The work of H. Hundal (Nonlinear Anal. 57 (2004), 35-61) has revealed that the sequence generated by the method of alternating projections converges weakly, but not strongly in general. This paper seeks to design strongly convergent algorithms by means of alternating the resolvents of two maximal monotone operators, A and B, that can be used to approximate common zeroes of A and B. In particular, methods of alternating projections which generate sequences that converge strongly are obtained. A particular case of such algorithms enables one to approximate minimum values of certain convex functionals under less restrictive conditions on the regularization parameters involved.