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Keywords:
convergent nonlinear splitting; orthogonal projection; iterations; Hilbert space; fixed point
convergent nonlinear splitting; orthogonal projection; iterations; Hilbert space; fixed point
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 $\left\|(I-P_2)W(P_2)\right\|\leq \left\|(I-P_1)W(P_1)\right\|$.
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