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An iterative process for nonlinear monotonic nonexpansive operators in Hilbert space


Author: W. G. Dotson
Journal: Math. Comp. 32 (1978), 223-225
MSC: Primary 47H15; Secondary 65J05
DOI: https://doi.org/10.1090/S0025-5718-1978-0470779-8
MathSciNet review: 0470779
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Abstract: The following theorem is proved: Suppose H is a complex Hilbert space, and $ T:H \to H$ is a monotonic, nonexpansive operator on H, and $ f \in H$. Define $ S:H \to H$ by $ Su = - Tu + f$ for all $ u \in H$. Suppose $ 0 \leqslant {t_n} \leqslant 1$ for all $ n = 1,2,3, \ldots ,$ and $ \Sigma _{n = 1}^\infty \;{t_n}(1 - {t_n})$ diverges. Then the iterative process $ {V_{n + 1}} = (1 - {t_n}){V_n} + {t_n}S{V_n}$ converges to the unique solution $ u = p$ of the equation $ u + Tu = f$.


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

  • [1] W. G. DOTSON, JR., "On the Mann iterative process," Trans. Amer. Math. Soc., v. 149, 1970, pp. 65-73. MR 0257828 (41:2477)
  • [2] C. W. GROETSCH, "A note on segmenting Mann iterates," J. Math. Anal. Appl., v. 40, 1972, pp. 369-372. MR 0341204 (49:5954)
  • [3] E. H. ZARANTONELLO, Solving Functional Equations by Contractive Averaging, Technical Report No. 160, U. S. Army Math. Res. Center, Madison, Wisc., 1960.
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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1978-0470779-8
Keywords: Iteration, monotonic operators, nonexpansive operators
Article copyright: © Copyright 1978 American Mathematical Society

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