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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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