An iterative process for nonlinear monotonic nonexpansive operators in Hilbert space
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- by W. G. Dotson PDF
- Math. Comp. 32 (1978), 223-225 Request permission
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
- W. G. Dotson Jr., On the Mann iterative process, Trans. Amer. Math. Soc. 149 (1970), 65–73. MR 257828, DOI 10.1090/S0002-9947-1970-0257828-6
- C. W. Groetsch, A note on segmenting Mann iterates, J. Math. Anal. Appl. 40 (1972), 369–372. MR 341204, DOI 10.1016/0022-247X(72)90056-X 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
- © Copyright 1978 American Mathematical Society
- 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