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Zbl 0928.47040
Jung, Jong Soo; Kim, Tae Hwa
Strong convergence of approximating fixed points for nonexpansive nonself-mappings in Banach spaces.
(English)
[J] Kodai Math. J. 21, No.3, 259-272 (1998). ISSN 0386-5991

Summary: Let $E$ be a reflexive Banach space with a uniformly Gâteaux differentiable norm, $C$ a nonempty closed convex subset of $E$, and $T: C\to E$ a nonexpansive mapping satisfying the inwardness condition. Assume that every weakly compact convex subset of $E$ has the fixed point property. For $u\in C$ and $t\in (0,1)$, let $x_t$ be a unique fixed point of a contraction $G_t: C\to E$, defined by $G_tx= tTx+(1- t)u$, $x\in C$. It is proved that if $\{x_t\}$ is bounded, then the strong $\lim_{t\to 1}x_t$ exists and belongs to the fixed point set of $T$. Furthermore, the strong convergence of other two schemes involving the sunny nonexpansive retraction is also given in a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm.
MSC 2000:
*47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: reflexive Banach space; uniformly Gâteaux differentiable norm; nonexpansive mapping; inwardness condition; fixed point property; fixed point set; strictly convex Banach space

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