Recently in Castellani-Guili (J. Optim. Th. Appl., 147 (2010), 157-168), it has been showed that the proof of the existence result for quasimonotone Stampacchia variational inequalities developed in Aussel-Hadjisavvas (J. Optim. Th. Appl., 121 (2004), 445-450) can be adapted to the case of equilibrium problem. This proof was based on KKM techniques. In this paper we define and study the so called quasi-equilibrium problem, that is an equilibrium problem with a constraint set depending of the current point. Our main contribution consists of an existence result combining fixed point techniques with stability analysis of perturbed equilibrium problems.