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.