Fixed points by a new iteration method

Ishikawa, Shiro

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

Fixed points by a new iteration method
HTML articles powered by AMS MathViewer

by Shiro Ishikawa PDF

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

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.

References

Gordon G. Johnson, Fixed points by mean value iterations, Proc. Amer. Math. Soc. 34 (1972), 193–194. MR 291918, DOI 10.1090/S0002-9939-1972-0291918-4
W. Robert Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510. MR 54846, DOI 10.1090/S0002-9939-1953-0054846-3
F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197–228. MR 217658, DOI 10.1016/0022-247X(67)90085-6
R. L. Franks and R. P. Marzec, A theorem on mean-value iterations, Proc. Amer. Math. Soc. 30 (1971), 324–326. MR 280656, DOI 10.1090/S0002-9939-1971-0280656-9