# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Approximation of fixed points of strongly pseudocontractive mappingsHTML articles powered by AMS MathViewer

by C. E. Chidume
Proc. Amer. Math. Soc. 120 (1994), 545-551 Request permission

## Abstract:

Let $E$ be a real Banach space with a uniformly convex dual, and let $K$ be a nonempty closed convex and bounded subset of $E$. Let $T:K \to K$ be a continuous strongly pseudocontractive mapping of $K$ into itself. Let $\{ {c_n}\} _{n = 1}^\infty$ be a real sequence satisfying: (i) $0 < {c_n} < 1$ for all $n \geqslant 1$; (ii) $\sum \nolimits _{n = 1}^\infty {{c_n} = \infty }$; and (iii) $\sum \nolimits _{n = 1}^\infty {{c_n}b({c_n}) < \infty }$, where $b:[0,\infty ) \to [0,\infty )$ is some continuous nondecreasing function satisfying $b(0) = 0, b(ct) \leqslant cb(t)$ for all $c \geqslant 1$. Then the sequence $\{ {x_n}\} _{n = 1}^\infty$ generated by ${x_1} \in K$, ${x_{n + 1}} = (1 - {c_n}){x_n} + {c_n}T{x_n},\qquad n \geqslant 1,$ converges strongly to the unique fixed point of $T$. A related result deals with the Ishikawa iteration scheme when $T$ is Lipschitzian and strongly pseudocontractive.
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