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

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