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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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