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Title: A convergent nonlinear splitting via orthogonal projection (English)
Author: Mandel, Jan
Language: English
Journal: Aplikace matematiky
ISSN: 0373-6725
Volume: 29
Issue: 4
Year: 1984
Pages: 250-257
Summary lang: English
Summary lang: Czech
Summary lang: Russian
Category: math
Summary: We study the convergence of the iterations in a Hilbert space $V,x_{k+1}=W(P)x_k, W(P)z=w=T(Pw+(I-P)z)$, where $T$ maps $V$ into itself and $P$ is a linear projection operator. The iterations converge to the unique fixed point of $T$, if the operator $W(P)$ is continuous and the Lipschitz constant $\left\|(I-P)W(P)\right\|<1$. If an operator $W(P_1)$ satisfies these assumptions and $P_2$ is an orthogonal projection such that $P_1P_2=P_2P_1=P_1$, then the operator $W(P_2)$ is defined and continuous in $V$ and satisfies $\left\|(I-P_2)W(P_2)\right\|\leq \left\|(I-P_1)W(P_1)\right\|$. (English)
Keyword: convergent nonlinear splitting
Keyword: orthogonal projection
Keyword: iterations
Keyword: Hilbert space
Keyword: fixed point
MSC: 47H10
MSC: 47H17
MSC: 47J25
MSC: 65J15
idZBL: Zbl 0613.65060
idMR: MR0754077
DOI: 10.21136/AM.1984.104093
Date available: 2008-05-20T18:25:11Z
Last updated: 2020-07-28
Stable URL:
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