∑Notes

T6 — Closure Fixed Point (Banach Contraction)

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#1
Let be the sigmoid closure operator
$Cl_{t} (u) (x) = σ (a_{cl} ((1 - η_{cl}) u (x) + η_{cl} (P_{t} u) (x) - τ_{cl})) .$
#2
T6b. Banach contraction mapping theorem. The Lipschitz constant of is
$L \leq (max σ^{'}) \cdot a_{cl} \cdot max (1 - η_{cl}, η_{cl}) \leq \frac{a _{cl}}{4},$