摘要:
设~$X, Y$~是~Banach~空间, ~$T$~是\ $\mathcal{D}(T)\subset X$~%到\ $Y$~的稠定闭线性算子而且它的值域在\ $Y$~闭.~设相容算子方程~$Tx=b$~的非相容 扰动为\ $ \|(T+\delta T)x-\barb\|=\min\limits_{z\in\mathcal{D}(T)}\|(T+\delta T)z-\bar b\|,$~%这里\ $\delta T$~是\ $X\to Y$~的有界线性算子. ~在某些条件下\ (比如\$X, \, Y$~是自反的), ~设上述方程的最小范数 解为\ $\bar x_m$, 并 设\$Tx=b$~的解集\ $S(T, b)$~中的最小范数解为\ $x_m$. ~本文给出了当\$\delta(\Ker T, \Ker(T+\delta T))$~较小时, $\dfrac{\dist(\bar x_m,S(T, b))}{\|x_m\|}$~的上界估计式.
Abstract:
Let~$X, Y$~ be Banach spaces and let $T$ be adensely--defined closed linear operator from $\mathcal{D}(T)\subset$to $Y$ with closed range. Suppose the non-consistent perturbationof the consistent equation $Tx=b$ is $ \|(T+\delta T)x-\barb\|=\min\limits_{z\in\mathcal{D}(T)}\|(T+\delta T)z-\bar b\|, $where $\delta T$ is a bounded linear operator from $X$ to $Y$. Undercertain conditions (e. g. $X$ and $Y$ are reflexive Banach spaces),let $\bar x_m$ be the minimal norm solution of above equation andlet $x_m$ be minimal norm solution of the set $S(T,b)=\{x\in\mathcal{D}(T)\vert\, Tx=b\}$. In this paper, we give anestimation of the upper bound of $\dfrac{\dist(\bar x_m, S(T,b))}{\|x_m\|}$ when $\delta(\Ker T, \Ker(T+\delta T))$ is smallenough.
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