  Let $X$ be a smooth projective surface over an algebraically closed field $k$
of characteristic $p> 0$ with $\Omega_{X}^{1}$ semistable and
$\mu(\Omega_{X}^{1})>0$. For any semistable (resp. stable) bundle $W$ of rank
$r$, we prove that $F_*W$ is semistable (resp. stable) when $p\geq r(r-1)^2+1$.
