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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 typedoi:10.1017/s000497270900046x fatcat:47p6gijc6vdt7fht74n57wciiq