Demiclosedness Principles for Generalized Nonexpansive Mappings
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Demiclosedness Principles for Generalized Nonexpansive Mappings Sedi Bartz1 · Rubén Campoy1
· Hung M. Phan1
Received: 29 April 2020 / Accepted: 28 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Demiclosedness principles are powerful tools in the study of convergence of iterative methods. For instance, a multi-operator demiclosedness principle for firmly nonexpansive mappings is useful in obtaining simple and transparent arguments for the weak convergence of the shadow sequence generated by the Douglas–Rachford algorithm. We provide extensions of this principle, which are compatible with the framework of more general families of mappings such as cocoercive and conically averaged mappings. As an application, we derive the weak convergence of the shadow sequence generated by the adaptive Douglas–Rachford algorithm. Keywords Demiclosedness principle · Cocoercive mapping · Conically averaged mapping · Weak convergence · Douglas–Rachford algorithm · Adaptive Douglas–Rachford algorithm Mathematics Subject Classification 47H05 · 47J25 · 49M27
1 Introduction Demiclosedness principles play an important role in convergence analysis of fixed point algorithms. The concept of demiclosedness sheds light on topological properties
Communicated by Michel Théra.
B
Rubén Campoy [email protected] Sedi Bartz [email protected] Hung M. Phan [email protected]
1
Department of Mathematical Sciences, Kennedy College of Sciences, University of Massachusetts Lowell, Lowell, MA, USA
123
Journal of Optimization Theory and Applications
of mappings, in particular, in the case where a weak topology is considered. More precisely, given a weakly sequentially closed subset D of a Hilbert space H, the mapping T : D → H is said to be demiclosed at x ∈ D, if for every sequence (xk ) in D such that (xk ) converges weakly to x and T (xk ) converges strongly, say, to u, it follows that T (x) = u. By its definition, demiclosedness holds trivially whenever T is weakly sequentially continuous; however, it does not hold in general. Let Id denote the identity mapping on H. A fundamental result in the theory of nonexpansive mappings is Browder’s celebrated demiclosedness principle [1], which asserts that, if T is nonexpansive, then the mapping Id −T is demiclosed at every point in D. Browder’s result holds in more general settings and, by now, has become a key tool in the study of asymptotic and ergodic properties of nonexpansive mappings; see [2–5], for example. In [6], Browder’s demiclosedness principle was extended and a version for finitely many firmly nonexpansive mappings was provided. As an application, a simple proof of the weak convergence of the Douglas–Rachford (DR) algorithm [7,8] was also provided in [6]: The DR algorithm belongs to the class of splitting methods for the problem of finding a zero of the sum of two maximally monotone operators A, B : H ⇒ H; see (25). The DR algorithm generates a sequence by an iterative application of the DR operator (see (26), (27) and the comment thereaf
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