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An iterative algorithm on approximating fixed points of pseudocontractive mappings. (English) Zbl 1295.47104
Summary: Let $$E$$ be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let $$K$$ be a nonempty bounded closed convex subset of $$E$$, and assume that every nonempty closed convex bounded subset of $$K$$ has the fixed point property for non-expansive self-mappings. Let $$f : K \rightarrow K$$ be a contractive mapping and $$T : K \rightarrow K$$ be a uniformly continuous pseudocontractive mapping with $$F(T) \neq \emptyset$$. Let $$\{\lambda_n\} \subset (0, 1/2)$$ be a sequence satisfying the following conditions: (i) $$\lim_{n \rightarrow \infty} \lambda_n = 0$$; (ii) $$\sum^\infty_{n=0} \lambda_n = \infty$$. Define the sequence $$\{x_n\}$$ in $$K$$ by $$x_0 \in K$$, $$x_{n+1} = \lambda_n f(x_n) + (1 - 2\lambda_n) x_n + \lambda_n Tx_n$$ for all $$n \geq 0$$. Under some appropriate assumptions, we prove that the sequence $$\{x_n\}$$ converges strongly to a fixed point $$p \in F(T)$$ which is the unique solution of the variational inequality $$\langle f(p) - p, j(z - p)\rangle \leq 0$$ for all $$z \in F(T)$$.
Reviewer: Reviewer (Berlin)

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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##### References:
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