On ๐ผ-Firmly Nonexpansive Operators in r-Uniformly Convex Spaces
Bรซrdรซllima, Arian; Steidl, Gabriele
We introduce the class of ๐ผ-firmly nonexpansive and quasi ๐ผ-firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where ๐ผ-firmly nonexpansive operators coincide with so-called ๐ผ-averaged operators. For our more general setting, we show that ๐ผ-averaged operators form a subset of ๐ผ-firmly nonexpansive operators. We develop some basic calculus rules for (quasi) ๐ผ-firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) ๐ผ-firmly nonexpansive. Moreover, we will see that quasi ๐ผ-firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browderโs demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates ๐ฅ๐+1:=๐๐ฅ๐ belong to the fixed point set Fix๐ whenever the operator T is nonexpansive and quasi ๐ผ-firmly. If additionally the space has a Frรฉchet differentiable norm or satisfies Opialโs property, then these iterates converge weakly to some element in Fix๐. Further, the projections ๐Fix๐๐ฅ๐ converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in ๐ฟ๐, ๐โ(1,โ)โ{2} spaces on probability measure spaces.
