We prove the existence of a weak solution for the quasilinear parabolic initial boundary value problem associated to the equation $$ u_{t}-\Delta_{p(x)}u=h, $$ by using the Topological degree theory for operators of the form $L+S$, where $L$ is a linear densely defined maximal monotone map and $S$ is a bounded demicontinuous map of class $(S_+)$ with respect to the domain of $L$.