Fixed points by mean value iterations

Johnson, Gordon G.

doi:10.1090/S0002-9939-1972-0291918-4

Fixed points by mean value iterations
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by Gordon G. Johnson

Proc. Amer. Math. Soc. 34 (1972), 193-194

DOI: https://doi.org/10.1090/S0002-9939-1972-0291918-4

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Abstract:

If E is a convex compact subset of a Hilbert space, T is a strictly pseudocontractive function from E into E and ${x_1}$ is a point in E, then the point sequence $\{ {x_1}\} _{i = 1}^\infty$ converges to a fixed point of T, where for each positive integer n, \[ {x_{n + 1}} = [1/(n + 1)][T{x_n} + n{x_n}].\]

References

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

Der fixpunktsatz in funktionalraumen

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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