Abstract. In this paper, we present a generalization of the method of alter- nating resolvents introduced in the previous authors' paper [On the method of alternating resolvents, Nonlinear Anal. 74 (2011), 5147-5160]. It is shown that the sequence generated by this method converges strongly under weaker condi- tions on the control parameters. Concerning the error sequences, many more conditions are used here as compared to the above quoted paper.