densely--defined closed linear operator from $\mathcal{D}(T)\subset$

to $Y$ with closed range. Suppose the non-consistent perturbation

of the consistent equation $Tx=b$ is $ \|(T+\delta T)x-\bar

b\|=\min\limits_{z\in\mathcal{D}(T)}\|(T+\delta T)z-\bar b\|, $

where $\delta T$ is a bounded linear operator from $X$ to $Y$. Under

certain conditions (e. g. $X$ and $Y$ are reflexive Banach spaces),

let $\bar x_m$ be the minimal norm solution of above equation and

let $x_m$ be minimal norm solution of the set $S(T,

b)=\{x\in\mathcal{D}(T)\vert\, Tx=b\}$. In this paper, we give an

estimation of the upper bound of $\dfrac{\dist(\bar x_m, S(T,

b))}{\|x_m\|}$ when $\delta(\Ker T, \Ker(T+\delta T))$ is small

enough.