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

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.**149**(1970), 65–73. MR**0257828**, 10.1090/S0002-9947-1970-0257828-6**[2]**C. W. Groetsch,*A note on segmenting Mann iterates*, J. Math. Anal. Appl.**40**(1972), 369–372. MR**0341204****[3]**E. H. ZARANTONELLO,*Solving Functional Equations by Contractive Averaging*, Technical Report No. 160, U. S. Army Math. Res. Center, Madison, Wisc., 1960.**[4]**Eduardo H. Zarantonello,*The closure of the numerical range contains the spectrum*, Pacific J. Math.**22**(1967), 575–595. MR**0229079**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1978-0470779-8

Keywords:
Iteration,
monotonic operators,
nonexpansive operators

Article copyright:
© Copyright 1978
American Mathematical Society