On alpha -Firmly Nonexpansive Operators in r-Uniformly Convex Spaces

Abstract

We introduce the class of alpha -firmly nonexpansive and quasi alpha -firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where alpha -firmly nonexpansive operators coincide with so-called alpha -averaged operators. For our more general setting, we show that alpha -averaged operators form a subset of alpha -firmly nonexpansive operators. We develop some basic calculus rules for (quasi) alpha -firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) alpha -firmly nonexpansive. Moreover, we will see that quasi alpha -firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder’s demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates x_{n+1}:=Tx_{n} belong to the fixed point set {{,mathrm{Fix},}}T whenever the operator T is nonexpansive and quasi alpha -firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial’s property, then these iterates converge weakly to some element in {{,mathrm{Fix},}}T. Further, the projections P_{{{,mathrm{Fix},}}T}x_n 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 L_p, p in (1,infty ) backslash {2} spaces on probability measure spaces.