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Approximating fixed points of Lipschitzian pseudocontractive mappings. (English) Zbl 0804.47057
Summary: Let $$T$$ be a Lipschitzian pseudocontractive selfmapping of a nonempty closed bounded and convex subset $$A$$ of a Hilbert space $$(E,(\cdot,\cdot))$$ and let $$w$$ be an arbitrary point of $$A$$. Then the iteration procedure $$z_{n+1}:= \mu_{n+1}(a_ n T(z_ n)+ (1- a_ n)z_ n)+ (1- \mu_{n+1})w$$ converges strongly to the unique fixed point of $$T$$ which is closest to $$w$$, provided $$(\mu_ n)$$ and $$(a_ n)$$ have certain properties. No compactness assumption is made on $$A$$.

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