)<br />
If d(C) := diam(C) < ∞, then xn − f(xn) n→∞<br />
→ 0.<br />
<strong>Proof</strong>s based on (xn − f(xn)) being non-increasing!<br />
• Also for hyperbolic spaces <strong>and</strong> directionally n.e. functions.<br />
• For uniformly convex spaces: even asymptotically<br />
(quasi-)nonexpansive mappings.<br />
Ulrich Kohlenbach 10
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