Using a variational method introduced in [D. Azé and J.-N. Corvellec, ‘A variational method in fixed point results with inwardness conditions’, Proc. Amer. Math. Soc.134(12) (2006), 3577–3583], deriving directly from the Ekeland principle, we give a general result on the existence of a fixed point for a very general class of multifunctions, generalizing the recent results of [Y. Feng and S. Liu, ‘Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings’, J. Math. Anal. Appl.317(1) (2006), 103–112; D. Klim and D. Wardowski, ‘Fixed point theorems for set-valued contractions in complete metric spaces’, J. Math. Anal. Appl.334(1) (2007), 132–139]. Moreover, we give a sharp estimate for the distance to the fixed-points set.
