The authors provide some fundamental properties of resolvents of maximal monotone operators in Banach spaces. The results are used for the study of the asymptotic behavior of the sequences generated by two modifications of the proximal point algorithm. By this, known convergence theorems of R. T. Rockafellar [SIAM J. Control Optimization 14, 877–898 (1976; Zbl 0358.90053)] and the authors [e.g., S. Kamimura, F. Kohsaka and W. Takahashi, Set-Valued Anal. 12, No. 4, 417–429 (2004; Zbl 1078.47050)] can be generalized.
Using the subdifferential mapping of convex functions, the approach can be applied to find minimizers of convex optimization problems. Another application concerns the problem of finding fixed points of nonexpansive mappings in Hilbert spaces.