  In this paper, we show that, under the assumption that $\|\e\|_2\leq
\epsilon$, every $k-$sparse signal $\x\in \mathbb{R}^n$ can be stably
($\epsilon\neq0$) or exactly recovered ($\epsilon=0$) from $\y=\A\x+\e$ via
$l_p-$mnimization with $p\in(0, \bar{p}]$, where \beqnn \bar{p}= \begin{cases}
\frac{50}{31}(1-\delta_{2k}), &\delta_{2k}\in[\frac{\sqrt{2}}{2}, 0.7183)\cr
0.4541, &\delta_{2k}\in[0.7183,0.7729)\cr 2(1-\delta_{2k}),
&\delta_{2k}\in[0.7729,1) \end{cases}, \eeqnn even if the restricted isometry
constant of $\A$ satisfies $\delta_{2k}\in[\frac{\sqrt{2}}{2}, 1)$.
Furthermore, under the assumption that $n\leq 4k$, we show that the range of
$p$ can be further improved to $p\in(0,\frac{3+2\sqrt{2}}{2}(1-\delta_{2k})]$.
This not only extends some discussions of only the noiseless recovery (Lai et
al. and Wu et al.) to the noise recovery, but also greatly improves the best
existing results where $p\in(0,\min\{1, 1.0873(1-\delta_{2k}) \})$ (Wu et al.).
