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Zbl 1089.47058
Approximating fixed points of non-self nonexpansive mappings in Banach spaces.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 61, No. 6, A, 1031-1039 (2005). ISSN 0362-546X

Let $K$ be a nonempty closed convex subset of a real uniformly convex Banach space $E$, which is also a nonexpansive retract of $E$ $(P$ is a nonexpansive retraction of $E$ onto $K)$. Let $\{x_n\}$ be the sequence defined by $x_1=x\in K$, $x_{n+1}=P((1-\alpha_n)x_n+ \alpha_nTP[(1-\beta_n)x_n+\beta_nTx_n])$, $n\ge 1$, where $T:E\to K$ is a nonexpansive mapping and $\{\alpha_n\}$, $\{\beta_n\}$ are sequences in $[\varepsilon,1-\varepsilon]$ for some $\varepsilon\in(0,1)$. In the present paper, the author proves the following: (1) If $F(T)\ne \emptyset$ and the dual $E^*$ of $E$ has the Kadec-Klee property, then $\{x_n\}$ convergence weakly to some fixed point of $T$. (2) If $F(T)\ne \emptyset$ and if there is a nondecreasing function $f:[0,+\infty) \to[0,+\infty)$ with $f(0)=0$ and $f(r)>0$ for all $r>0$ such that for all $x\in K$, $\Vert x-Tx\Vert\ge f(d(x,F(T)))$, then $\{x_n\}$ converges strongly to some fixed point of $T$.
[Jarosław Górnicki (Rzeszów)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties

Keywords: strong convergence; weak convergence; Kadec-Klee property; nonexpansive non-self map; demiclosed map

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