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 | References | Similar Articles | Additional Information

Abstract: The following theorem is proved: Suppose *H* is a complex Hilbert space, and is a monotonic, nonexpansive operator on *H*, and . Define by for all . Suppose for all and diverges. Then the iterative process converges to the unique solution of the equation .

**[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.**[4]**E. H. ZARANTONELLO, "The closure of the numerical range contains the spectrum,"*Pacific J. Math.*, v. 22, 1967, pp. 575-595. MR 37 #4657. MR**0229079 (37:4657)**

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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