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

##### MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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