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Proceedings of the American Mathematical Society

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Fixed points by mean value iterations

Author: Gordon G. Johnson
Journal: Proc. Amer. Math. Soc. 34 (1972), 193-194
MSC: Primary 47H10
MathSciNet review: 0291918
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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,

$\displaystyle {x_{n + 1}} = [1/(n + 1)][T{x_n} + n{x_n}].$

References [Enhancements On Off] (What's this?)

  • [1] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. MR 14, 988. MR 0054846 (14:988f)
  • [2] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967), 197-228. MR 36 #747. MR 0217658 (36:747)
  • [3] J. Schauder, Der fixpunktsatz in funktionalraumen, Studia Math. 2 (1970), 171-180.
  • [4] R. L. Franks and R. P. Marzec, A theorem on mean value iterations, Proc. Amer. Math. Soc. 30 (1971), 324-326. MR 0280656 (43:6375)

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