  In this work we study a Sznajd-like opinion dynamics on a square lattice of
linear size $L$. For this purpose, we consider that each agent has a convincing
power $C$, that is a time-dependent quantity. Each high convincing power group
of four agents sharing the same opinion may convince its neighbors to follow
the group opinion, which induces an increase of the group's convincing power.
In addition, we have considered that a group with a local majority opinion (3
up/1 down spins or 1 up/3 down spins) can persuade the agents neighboring the
group with probability $p$, since the group's convincing power is high enough.
The two mechanisms (convincing powers and probability $p$) lead to an increase
of the competition among the opinions, which avoids dictatorship (full
consensus, all spins parallel) for a wide range of model's parameters, and
favors the occurrence of democratic states (partial order, the majority of
spins pointing in one direction). We have found that the relaxation times of
the model follow log-normal distributions, and that the average relaxation time
$\tau$ grows with system size as $\tau \sim L^{5/2}$, independent of $p$. We
also discuss the occurrence of the usual phase transition of the Sznajd model.
