Fixed points by a new iteration method

Ishikawa, Shiro

doi:10.1090/S0002-9939-1974-0336469-5

Fixed points by a new iteration method
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by Shiro Ishikawa

Proc. Amer. Math. Soc. 44 (1974), 147-150

DOI: https://doi.org/10.1090/S0002-9939-1974-0336469-5

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Abstract:

The following result is shown. If $T$ is a lipschitzian pseudo-contractive map of a compact convex subset $E$ of a Hilbert space into itself and ${x_1}$ is any point in $E$, then a certain mean value sequence defined by ${x_{n + 1}} = {\alpha _n}T[{\beta _n}T{x_n} + (1 - {\beta _n}){x_n}] + (1 - {\alpha _n}){x_n}$ converges strongly to a fixed point of $T$, where $\{ {\alpha _n}\}$ and $\{ {\beta _n}\}$ are sequences of positive numbers that satisfy some conditions.

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