This article deals with an infinitely dimensional nonlinear dynamical systems given in a state space form. Among such systems we distinguish a wide class of semilinear systems, for which we present a set of controllability conditions. These conditions for controllability are based on a fixed point theorems and measures of noncompactness. We perform a survey for the cases where a specific Picard type integral operator, under appropriate assumptions, has a fixed point in a given function space. The controllability conditions follow from it.