  We analyze a nonlinear $q$-voter model with stochastic noise, interpreted in
the social context as independence, on a duplex network. The size of the lobby
$q$ (i.e., the pressure group) is a crucial parameter that changes the behavior
of the system. The $q$-voter model has been applied on multiplex networks in a
previous work [Phys. Rev E. 92. 052812. (2015)], and it has been shown that the
character of the phase transition depends on the number of levels in the
multiplex network as well as the value of $q$. Here we study phase transition
character in the case when on each level of the network the lobby size is
different, resulting in two parameters $q_1$ and $q_2$. We find evidence of
successive phase transitions when a continuous phase transition is followed by
a discontinuous one or two consecutive discontinuous phases appear, depending
on the parameter. When analyzing this system, we even encounter mixed-order (or
hybrid) phase transition. We perform simulations and obtain supporting
analytical solutions on a simple multiplex case - a duplex clique, which
consists of two fully overlapped complete graphs (cliques).
