Extensions of Browder’s demiclosedness principle and Reich’s lemma and their applications. (English) Zbl 1384.47023

Kubota, Rieko; Takahashi, Wataru; Takeuchi, Yukio

Pure Appl. Funct. Anal. 1, No. 1, 63-84 (2016).

Let \(E\) be a real Banach space and \(C\) be a closed subset of \(E\). A mapping \(T : C\rightarrow E\) is said to be demi-closed (at \(y\)) if, for any sequence \(\{x_n\}\) in \(C\), the conditions \(x_n \rightarrow \overline{x}\) weakly and \(Tx_n \rightarrow y\) strongly imply that \(T\overline{x} = y\).
One of the fundamental results in the theory of nonexpansive mappings is F. E. Browder’s demiclosedness principle [Bull. Am. Math. Soc. 74, 660–665 (1968; Zbl 0164.44801)] which states that, if \(E\) is a uniformly convex Banach space, \(C\) is a closed convex subset of \(E\) and \(T : C\rightarrow E\) is nonexpansive, then \(I-T\) is demiclosed at \(0\).
In the paper under review, the authors study some fundamental properties of nonexpansive mappings and obtain extensions of Browder’s demiclosedness principle and S. Reich’s lemma [J. Math. Anal. Appl. 67, 274–276 (1979; Zbl 0423.47026)]. Using these results, they also obtain extensions of some weak convergence theorems due to Reich as well as S. H. Khan and T. Suzuki [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 80, 211–215 (2013; Zbl 1258.47069)].

Reviewer: Vasile Berinde (Baia Mare)

MSC:

47H10	Fixed-point theorems
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

Banach space; nonexpansive mapping; Browder’s demiclosedness principle; weak convergence; Mann iteration; attractive point; condition (C)