Abstract: Let~$X, Y$~ be Banach spaces and let $T$ be a
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.

Key words: reduced minimum modulus, minimal norm solution, closed range, reduced minimum modulus, minimal norm solution