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Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces. (English) Zbl 1102.47048
Summary: Let \(E\) be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let \(K\) be a nonempty closed convex subset of \(E\), and let \(T: K\rightarrow K\) be a uniformly continuous pseudocontraction. If \(f: K\rightarrow K\) is any contraction map on \(K\) and if every nonempty closed convex and bounded subset of \(K\) has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers \(\{\alpha_n\}\), \(\left\{ {\mu _n } \right\}\), that the iteration process \(z_1\in K\), \(z_{n+1}=\mu_n(\alpha_nTz_n+(1-\alpha_n)z_n)+(1-\mu_n)f(z_n)\), \(n\in\mathbb{N}\), strongly converges to the fixed point of \(T\), which is the unique solution of some variational inequality, provided that \(K\) is bounded.

MSC:
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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