scholarly journals New Demiclosedness Principles for (Firmly) Nonexpansive Operators

2013 ◽  
pp. 19-28 ◽  
Author(s):  
Heinz H. Bauschke
Keyword(s):  
Firmly Nonexpansive ◽  
Nonexpansive Operators

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

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Arian Bërdëllima ◽  
Gabriele Steidl
Keyword(s):  
Banach Spaces ◽  
Semigroup Theory ◽  
General Setting ◽  
Weak Limit ◽  
Uniformly Convex ◽  
Fixed Point Set ◽  
Uniformly Convex Banach Spaces ◽  
Differentiable Norm ◽  
Firmly Nonexpansive ◽  
Nonexpansive Operators

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

scholarly journals On compositions of special cases of Lipschitz continuous operators

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Pontus Giselsson ◽  
Walaa M. Moursi
Keyword(s):  
Optimization Problems ◽  
Nonexpansive Mappings ◽  
Structure And Properties ◽  
Monotone Inclusion ◽  
Lipschitz Continuous ◽  
Iterative Optimization ◽  
Special Cases ◽  
Firmly Nonexpansive ◽  
Nonexpansive Operators ◽  
Continuous Operators

AbstractMany iterative optimization algorithms involve compositions of special cases of Lipschitz continuous operators, namely firmly nonexpansive, averaged, and nonexpansive operators. The structure and properties of the compositions are of particular importance in the proofs of convergence of such algorithms. In this paper, we systematically study the compositions of further special cases of Lipschitz continuous operators. Applications of our results include compositions of scaled conically nonexpansive mappings, as well as the Douglas–Rachford and forward–backward operators, when applied to solve certain structured monotone inclusion and optimization problems. Several examples illustrate and tighten our conclusions.

scholarly journals Regular Sequences of Quasi-Nonexpansive Operators and Their Applications

2018 ◽  
Vol 28 (2) ◽  
pp. 1508-1532 ◽  
Author(s):  
Andrzej Cegielski ◽  
Simeon Reich ◽  
Rafał Zalas
Keyword(s):  
Regular Sequences ◽  
Nonexpansive Operators

A limit theorem for order preserving nonexpansive operators inL 1

1990 ◽  
Vol 71 (2) ◽  
pp. 181-191 ◽  
Author(s):  
Ulrich Krengel ◽  
Michael Lin ◽  
Rainer Wittmann
Keyword(s):  
Limit Theorem ◽  
Nonexpansive Operators

Order preserving nonexpansive operators inL 1

1987 ◽  
Vol 58 (2) ◽  
pp. 170-192 ◽  
Author(s):  
Ulrich Krengel ◽  
Michael Lin
Keyword(s):  
Nonexpansive Operators

Existence and Approximation of Fixed Points of Firmly Nonexpansive-Type Mappings in Banach Spaces

2008 ◽  
Vol 19 (2) ◽  
pp. 824-835 ◽  
Author(s):  
Fumiaki Kohsaka ◽  
Wataru Takahashi
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Firmly Nonexpansive
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