## Strong convergence of approximating fixed points for nonexpansive nonself-mappings in Banach spaces.(English)Zbl 0928.47040

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:

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

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