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Mathematics of Computation

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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
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$.

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Keywords: Iteration, monotonic operators, nonexpansive operators
Article copyright: © Copyright 1978 American Mathematical Society

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