  We generalize some of those results reported by Gonz\'{a}lez-D\'{i}az by
further tuning the parameter ($\beta$) which is closely related to the
canonical kinetic term in $k$-essence formalism. The scale factor $a(t)$ could
be negative and decreasing within a specific range of $\beta$ ($\equiv
-1/\omega$, $\omega$ : the equation-of-state parameter) during the initial
evolutional period.
