Summary: Compressed sensing is a new scheme which shows the ability to recover sparse signals from fewer measurements, using \(l _1\)-minimization. Recently, R. Chartrand and V. Staneva showed in [Inverse Probl. 24, No. 3, Article ID 035020, 14 p. (2008; Zbl 1143.94004)] that \(l_p\)-minimization with \(0<p<1\) recovers sparse signals from fewer linear measurements than does \(l_1\)-minimization. They proved that \(l_p\)-minimization with \(0<p<1\) recovers \(S\)-sparse signals \(x \in \mathbb R^{N }\) from fewer Gaussian random measurements for some smaller \(p\) with probability exceeding \(1-1/ {N\choose S}\). The first aim of this paper is to show that the above result is right for the case of Gaussian random measurements with probability exceeding \(1-2e^{-c(p)M}\), where \(M\) is the numbers of rows of Gaussian random measurements and \(c(p)\) is a positive constant that guarantees \(1-2e^{-c(p)M}>1-1/{N\choose S}\) for \(p\) smaller. The second purpose of the paper is to show that under certain weaker conditions, decoders \(\Delta_p\) are stable in the sense that they are \((2,p)\) instance optimal for a large class of encoders for \(0<p<1\).