International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 691952, 11 pages
Research Article

Relatively Inexact Proximal Point Algorithm and Linear Convergence Analysis

Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

Received 30 July 2009; Accepted 9 November 2009

Academic Editor: Petru Jebelean

Copyright © 2009 Ram U. Verma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Based on a notion of relatively maximal (m)-relaxed monotonicity, the approximation solvability of a general class of inclusion problems is discussed, while generalizing Rockafellar's theorem (1976) on linear convergence using the proximal point algorithm in a real Hilbert space setting. Convergence analysis, based on this new model, is simpler and compact than that of the celebrated technique of Rockafellar in which the Lipschitz continuity at 0 of the inverse of the set-valued mapping is applied. Furthermore, it can be used to generalize the Yosida approximation, which, in turn, can be applied to first-order evolution equations as well as evolution inclusions.